# Seaweed algebras and the unimodal spectrum property

Nicholas Mayers  Department of Mathematics, North Carolina State University,
Raleigh, NC 27695  Nicholas Russoniello  Department of Mathematics, College of
William & Mary, Williamsburg, VA 23185

###### Abstract

If $\mathfrak{g}$ is a Frobenius Lie algebra, then the spectrum of
$\mathfrak{g}$ is an algebraic invariant equal to the multiset of eigenvalues
corresponding to a particular operator acting on $\mathfrak{g}$. In the case
of Frobenius seaweed subalgebras of $A_{n-1}=\mathfrak{sl}(n)$, or type-A
seaweeds for short, it has been shown that the spectrum can be computed
combinatorially using an attendant graph. With the aid of such graphs, it was
further shown that the spectrum of a type-A seaweed consists of an unbroken
sequence of integers centered at $\frac{1}{2}$. It has been conjectured that
if the eigenvalues are arranged in increasing order, then the sequence of
multiplicities forms a unimodal sequence about $\frac{1}{2}$. Here, we
establish this conjecture for certain families of Frobenius type-A seaweeds by
finding explicit formulas for their spectra; in fact, for some families we are
able to show that the corresponding sequences of multiplicities form log-
concave sequences. All arguments are combinatorial.

Mathematics Subject Classification 2020: 05E16, 05C25, 17B45
Key Words and Phrases: unimodal, log-concave, spectrum, meander, seaweed,
Frobenius Lie algebra

##  1 Introduction

A biparabolic (seaweed) subalgebra of a complex reductive Lie algebra
$\mathfrak{r}$ is the intersection of two parabolic subalgebras whose sum is
$\mathfrak{r}$. They – along with certain associated planar graphs called
“meanders” – were first introduced in the case $\mathfrak{r}=\mathfrak{gl}(n)$
by Dergachev and Kirillov ([6], 2000). One of the main results of [6] is that
the algebra’s “index,” a notoriously difficult-to-compute Lie-algebraic
invariant of recent interest (see [3, 5, 15, 17]), can be found by counting
the number of paths and cycles in its associated meander. Of particular
significance are those seaweed subalgebras of $\mathfrak{gl}(n)$ whose
meanders consist of a single path and no cycles. For such algebras, imposing a
vanishing trace condition results in a seaweed subalgebra of
$\mathfrak{sl}(n)$ with index zero. In general, algebras with index zero are
called “Frobenius” and have been studied extensively in the context of
invariant theory (see [13]) and are connected to the classical Yang-Baxter
equation (see [10] and [11]). Here, we are concerned with the “spectrum” of a
Frobenius Lie algebra, which is an invariant multiset of eigenvalues arising
from a certain operator’s action on the algebra. In the case of a Frobenius
seaweed subalgebra of $\mathfrak{sl}(n)$, it is conjectured that the
multiplicities of eigenvalues in the spectrum form a unimodal sequence. We
establish this conjecture for particular families of Frobenius seaweed
subalgebras of $\mathfrak{sl}(n)$ by utilizing the seaweeds’ meanders to find
explicit formulas for their spectra.

To fix the notation, let $\mathfrak{g}$ be a Lie algebra over $\mathbb{C}$.
From any linear form $F\in\mathfrak{g}^{*}$ arises a skew-symmetric, bilinear
two-form $B_{F}(-,-)=F([-,-])$, called the Kirillov form. The index of
$\mathfrak{g}$ is then given by

${\rm ind\hskip
2.84544pt}\mathfrak{g}=\min_{F\in\mathfrak{g}^{*}}\dim\ker(B_{F})$

(see [7]). The Lie algebra $\mathfrak{g}$ is called Frobenius if its index is
zero, or equivalently, if there exists a linear form $F\in\mathfrak{g}^{*}$
such that $B_{F}$ is non-degenerate. We call such an $F$ a Frobenius (linear)
form, and the natural map $\mathfrak{g}\to\mathfrak{g}^{*}$ defined by
$x\mapsto F[x,-]$ is an isomorphism. The inverse image of $F$ under this map
is called a principal element of $\mathfrak{g}$ and will be denoted
$\widehat{F}$; that is, $\widehat{F}$ is the unique element of $\mathfrak{g}$
such that

$F\circ\text{ad }\widehat{F}=F([\widehat{F},-])=F.$

Ooms ([14], 1980) showed that the spectrum of the adjoint action of a
principal element acting on its Frobenius Lie algebra is independent of the
choice of principal element (see also [12], Theorem 3). Consequently, if
$\mathfrak{g}$ is Frobenius and $\widehat{F}\in\mathfrak{g}$ is any principal
element, then the multiset of eigenvalues of
$ad_{\widehat{F}}~{}:~{}\mathfrak{g}\to\mathfrak{g}$ is an invariant of
$\mathfrak{g}$, and so we call such a multiset the spectrum of $\mathfrak{g}$.
In the case that $\mathfrak{g}$ is a seaweed subalgebra of $\mathfrak{sl}(n),$
Gerstenhaber and Giaquinto ([12], 2009) asserted that the spectrum consists of
an unbroken set of integers. Coll et al. ([1], 2016) were able to prove the
unbrokenness property initially stated in [12], showing that the distinct
eigenvalues in the spectrum of a Frobenius seaweed subalgebra of
$\mathfrak{sl}(n)$ forms an interval of integers centered at $\frac{1}{2}$.
Moreover, the authors of [1] conjectured that if the eigenvalues are arranged
in increasing order, then the corresponding sequence of multiplicities form a
unimodal sequence centered at $\frac{1}{2}$. The primary goal of this article
is to establish this unimodality conjecture for certain families of seaweed
subalgebras of $\mathfrak{sl}(n)$; we do so by finding explicit formulas for
the spectra by combinatorial means.

The methods used in this paper make use of a family of graphs, called
“meanders,” which can be associated to seaweed subalgebras of
$\mathfrak{sl}(n)$ via the seaweed’s defining compositions. See Section 2 for
the relationship between seaweed subalgebras of $\mathfrak{sl}(n)$ and
compositions, as well as a detailed construction of a meander. The raison
d’etre for the introduction of meanders into the theory of seaweed subalgebras
was the ability to implement them in the computation of algebraic invariants,
including index and spectrum. With this in mind, we are able to translate
questions of spectrum to questions about graphs.

The outline of the paper is as follows. In Section 2, we outline the necessary
preliminaries concerning seaweed subalgebras of $\mathfrak{sl}(n)$ and
meanders. In Section 3, we develop lemmas concerning the spectra of Frobenius
seaweed subalgebras of $\mathfrak{sl}(n)$ associated with ordered pairs of
compositions containing three parts in total; these lemmas are then used to
find explicit formulas for the spectra of such algebras. In Section 4, we
consider the spectra of families of Frobenius seaweed subalgebras of
$\mathfrak{sl}(n)$ parametrized by the number of occurrences of a fixed value
in the defining compositions; for such families, we observe new behavior
concerning the stability of the set of distinct eigenvalues. Section 5
consists of a discussion of our results, a number of questions arising from
our work, and directions for further research.

##  2 Preliminaries

Given a reductive Lie algebra $\mathfrak{r},$ a seaweed subalgebra of
$\mathfrak{r}$ is the intersection of two parabolic subalgebras
$\mathfrak{p},\mathfrak{p}^{\prime}\subset\mathfrak{r}$ satisfying
$\mathfrak{p}+\mathfrak{p}^{\prime}=\mathfrak{r}.$ While seaweed subalgebras
are defined generally as above, when restricting to seaweed subalgebras of
$\mathfrak{gl}(n,\mathbb{C})=\mathfrak{gl}(n)$ and
$\mathfrak{sl}(n,\mathbb{C})=\mathfrak{sl}(n)$, an equivalent definition in
terms of compositions is available. In particular, we define a seaweed
subalgebra of $\mathfrak{gl}(n)$, or simply a seaweed, to be any matrix Lie
algebra constructed from two compositions of $n$ as follows. Let $V$ be a
complex $n$-dimensional vector space with basis $\\{e_{1},\dots,e_{n}\\},$ let
$\underline{a}=(a_{1},\dots,a_{m})$ and $\underline{b}=(b_{1},\dots,b_{t})$ be
two compositions of $n,$ and consider the flags

$\mathscr{V}=\big{\\{}\\{0\\}\subset V_{1}\subset\dots\subset V_{m-1}\subset
V_{m}=V\big{\\}}\text{ and }\mathscr{W}=\big{\\{}V=W_{0}\supset
W_{1}\supset\dots\supset W_{t}=\\{0\\}\big{\\}},$

where $V_{i}=span\\{e_{1},\dots,e_{a_{1}+\dots+a_{i}}\\}$ and
$W_{j}=span\\{e_{b_{1}+\dots+b_{j}+1},\dots,e_{n}\\}.$ Then the seaweed
$\mathfrak{p}(\underline{a}|\underline{b})=\mathfrak{p}\frac{a_{1}|\dots|a_{m}}{b_{1}|\dots|b_{t}}$
is the subalgebra of $\mathfrak{gl}(n)$ that preserves the flags $\mathscr{V}$
and $\mathscr{W}.$ Similarly, the seaweed subalgebra of
$A_{n-1}=\mathfrak{sl}(n)$, or type-A seaweed,
$\mathfrak{p}^{A}(\underline{a}|\underline{b})=\mathfrak{p}^{A}\frac{a_{1}|\dots|a_{m}}{b_{1}|\dots|b_{t}}$
is the subalgebra of $\mathfrak{sl}(n)$ that preserves the flags $\mathscr{V}$
and $\mathscr{W}$. One special case is of note: if $\underline{a}=(n)$ and
$\underline{b}=(b_{1},b_{2})$, then the type-A seaweeds
$\mathfrak{p}^{A}(\underline{a}|\underline{b})$ and
$\mathfrak{p}^{A}(\underline{b}|\underline{a})$ are called maximal parabolic.

######  Remark 1.

Typically, one includes $n$ in the notation for seaweed subalgebras of
$\mathfrak{gl}(n)$ and $\mathfrak{sl}(n)$, writing
$\mathfrak{p}_{n}(\underline{a}|\underline{b})=\mathfrak{p}_{n}\frac{a_{1}|\dots|a_{m}}{b_{1}|\dots|b_{t}}$
and
$\mathfrak{p}^{A}_{n}(\underline{a}|\underline{b})=\mathfrak{p}^{A}_{n}\frac{a_{1}|\dots|a_{m}}{b_{1}|\dots|b_{t}}$
for the corresponding (type-A) seaweeds; however, since $n$ is encoded in the
compositions $\underline{a}$ and $\underline{b},$ we omit it for ease of
notation.

The evocative “seaweed” is descriptive of the shape of the algebra when
exhibited in matrix form. For example, the seaweed algebra
$\mathfrak{p}^{A}\frac{2|4}{1|2|3}$ consists of traceless matrices of the form
depicted in Figure 1 (left), where $\ast$’s indicate potential non-zero
entries.

*****************12324 $v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$ Figure 1:
The seaweed $\mathfrak{p}\frac{2|4}{1|2|3}$ (left) and its associated meander
(right)

In [6], Dergachev and Kirillov assign to each seaweed
$\mathfrak{g}=\mathfrak{p}\frac{a_{1}|\cdots|a_{m}}{b_{1}|\cdots|b_{t}}$ with
$\sum_{i=1}^{m}a_{i}=\sum_{j=1}^{t}b_{j}=n$ a planar graph $M(\mathfrak{g})$
on $n$ vertices, called a meander, as follows. Begin by placing $n$ labeled
vertices $v_{1},v_{2},\dots,v_{n}$ from left to right along a horizontal line.
Next, partition the vertices by grouping together the first $a_{1}$ vertices,
then the next $a_{2}$ vertices, and so on, lastly grouping together the final
$a_{m}$ vertices. We call each set of vertices formed a block. For each block
in the prescribed set partition, add a concave-down edge, called a top edge,
from the first vertex of the block to the last vertex of the block, then add a
top edge between the second vertex of the block and the second-to-last vertex
of the block, and so on within each block, assuming that the vertices being
connected are distinct. In a similar way, partition the vertices according to
the composition $\underline{b},$ and then place bottom edges, i.e., concave-up
edges, between vertices in each block. See Figure 1 (right).

The main theorem of [6] establishes the combinatorial formula for the index of
a seaweed given in Theorem 2 below.

######  Theorem 2.

If $\mathfrak{g}$ is a seaweed, then

${\rm ind\hskip 2.84544pt}\mathfrak{g}=2C+P,$

where $C$ is the number of cycles and $P$ is the number of paths in
$M(\mathfrak{g})$.

Here, a path is any acyclic connected component of a meander; that is, in
addition to traditional path graphs, connected components consisting of a
single vertex are considered paths. Utilizing Theorem 2, the corollary below
follows immediately.

######  Corollary 3.

If $\mathfrak{g}$ is a type-A seaweed, then

${\rm ind\hskip 2.84544pt}\mathfrak{g}=2C+P-1,$

where $C$ is the number of cycles and $P$ is the number of paths in
$M(\mathfrak{g}).$

Interestingly, in ([11], 2008) and [12], Gerstenhaber and Giaquinto show, in
particular, that one can also determine the spectrum of a Frobenius, type-A
seaweed $\mathfrak{g}$ using $M(\mathfrak{g})$. The result is based on the
construction of a principal element $\widehat{F}\in\mathfrak{g}$ for which
each matrix unit $e_{i,j}\in\mathfrak{g}$ and each
$e_{i,i}-e_{i+1,i+1}\in\mathfrak{g}$ is an eigenvector of $\text{ad
}\widehat{F}$. To describe their choice of $\widehat{F}$, we first need to
decorate $M(\mathfrak{g})$ by adding an orientation to its edges: top edges
are oriented from right to left, while bottom edges are oriented from left to
right. We refer to a meander with such an orientation as the oriented meander,
denoted $\overrightarrow{M}(\mathfrak{g}).$ Considering Corollary 3,
$M(\mathfrak{g})$ must consist of a single path. Thus, for all $1\leq i,j\leq
n,$ there is a unique path from vertex $v_{i}$ to vertex $v_{j}$ in
$M(\mathfrak{g})$, denoted $P_{i,j}(\mathfrak{g})$. Define the weight
$w(P_{i,j}(\mathfrak{g}))$ of the path $P_{i,j}(\mathfrak{g})$ in
$M(\mathfrak{g})$ from $v_{i}$ to $v_{j}$ to be the number of forward edges
minus the number of backward edges encountered when moving along
$P_{i,j}(\mathfrak{g})$ from $v_{i}$ to $v_{j}$ in
$\overrightarrow{M}(\mathfrak{g})$.

The following results appear in [1] as a consequence of a result from
Gerstenhaber and Giaquinto [11]; however, we include proofs for completeness.

######  Lemma 4.

Let $\mathfrak{g}\subset\mathfrak{gl}(n)$ be a seaweed with oriented meander
$\overrightarrow{M}(\mathfrak{g})=(V,E)$ consisting of one path and no cycles,
where $V=\\{v_{1},\dots,v_{n}\\}$ is the vertex set and $E$ is the edge set of
$\overrightarrow{M}(\mathfrak{g}).$ If

$F=\sum_{(v_{i},v_{j})\in E}e_{i,j}^{*}\in\mathfrak{g}^{*}$

and

$\widehat{F}=\sum_{i=1}^{n}w(P_{i,n}(\mathfrak{g}))e_{i,i}\in\mathfrak{g},$

then $F\left(\left[\widehat{F},x\right]\right)=F(x),$ for all
$x\in\mathfrak{g}.$

###### Proof.

If $P_{i,j}(\mathfrak{g})$ is the unique path from $v_{i}$ to $v_{j}$ in
$\overrightarrow{M}(\mathfrak{g}),$ then it immediately follows that
$w(P_{i,j}(\mathfrak{g}))=-w(P_{j,i}(\mathfrak{g}))$ and
$w(P_{i,j}(\mathfrak{g}))+w(P_{j,k}(\mathfrak{g}))=w(P_{i,k}(\mathfrak{g})),$
for all $1\leq i,j,k\leq n.$ Now, if
$\widehat{F}=\sum_{i=1}^{n}w(P_{i,n}(\mathfrak{g}))e_{i,i},$ then we have that

$\left[\widehat{F},e_{i,j}\right]=\big{(}w(P_{i,n}(\mathfrak{g}))-w(P_{j,n}(\mathfrak{g}))\big{)}e_{i,j}=\big{(}w(P_{i,n}(\mathfrak{g}))+w(P_{n,j}(\mathfrak{g}))\big{)}e_{i,j},$

for all $1\leq i,j\leq n$. Note that if $i=j,$ then
$F\left(\left[\widehat{F},e_{i,j}\right]\right)=\left[\widehat{F},e_{i,j}\right]=0.$
On the other hand, if $i\neq j,$ then

$\left[\widehat{F},e_{i,j}\right]=\big{(}w(P_{i,n}(\mathfrak{g}))+w(P_{n,j}(\mathfrak{g}))\big{)}e_{i,j}=\big{(}w(P_{i,j}(\mathfrak{g}))\big{)}e_{i,j}.$

Therefore, since $w(P_{i,j}(\mathfrak{g}))=1$ for all $(v_{i},v_{j})\in E,$ we
have that

$F\left(\left[\widehat{F},e_{i,j}\right]\right)=\begin{cases}1,&\text{ if
}(v_{i},v_{j})\in E;\\\ 0,&\text{ otherwise}.\end{cases}$

The result follows from the linearity of $F$ and the fact that the set of all
$e_{i,j}$ such that $(i,j)$ is a potentially nonzero entry in the matrix form
of $\mathfrak{g}$ is a basis for $\mathfrak{g}.$ ∎

######  Theorem 5.

Let $\mathfrak{g}\subset\mathfrak{sl}(n)$ be a Frobenius, type-A seaweed with
oriented meander $\overrightarrow{M}(\mathfrak{g})=(V,E),$ where
$V=\\{v_{1},\dots,v_{n}\\}$ is the vertex set and $E$ is the edge set of
$\overrightarrow{M}(\mathfrak{g}).$ If

$F=\sum_{(v_{i},v_{j})\in E}e_{i,j}^{*}\in\mathfrak{g}^{*},$

is a Frobenius form on $\mathfrak{g},$ then the principal element
corresponding to $F$ is given by

$\widehat{F}=\sum_{i=1}^{n}\left(w(P_{i,n}(\mathfrak{g}))-\frac{\sum_{j=1}^{n}w(P_{j,n}(\mathfrak{g}))}{n}\right)e_{i,i}\in\mathfrak{g}.$

###### Proof.

First, note that since $\mathfrak{g}$ is a Frobenius, type-A seaweed, the
oriented meander $\overrightarrow{M}(\mathfrak{g})$ consists of one path and
no cycles. Now, if we view $F=\displaystyle\sum_{(v_{i},v_{j})\in
E}e_{i,j}^{*}\in\mathfrak{g}^{*}$ as an element of $(\mathfrak{gl}(n))^{*},$
then we may apply Lemma 4 to conclude that

$\displaystyle F\left(\left[\widehat{F},e_{i,j}\right]\right)$
$\displaystyle=F\left(\left[\sum_{k=1}^{n}\left(w(P_{k,n}(\mathfrak{p}))-\frac{\sum_{\ell=1}^{n}w(P_{\ell,n}(\mathfrak{g}))}{n}\right)e_{k,k},e_{i,j}\right]\right)$
$\displaystyle=F\left(\left[\sum_{k=1}^{n}w(P_{k,n}(\mathfrak{g}))e_{k,k},e_{i,j}\right]\right)$
$\displaystyle=F(e_{i,j}),$

for all $e_{i,j}$ such that $(i,j)$ is a potentially nonzero entry in the
matrix form of $\mathfrak{g}.$ Therefore, $\widehat{F}$ is an element of
$\mathfrak{g}$ – in particular, $\widehat{F}$ has trace zero – with the
property that $F\left(\left[\widehat{F},x\right]\right)=F(x),$ for all
$x\in\mathfrak{g}.$ Since $F$ is Frobenius, we have that $\widehat{F}$ is a
principal element of $F.$ The result follows from uniqueness of the principal
element. ∎

In (Dougherty [8], 2019), it is shown that the $F$ in Theorem 5 is Frobenius.
Consequently, to determine the spectrum of a Frobenius, type-A seaweed
$\mathfrak{g}$, it suffices to compute the eigenvalues of $ad~{}\widehat{F}$,
for $\widehat{F}$ as described in Theorem 5. Since $\widehat{F}$ is diagonal,
the eigenvalues corresponding to $e_{i,i}-e_{i+1,i+1}\in\mathfrak{g}$ are 0.
As for basis elements of the form $e_{i,j}\in\mathfrak{g}$, the proof of
Theorem 5 shows that the eigenvalue is given by $w(P_{i,j}(\mathfrak{g}))$.
The collection of such values can be nicely organized to form the spectrum
matrix of a Frobenius, type-A seaweed $\mathfrak{g}$, denoted
$\Sigma(\mathfrak{g})$. The matrix $\Sigma(\mathfrak{g})$ is the element of
$\mathfrak{g}$ whose $i,j$ entry is equal to $w(P_{i,j}(\mathfrak{g})).$ See
Example 6.

######  Example 6.

In Figure 2, we illustrate the spectrum matrix (left) and oriented meander
(right) corresponding to the Frobenius, type-A seaweed
$\mathfrak{g}=\mathfrak{p}^{A}\frac{2|4}{1|2|3}$. Using
$\Sigma(\mathfrak{g})$, we find that the spectrum of $\mathfrak{g}$ is equal
to $\\{-2,-1^{2},0^{5},1^{5},2^{2},3\\}$.

0101023101-1-1012-2012324 $v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$ Figure 2:
The spectrum matrix $\Sigma(\mathfrak{p})$ (left) and oriented meander
$\overrightarrow{M}(\mathfrak{p})$ (right)

Given a Frobenius, type-A seaweed $\mathfrak{g}$, care must be taken when
using $\Sigma(\mathfrak{g})$ to compute its spectrum. Since we are working
with seaweed subalgebras of $\mathfrak{sl}(n)$, we do not have basis elements
of the form $e_{i,i}$. Instead, basis elements corresponding to diagonal
matrices are of the form $e_{i,i}-e_{i+1,i+1}$. Consequently, the multiset of
values contained in $\Sigma(\mathfrak{g})$ is equal to the spectrum of
$\mathfrak{g}$ with exactly one additional 0. It would be more precise to
exclude the 0 in the $n,n$-location of $\Sigma(\mathfrak{g})$ and interpret a
0 in the $i,i$-location of $\Sigma(\mathfrak{g})$ as the eigenvalue
corresponding to the basis element $e_{i,i}-e_{i+1,i+1}$, for $1\leq i<n$.
However, for the sake of arguments in the following sections, we leave the 0
in the bottom right corner. So that we may recall this subtlety later, we
include the remark below.

######  Remark 7.

Let $\mathfrak{g}$ be a Frobenius, type-A seaweed. Then the multiset of values
contained in $\Sigma(\mathfrak{g})$ is equal to the spectrum of $\mathfrak{g}$
with exactly one additional value of 0.

Utilizing the relationship between the spectrum of a Frobenius, type-A seaweed
$\mathfrak{g}$ and its oriented meander, the authors of [1] were able to show
that the set of distinct eigenvalues in the spectrum of $\mathfrak{g}$
consists of an unbroken sequence of integers centered at one-half. Moreover,
the authors supplied the following conjecture.

######  Conjecture 8 (Coll et al. [1], 2016).

Let $\mathfrak{g}$ be a Frobenius, type-A seaweed. If the eigenvalues in the
spectrum of $\mathfrak{g}$ are written in increasing order, then the
corresponding sequence of multiplicities forms a unimodal sequence about one-
half.

Recall that a sequence $a_{0},\ldots,a_{n}$ of positive integers is called
unimodal if there exists $0\leq i\leq n$ such that $a_{0}\leq
a_{1}\leq\ldots\leq a_{i}\geq\ldots\geq a_{n-1}\geq a_{n}$ and is called log-
concave if $a_{i}^{2}\geq a_{i+1}a_{i-1}$, for $0<i<n$. Ongoing, we will say
that a Frobenius, type-A seaweed $\mathfrak{g}$ satisfying Conjecture 8 has
the unimodal spectrum property. Similarly, if the sequence of mulitplicities,
ordered as in Conjecture 8, forms a log-concave sequence, then we say that
$\mathfrak{g}$ has the log-concave spectrum property. Note that the log-
concave spectrum property implies the unimodal spectrum property.

######  Example 9.

Taking $\mathfrak{g}$ to be the type-A seaweed of Example 6, the sequence of
multiplicities associated with the spectrum of $\mathfrak{g}$ is equal to
$1,2,5,5,2,1$. Clearly, $\mathfrak{g}$ has the log-concave (and, consequently,
the unimodal) spectrum property.

One of the main goals of this paper is to show that certain families of
Frobenius, type-A seaweeds have the unimodal spectrum property. In this
pursuit, an extended notion of spectrum will prove helpful. Extending the
spectrum matrix to contain all weights of paths between vertices $v_{i}$ and
$v_{j}$ of $M(\mathfrak{g})$ results in what we will call the extended
spectrum matrix of $\mathfrak{g}$, denoted $\widehat{\Sigma}(\mathfrak{g})$.
Analogous to spectrum, excluding a single value of 0, we refer to the
remaining multiset of values occurring in the extended spectrum matrix of a
Frobenius, type-A seaweed $\mathfrak{g}$ as the extended spectrum of
$\mathfrak{g}$. See Example 10.

######  Example 10.

In Figure 3, we illustrate the extended spectrum matrix corresponding to the
Frobenius, type-A seaweed $\mathfrak{g}=\mathfrak{p}^{A}\frac{2|4}{1|2|3}$.
Using $\widehat{\Sigma}(\mathfrak{g})$, we find that the extended spectrum of
$\mathfrak{g}$ is equal to
$\\{-3^{2},-2^{4},-1^{7},0^{9},1^{7},2^{4},3^{2}\\}$.

0-10-2-3-1101-1-200-10-2-3-122132310101-1-1012-20 Figure 3: The extended
spectrum matrix $\widehat{\Sigma}(\mathfrak{p})$

######  Remark 11.

Let $\mathfrak{g}$ be a Frobenius, type-A seaweed. Since
$w(P_{i,j}(\mathfrak{g}))=-w(P_{j,i}(\mathfrak{g}))$, it follows that
$\widehat{\Sigma}(\mathfrak{g})$ is skew-symmetric.

##  3 Maximal parabolic

In this section, we consider the spectra of Frobenius, maximal parabolic,
type-A seaweeds, i.e., Frobenius, type-A seaweeds of the form
$\mathfrak{p}^{A}\frac{a|b}{a+b}$ or $\mathfrak{p}^{A}\frac{a+b}{a|b}$. It is
well known that such algebras are Frobenius if and only if $\gcd(a,b)=1$.

Before calculating any spectra, we prove some structural lemmas. The first
group, beginning with Lemma 12 and ending with Corollary 18, concern general
Frobenius, type-A seaweeds. In particular, given a Frobenius, type-A seaweed,
we show that switching and reversing the defining compositions does not affect
the algebra’s (extended) spectrum. The final group of structural lemmas –
consisting of Lemmas 21 through 32– concern Frobenius, maximal parabolic,
type-A seaweeds.

######  Lemma 12.

Let
$\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}}{b_{1}|\ldots|b_{t}}$
and
$\mathfrak{g}_{2}=\mathfrak{p}^{A}\frac{b_{1}|\ldots|b_{t}}{a_{1}|\ldots|a_{m}}$
be Frobenius, type-A seaweeds, where
$n=\sum_{i=1}^{m}a_{i}=\sum_{j=1}^{t}b_{j}$. Then
$w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{j,i}(\mathfrak{g}_{2}))$, for all $1\leq
i,j\leq n.$

###### Proof.

Take an arbitrary path $P_{i,j}(\mathfrak{g}_{1})$ in $M(\mathfrak{g}_{1})$.
Recall that $P_{i,j}(\mathfrak{g}_{1})$ is the unique path from $v_{i}$ to
$v_{j}$ in $M(\mathfrak{g}_{1})$. Let $l$ denote a line parallel to the line
through the vertices of $M(\mathfrak{g}_{1}).$ Note that reflecting
$M(\mathfrak{g}_{1})$ about $l$ results in $M(\mathfrak{g}_{2})$ with each
vertex remaining fixed; that is, vertex $v_{i}$ of $M(\mathfrak{g}_{1})$
becomes vertex $v_{i}$ of $M(\mathfrak{g}_{2})$, for $1\leq i\leq n$.
Moreover, reflecting $\overrightarrow{M}(\mathfrak{g}_{1})$ about $l$ results
in the meander $M(\mathfrak{g}_{2})$ with each edge oriented in the opposite
direction to that in $\overrightarrow{M}(\mathfrak{g}_{2})$. Thus,
$w(P_{i,j}(\mathfrak{g}_{1}))=-w(P_{i,j}(\mathfrak{g}_{2}))=w(P_{j,i}(\mathfrak{g}_{2}))$,
for all $1\leq i,j\leq n.$ ∎

######  Corollary 13.

Let
$\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}}{b_{1}|\ldots|b_{t}}$
and
$\mathfrak{g}_{2}=\mathfrak{p}^{A}\frac{b_{1}|\ldots|b_{t}}{a_{1}|\ldots|a_{m}}$
be Frobenius, type-A seaweeds. Then
$\Sigma(\mathfrak{g}_{1})=\Sigma(\mathfrak{g}_{2})^{t}$ and
$\widehat{\Sigma}(\mathfrak{g}_{1})=\widehat{\Sigma}(\mathfrak{g}_{2})^{t}$.

###### Proof.

Applying Lemma 12, it follows that the $i,j$-entry of
$\widehat{\Sigma}(\mathfrak{g}_{1})$ is equal to the $j,i$-entry of
$\widehat{\Sigma}(\mathfrak{g}_{2})$, i.e.,
$\widehat{\Sigma}(\mathfrak{g}_{1})=\widehat{\Sigma}(\mathfrak{g}_{2})^{t}$.
Consequently, since $e_{i,j}\in\mathfrak{g}_{1}$ if and only if
$e_{j,i}\in\mathfrak{g}_{2}$,
$\Sigma(\mathfrak{g}_{1})=\Sigma(\mathfrak{g}_{2})^{t}$. ∎

######  Corollary 14.

Let
$\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}}{b_{1}|\ldots|b_{t}}$
and
$\mathfrak{g}_{2}=\mathfrak{p}^{A}\frac{b_{1}|\ldots|b_{t}}{a_{1}|\ldots|a_{m}}$
be Frobenius, type-A seaweeds. Then the (extended) spectra of
$\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ are equal.

######  Lemma 15.

Let
$\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}}{b_{1}|\ldots|b_{t}}$
and
$\mathfrak{g}_{2}=\mathfrak{p}^{A}\frac{a_{m}|\ldots|a_{1}}{b_{t}|\ldots|b_{1}}$
be Frobenius, type-A seaweeds, where
$n=\sum_{i=1}^{m}a_{i}=\sum_{j=1}^{t}b_{j}$. Then
$w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{n-j+1,n-i+1}(\mathfrak{g}_{2}))$, for all
$1\leq i,j\leq n.$

###### Proof.

Take an arbitrary path $P_{i,j}(\mathfrak{g}_{1})$ in $M(\mathfrak{g}_{1})$.
Let $l$ denote a line perpendicular to the line through the vertices of
$M(\mathfrak{g}_{1})$. Note that reflecting $M(\mathfrak{g}_{1})$ about $l$
results in $M(\mathfrak{g}_{2})$ with vertex $v_{i}$ becoming vertex
$v_{n-i+1}$, for $1\leq i\leq n$. Moreover, reflecting
$\overrightarrow{M}(\mathfrak{g}_{1})$ about $l$ results in the meander
$M(\mathfrak{g}_{2})$ with each edge oriented in the opposite direction to
that in $\overrightarrow{M}(\mathfrak{g}_{2})$. Thus,

$w(P_{i,j}(\mathfrak{g}_{1}))=-w(P_{n-i+1,n-j+1}(\mathfrak{g}_{2}))=w(P_{n-j+1,n-i+1}(\mathfrak{g}_{2})).$

∎

######  Corollary 16.

Let
$\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{a_{1}|\dots|a_{m}}{b_{1}|\dots|b_{t}}$
and $\mathfrak{g}_{2}=\frac{a_{m}|\dots|a_{1}}{b_{t}|\dots|b_{1}}$ be
Frobenius, type-A seaweeds. Then
$\Sigma(\mathfrak{g}_{1})=\Sigma(\mathfrak{g}_{2})^{\tau}$ and
$\widehat{\Sigma}(\mathfrak{g}_{1})=\widehat{\Sigma}(\mathfrak{g}_{2})^{\tau},$
where $\tau$ denotes transpose with respect to the antidiagonal.

###### Proof.

Applying Lemma 15, it follows that the $i,j$-entry of
$\widehat{\Sigma}(\mathfrak{g}_{1})$ is equal to the $n-j+1,n-i+1$-entry of
$\widehat{\Sigma}(\mathfrak{g}_{2}),$ i.e.,
$\widehat{\Sigma}(\mathfrak{g}_{1})=\widehat{\Sigma}(\mathfrak{g}_{2}).$
Consequently, since $e_{i,j}\in\mathfrak{g}$ if and only if $e_{n-j+1,n-i+1},$
$\Sigma(\mathfrak{g}_{1})=\Sigma(\mathfrak{g}_{2})^{\tau}.$ ∎

######  Corollary 17.

Let
$\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}}{b_{1}|\ldots|b_{t}}$
and
$\mathfrak{g}_{2}=\mathfrak{p}^{A}\frac{a_{m}|\ldots|a_{1}}{b_{t}|\ldots|b_{1}}$
be Frobenius, type-A seaweeds. Then the (extended) spectra of
$\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ are equal.

######  Corollary 18.

Let
$\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}}{b_{1}|\ldots|b_{t}}$
and
$\mathfrak{g}_{2}=\mathfrak{p}^{A}\frac{b_{t}|\ldots|b_{1}}{a_{m}|\ldots|a_{1}}$
be Frobenius, type-A seaweeds. Then the (extended) spectra of
$\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ are equal.

###### Proof.

The result follows upon combining Corollaries 14 and 17. ∎

######  Remark 19.

One can show that, for fixed compositions $(a_{1},\dots,a_{m})$ and
$(b_{1},\dots,b_{t})$ of $n,$ the type-A seaweeds
$\mathfrak{p}^{A}\frac{a_{1}|\dots|a_{m}}{b_{1}|\dots|b_{t}},$
$\mathfrak{p}^{A}\frac{b_{1}|\dots|b_{t}}{a_{1}|\dots|a_{m}},$
$\mathfrak{p}^{A}\frac{a_{m}|\dots|a_{1}}{b_{t}|\dots|b_{1}}$, and
$\mathfrak{p}^{A}\frac{b_{t}|\dots|b_{1}}{a_{m}|\dots|a_{1}}$ are isomorphic,
regardless of their (shared) respective index values.

Now, turning to Frobenius, maximal parabolic, type-A seaweeds, the following
lemmas relate the spectra of “bigger” algebras to naturally associated
“smaller” algebras. First, given such an algebra
$\mathfrak{g}=\mathfrak{p}^{A}\frac{a|b}{n}$ with $a>b$, Lemma 21 below
relates the values in the top left $a\times a$ block of $\Sigma(\mathfrak{g})$
to the values contained in $\widehat{\Sigma}(\mathfrak{g}^{\prime})$, where
$\mathfrak{g}^{\prime}$ is a seaweed subalgebra of $\mathfrak{sl}(a)$. To aid
the proof of Lemma 21, we first illustrate the result in Example 20.

######  Example 20.

Let $\mathfrak{g}=\mathfrak{p}^{A}\frac{5|2}{7}$ and
$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{3|2}{5}$. See Figure 4 for (a)
$\Sigma(\mathfrak{g})$ and (b) $M(\mathfrak{g})$, and see Figure 5 for (a)
$\widehat{\Sigma}(\mathfrak{g}^{\prime})$ and (b) $M(\mathfrak{g}^{\prime})$.
Note that the top left $5\times 5$ block of $\Sigma(\mathfrak{g})$ (to the
left of the vertical dashed line) consists of the same multiset of values as
$\widehat{\Sigma}(\mathfrak{g}^{\prime})$. Moreover, note that there is a copy
of $M(\mathfrak{g}^{\prime})$ inside of $M(\mathfrak{g})$. In particular,
identifying the vertices connected by dashed edges in Figure 4 (b) and then
rotating the resulting graph by 180 degrees yields $M(\mathfrak{g}^{\prime})$.

01-20-121-10-3-1-210230214301-20-12112-110320-110(a)
$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$v_{7}$(b) Figure 4: (a)
$\Sigma(\mathfrak{g})$ and (b) $M(\mathfrak{g})$
01-121-10-21012032-2-130-1-10-210(a) $v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$(b)
Figure 5: (a) $\widehat{\Sigma}(\mathfrak{g}^{\prime})$ and (b)
$M(\mathfrak{g}^{\prime})$

######  Lemma 21.

Let $k_{1},k_{2}\in\mathbb{Z}_{>0}$ satisfy $\gcd(k_{1},k_{2})=1$. If
$\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{mk_{1}+k_{2}|k_{1}}{(m+1)k_{1}+k_{2}}$
and
$\mathfrak{g}_{2}=\mathfrak{p}^{A}\frac{(m-1)k_{1}+k_{2}|k_{1}}{mk_{1}+k_{2}}$,
then the block corresponding to rows $\\{1,\ldots,mk_{1}+k_{2}\\}$ and columns
$\\{1,\ldots,mk_{1}+k_{2}\\}$ of $\Sigma(\mathfrak{g}_{1})$ consists of the
same multiset of values as $\widehat{\Sigma}(\mathfrak{g}_{2})$.

###### Proof.

Define the sets

$H=\\{1,\ldots,mk_{1}+k_{2}\\}\quad\text{and}\quad
T=\\{mk_{1}+k_{2}+1,\ldots,(m+1)k_{1}+k_{2}\\}.$

Note that $\\{v_{i}~{}|~{}i\in H\\}$ and $\\{v_{i}~{}|~{}i\in T\\}$ form a
partition of the vertices of $M(\mathfrak{g}_{1})$. The block $\Delta$ of
$\Sigma(\mathfrak{g}_{1})$ corresponding to rows $r\in H$ and columns $c\in H$
consists of the weights of paths between the vertices of $\\{v_{i}~{}|~{}i\in
H\\}$ in $M(\mathfrak{g}_{1})$. Clearly, both $\Delta$ and
$\widehat{\Sigma}(\mathfrak{g}_{2})$ have the same number, i.e.,
$mk_{1}+k_{2}$, of 0’s on their respective diagonals. Thus, we need to show
that $\Delta$ and $\widehat{\Sigma}(\mathfrak{g}_{2})$ have the same multisets
of values corresponding to off-diagonal entries. To do so, we define a weight-
preserving bijection

$\phi:~{}\\{P_{i,j}(\mathfrak{g}_{1})~{}|~{}i,j\in H,~{}i\neq
j\\}\to\\{P_{i,j}(\mathfrak{g}_{2})~{}|~{}i,j\in H,~{}i\neq j\\},$

i.e., a weight-preserving bijection from paths in
$\overrightarrow{M}(\mathfrak{g}_{1})$ between distinct vertices $v_{i}$ and
$v_{j}$ with $i,j\in H$ to all paths between distinct vertices in
$\overrightarrow{M}(\mathfrak{g}_{2})$. To aid in defining $\phi$, we make use
of a transformation $\mathcal{S}$ of $\overrightarrow{M}(\mathfrak{g}_{1})$
which is defined as follows. If $v_{h}$ is adjacent to $v_{t}$ in
$\overrightarrow{M}(\mathfrak{g}_{1}),$ for $h\in H$ and $t\in T,$ then
identify $v_{h}$ and $v_{t}$ and remove any edges between them. Let
$\mathfrak{g}_{3}=\mathfrak{p}^{A}\frac{mk_{1}+k_{2}}{k_{1}|(m-1)k_{1}+k_{2}}$.
Note that
$\mathcal{S}(\overrightarrow{M}(\mathfrak{g}_{1}))=\overrightarrow{M}(\mathfrak{g}_{3})$
with $\mathcal{S}$ mapping vertices $v_{i}$ and $v_{(m+1)k_{1}+k_{2}-i+1}$ in
$\overrightarrow{M}(\mathfrak{g}_{1})$ to $v_{i}$ in
$\overrightarrow{M}(\mathfrak{g}_{3})$, for $1\leq i\leq k_{1}$, and vertices
$v_{j}$ in $\overrightarrow{M}(\mathfrak{g}_{1})$ to $v_{j}$ in
$\overrightarrow{M}(\mathfrak{g}_{3})$, for $k_{1}+1\leq j\leq mk_{1}+k_{2}$.
We claim that $w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{i,j}(\mathfrak{g}_{3}))$, for
all distinct $i,j\in H$.

To establish the claim, take distinct $i,j\in H$. If
$P_{i,j}(\mathfrak{g}_{1})$ contains no $v_{t}$, for $t\in T$, then
$P_{i,j}(\mathfrak{g}_{1})$ along with its orientation in
$\overrightarrow{M}(\mathfrak{g}_{1})$ remain invariant under $\mathcal{S}$;
that is, $w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{i,j}(\mathfrak{g}_{3}))$.
Otherwise, $P_{i,j}(\mathfrak{g}_{1})$ contains a vertex $v_{t}$, for $t\in
T$. In this case, $P_{i,j}(\mathfrak{g}_{1})$ contains subpaths
$P_{h_{1},h_{2}}(\mathfrak{g}_{1})$ defined by sequences of vertices
$v_{h_{1}},v_{t_{1}},v_{t_{2}},v_{h_{2}}$, where

  * •

$h_{1},h_{2}\in H$ and $t_{1},t_{2}\in T$ are distinct,

  * •

$v_{h_{1}}$ is adjacent to $v_{t_{1}}$ via a bottom edge directed from
$v_{h_{1}}$ to $v_{t_{1}}$ in $\overrightarrow{M}(\mathfrak{g}_{1})$,

  * •

$v_{t_{1}}$ is adjacent to $v_{t_{2}}$ via a top edge in
$\overrightarrow{M}(\mathfrak{g}_{1})$, and

  * •

$v_{t_{2}}$ is adjacent to $v_{h_{2}}$ via a bottom edge directed from
$v_{h_{2}}$ to $v_{t_{2}}$ in $\overrightarrow{M}(\mathfrak{g}_{1})$.

To establish the claim in this case, it suffices to show that the weights of
such subpaths are invariant under $\mathcal{S}$. Note that

$\displaystyle w(P_{h_{1},h_{2}}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h_{1},t_{1}}(\mathfrak{g}_{1}))+w(P_{t_{1},t_{2}}(\mathfrak{g}_{1}))+w(P_{t_{2},h_{2}}(\mathfrak{g}_{1}))$
$\displaystyle=1+w(P_{t_{1},t_{2}}(\mathfrak{g}_{1}))-1$
$\displaystyle=w(P_{t_{1},t_{2}}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h_{1},h_{2}}(\mathfrak{g}_{3}))$
$\displaystyle=w(\mathcal{S}(P_{h_{1},h_{2}}(\mathfrak{g}_{1}))),$

where for the penultimate equality we have used the fact that $\mathcal{S}$
maps the vertices $v_{t_{i}}$ to $v_{h_{i}}$, for $i=1,2$, and the edge
connecting $v_{t_{1}}$ and $v_{t_{2}}$ in
$\overrightarrow{M}(\mathfrak{g}_{1})$ to the edge connecting $v_{h_{1}}$ and
$v_{h_{2}}$ in $\overrightarrow{M}(\mathfrak{g}_{3})$, preserving their
orientations. Thus,
$w(P_{h_{1},h_{2}}(\mathfrak{g}_{1}))=w(\mathcal{S}(P_{h_{1},h_{2}}(\mathfrak{g}_{1})))$,
and we may conclude that
$w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{i,j}(\mathfrak{g}_{3}))$. Consequently,
$w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{i,j}(\mathfrak{g}_{3}))$, for all distinct
$i,j\in H$, as claimed.

Now, applying Lemmas 12 and 15, it follows that
$w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{mk_{1}+k_{2}-i+1,mk_{1}+k_{2}-j+1}(\mathfrak{g}_{2}))$,
for all distinct $i,j\in H$. Considering our work above, it follows that we
can form our desired weight-preserving bijection $\phi$ by mapping the path
$P_{i,j}(\mathfrak{g}_{1})$ to
$P_{mk_{1}+k_{2}-i+1,mk_{1}+k_{2}-j+1}(\mathfrak{g}_{2})$, for all distinct
$i,j\in H$. The result follows. ∎

######  Remark 22.

In fact, for $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ as in Lemma 21, the
block corresponding to rows $\\{1,\ldots,k_{1}+k_{2}\\}$ and columns
$\\{1,\ldots,k_{1}+k_{2}\\}$ of $\Sigma(\mathfrak{g}_{1})$ is equal to
$-\widehat{\Sigma}(\mathfrak{g}_{2})^{t}$.

Next, given a Frobenius, maximal parabolic, type-A seaweed
$\mathfrak{g}=\mathfrak{p}^{A}\frac{a|b}{n}$ with $a>b$, we show how the
values in the bottom right $b\times b$ block of $\Sigma(\mathfrak{g})$ can be
related to the values contained in $\widehat{\Sigma}(\mathfrak{g}^{\prime})$,
where $\mathfrak{g}^{\prime}$ is a seaweed subalgebra of $\mathfrak{sl}(b)$.
Unlike in Lemma 21, we have to consider two separate cases corresponding to
$\lfloor\frac{a}{b}\rfloor=1$ and $\lfloor\frac{a}{b}\rfloor>1$. Lemma 24
addresses the case $\lfloor\frac{a}{b}\rfloor=1$. An illustration of Lemma 24
is given in Example 23.

######  Example 23.

Let $\mathfrak{g}=\mathfrak{p}^{A}\frac{5|3}{8}$ and
$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{1|2}{3}.$ See Figure 6 for (a)
$\Sigma(\mathfrak{g})$ and (b) $M(\mathfrak{g}),$ and see Figure 7 for (a)
$\widehat{\Sigma}(\mathfrak{g}^{\prime})$ and (b) $M(\mathfrak{g}^{\prime}).$
Note that the bottom right $3\times 3$ block (below the horizontal dashed
line) of $\Sigma(\mathfrak{g})$ consists of the same multiset of values as
$\widehat{\Sigma}(\mathfrak{g}^{\prime}).$ Moreover, note that there is a copy
of $M(\mathfrak{g}^{\prime})$ inside of $M(\mathfrak{g}).$ In particular,
identifying the vertices connected by dashed edges in Figure 6 (b) and then
rotating the resulting graph by 180 degrees yields $M(\mathfrak{g}^{\prime}).$

0-11-2-1201102-10312-1-20-3-21-1021301423102-103120-2-12011-10(a)
$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$v_{7}$$v_{8}$(b) Figure 6: (a)
$\Sigma(\mathfrak{g})$ and (b) $M(\mathfrak{g})$ 021-20-1-110(a)
$v_{1}$$v_{2}$$v_{3}$(b) Figure 7: (a)
$\widehat{\Sigma}(\mathfrak{g}^{\prime})$ and (b) $M(\mathfrak{g}^{\prime})$

######  Lemma 24.

Let $k_{1},k_{2}\in\mathbb{Z}_{>0}$ satisfy $k_{1}>k_{2}$ and
$\gcd(k_{1},k_{2})=1$. If
$\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{k_{1}+k_{2}|k_{1}}{2k_{1}+k_{2}}$ and
$\mathfrak{g}_{2}=\mathfrak{p}^{A}\frac{k_{1}-k_{2}|k_{2}}{k_{1}}$, then the
block corresponding to rows $\\{k_{1}+k_{2}+1,\ldots,2k_{1}+k_{2}\\}$ and
columns $\\{k_{1}+k_{2}+1,\ldots,2k_{1}+k_{2}\\}$ in
$\Sigma(\mathfrak{g}_{1})$ consists of the same multiset of values as
$\widehat{\Sigma}(\mathfrak{g}_{2})$.

###### Proof.

Define the sets

$H_{1}=\\{1,\ldots,k_{2}\\},\quad H_{2}=\\{k_{2}+1,\ldots,k_{1}\\},\quad
H_{3}=\\{k_{1}+1,\ldots,k_{1}+k_{2}\\},\quad\text{and}\quad
T=\\{k_{1}+k_{2}+1,\ldots,2k_{1}+k_{2}\\}.$

Note that $\\{v_{i}~{}|~{}i\in H_{1}\\},\\{v_{i}~{}|~{}i\in
H_{2}\\},\\{v_{i}~{}|~{}i\in H_{3}\\},$ and $\\{v_{i}~{}|~{}i\in T\\}$ form a
partition of the vertices of $M(\mathfrak{g}_{1})$ and are defined in such a
way that top edges in $M(\mathfrak{g}_{1})$ only connect pairs of vertices $u$
and $v$ with

  * •

$u\in\\{v_{i}~{}|~{}i\in H_{1}\\}$ and $v\in\\{v_{i}~{}|~{}i\in H_{3}\\}$,

  * •

$u,v\in\\{v_{i}~{}|~{}i\in H_{2}\\}$, or

  * •

$u,v\in\\{v_{i}~{}|~{}i\in T\\}$

and bottom edges in $M(\mathfrak{g}_{1})$ only connect pairs of vertices $u$
and $v$ with

  * •

$u\in\\{v_{i}~{}|~{}i\in H_{1}\cup H_{2}\\}$ and $v\in\\{v_{i}~{}|~{}i\in
T\\}$ or

  * •

$u,v\in\\{v_{i}~{}|~{}i\in H_{3}\\}$.

Now, the block $\Delta$ of $\Sigma(\mathfrak{g}_{1})$ corresponding to rows
$r\in T$ and columns $c\in T$ consists of the values
$w(P_{i,j}(\mathfrak{g}_{1}))$, for $i,j\in T$. Clearly, both $\Delta$ and
$\widehat{\Sigma}(\mathfrak{g}_{2})$ have the same number, i.e., $k_{1}$, of
0’s on their respective diagonals. Thus, we need to show that $\Delta$ and
$\widehat{\Sigma}(\mathfrak{g}_{2})$ have the same multisets of values
corresponding to off-diagonal entries. To do so, we define a weight-preserving
bijection

$\phi:~{}\\{P_{i,j}(\mathfrak{g}_{1})~{}|~{}i,j\in T,~{}i\neq
j\\}\to\\{P_{i,j}(\mathfrak{g}_{2})~{}|~{}i,j\in H_{1}\cup H_{2},~{}i\neq
j\\},$

i.e., a weight-preserving bijection from paths in
$\overrightarrow{M}(\mathfrak{g}_{1})$ between distinct vertices
$u,v\in\\{v_{i}~{}|~{}i\in T\\}$ to all paths between distinct vertices in
$\overrightarrow{M}(\mathfrak{g}_{2})$. To aid in defining our bijection, we
make use of a transformation $\mathcal{S}$ of
$\overrightarrow{M}(\mathfrak{g}_{1})$ which is defined as follows.

  1. 1.

If $u\in\\{v_{i}~{}|~{}i\in T\\}$ is adjacent to $v\in\\{v_{i}~{}|~{}i\in
H_{1}\cup H_{2}\\},$ then identify $u$ and $v$ while removing any edges
between them, and then

  2. 2.

if $u\in\\{v_{i}~{}|~{}i\in H_{3}\\}$ is adjacent to $v\in\\{v_{i}~{}|~{}i\in
H_{1}\\}$, then identify $u$ and $v$ while removing any edges between them.

Let $\mathfrak{g}_{3}=\mathfrak{p}^{A}\frac{k_{2}|k_{1}-k_{2}}{k_{1}}.$ Note
that
$\mathcal{S}(\overrightarrow{M}(\mathfrak{g}_{1}))=\overrightarrow{M}(\mathfrak{g}_{3})$
with $\mathcal{S}$ mapping vertices $v_{i}$ and $v_{2k_{1}+k_{2}-i+1}$ in
$M(\mathfrak{g}_{1})$ to $v_{i}$ in $M(\mathfrak{g}_{3})$, for $1\leq i\leq
k_{1}$, and vertices $v_{k_{1}+k_{2}-j+1}$ in $M(\mathfrak{g}_{1})$ to $v_{j}$
in $M(\mathfrak{g}_{3})$, for $1\leq j\leq k_{2}$. We claim that
$w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{2k_{1}+k_{2}-i+1,2k_{1}+k_{2}-j+1}(\mathfrak{g}_{3}))$,
for distinct $i,j\in T$.

To establish the claim, take distinct $i,j\in T$. If $v_{i}$ and $v_{j}$ are
adjacent in $M(\mathfrak{g}_{1})$, then $\mathcal{S}$ maps
$P_{i,j}(\mathfrak{g}_{1})$ along with its orientation in
$\overrightarrow{M}(\mathfrak{g}_{1})$ to
$P_{2k_{1}+k_{2}-i+1,2k_{1}+k_{2}-j+1}(\mathfrak{g}_{3})$ along with its
orientation in $\overrightarrow{M}(\mathfrak{g}_{3})$; that is,
$w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{i,j}(\mathfrak{g}_{3})).$ On the other
hand, if $v_{i}$ and $v_{j}$ are not adjacent in $M(\mathfrak{g}_{1})$, then
$P_{i,j}(\mathfrak{g}_{1})$ contains subpaths
$P_{t_{1},t_{2}}(\mathfrak{g}_{1})$ defined by sequences of vertices
$v_{t_{1}},v_{h_{1}},\ldots,v_{h_{\ell}},v_{t_{2}}$ with $t_{1},t_{2}\in T$
and $h_{1},\ldots,h_{\ell}\in H_{1}\cup H_{2}\cup H_{3}$. To establish the
claim, it suffices to show that the weights of such subpaths are invariant
under $\mathcal{S}$. There are two cases.

Case 1: Subpaths $P_{t_{1},t_{2}}$ defined by a sequence of vertices
$v_{t_{1}},v_{h_{1}^{1}},v_{h_{1}^{3}},v_{h_{2}^{3}},v_{h_{2}^{1}},v_{t_{2}}$,
where

  * •

$t_{1},t_{2}\in T$, $h_{1}^{1},h_{2}^{1}\in H_{1}$, and
$h_{1}^{3},h_{2}^{3}\in H_{3}$,

  * •

$v_{t_{1}}$ is adjacent to $v_{h_{1}^{1}}$ via a bottom edge directed from
$v_{h_{1}^{1}}$ to $v_{t_{1}}$ in $\overrightarrow{M}(\mathfrak{g}_{1})$,

  * •

$v_{h_{1}^{1}}$ is adjacent to $v_{h_{1}^{3}}$ via a top edge directed from
$v_{h_{1}^{3}}$ to $v_{h^{1}_{1}}$ in $\overrightarrow{M}(\mathfrak{g}_{1})$,

  * •

$v_{h_{1}^{3}}$ is adjacent to $v_{h^{3}_{2}}$ via a bottom edge in
$\overrightarrow{M}(\mathfrak{g}_{1})$,

  * •

$v_{h^{3}_{2}}$ is adjacent to $v_{h^{1}_{2}}$ via a top edge directed from
$v_{h^{3}_{2}}$ to $v_{h^{1}_{2}}$ in $\overrightarrow{M}(\mathfrak{g}_{1})$,
and

  * •

$v_{h^{1}_{2}}$ is adjacent to $v_{t_{2}}$ via a bottom edge directed from
$v_{h^{1}_{2}}$ to $v_{t_{2}}$ in $\overrightarrow{M}(\mathfrak{g}_{1})$.

Note that

$\displaystyle w(P_{t_{1},t_{2}}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{t_{1},h_{1}^{1}}(\mathfrak{g}_{1}))+w(P_{h^{1}_{1},h^{3}_{1}}(\mathfrak{g}_{1}))+w(P_{h^{3}_{1},h^{3}_{2}}(\mathfrak{g}_{1}))+w(P_{h^{3}_{2},h^{1}_{2}}(\mathfrak{g}_{1}))+w(P_{h^{1}_{2},t_{2}}(\mathfrak{g}_{1}))$
$\displaystyle=-1-1+w(P_{h^{3}_{1},h^{3}_{2}}(\mathfrak{g}_{1}))+1+1$
$\displaystyle=w(P_{h^{3}_{1},h^{3}_{2}}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h^{1}_{1},h^{1}_{2}}(\mathfrak{g}_{3}))$
$\displaystyle=w(\mathcal{S}(P_{t_{1},t_{2}}(\mathfrak{g}_{1}))),$

where for the penultimate equality we have used the fact that $\mathcal{S}$
maps $v_{h^{3}_{i}}$ to $v_{h^{1}_{i}}$, for $i=1,2$, as well as the edge
connecting $v_{h^{3}_{1}}$ and $v_{h^{3}_{2}}$ in
$\overrightarrow{M}(\mathfrak{g}_{1})$ to the edge connecting $v_{h^{1}_{1}}$
and $v_{h^{1}_{2}}$ in $\overrightarrow{M}(\mathfrak{g}_{1})$, preserving
their orientations. Thus, it follows that
$w(P_{t_{1},t_{2}}(\mathfrak{g}_{1}))$ is invariant under $\mathcal{S}$.

Case 2: Subpaths $P_{t_{1},t_{2}}(\mathfrak{g}_{1})$ defined by a sequence of
vertices $v_{t_{1}},v_{h_{1}^{2}},v_{h_{2}^{2}},v_{t_{2}}$, where

  * •

$t_{1},t_{2}\in T$ and $h_{1}^{2},h_{2}^{2}\in H_{2}$,

  * •

$v_{t_{1}}$ is adjacent to $v_{h_{1}^{2}}$ via a bottom edge directed from
$v_{h_{1}^{2}}$ to $v_{t_{1}}$ in $\overrightarrow{M}(\mathfrak{g}_{1})$,

  * •

$v_{h_{1}^{2}}$ is adjacent to $v_{h_{2}^{2}}$ via a top edge in
$\overrightarrow{M}(\mathfrak{g}_{1})$, and

  * •

$v_{h_{2}^{2}}$ is adjacent to $v_{t_{2}}$ via a bottom edge directed from
$v_{h_{2}^{2}}$ to $v_{t_{2}}$ in $\overrightarrow{M}(\mathfrak{g}_{1})$.

Note that

$\displaystyle w(P_{t_{1},t_{2}}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{t_{1},h_{1}^{2}}(\mathfrak{g}_{1}))+w(P_{h^{2}_{1},h^{2}_{2}}(\mathfrak{g}_{1}))+w(P_{h^{2}_{2},t_{2}}(\mathfrak{g}_{1}))$
$\displaystyle=-1+w(P_{h^{2}_{1},h^{2}_{2}}(\mathfrak{g}_{1}))+1$
$\displaystyle=w(P_{h^{2}_{1},h^{2}_{2}}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h^{2}_{1},h^{2}_{2}}(\mathfrak{g}_{3}))$
$\displaystyle=w(\mathcal{S}(P_{t_{1},t_{2}}(\mathfrak{g}_{1}))).$

Thus, it follows that $w(P_{t_{1},t_{2}}(\mathfrak{g}_{1}))$ is invariant
under $\mathcal{S}$.

Consequently,
$w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{2k_{1}+k_{2}-i+1,2k_{1}+k_{2}-j+1}(\mathfrak{g}_{3}))$,
for all distinct $i,j\in T$, as claimed.

Now, applying Lemma 15, it follows that

$w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{j-k_{1}-k_{2},i-k_{1}-k_{2}}(\mathfrak{g}_{2})),$

for all distinct $i,j\in T$. Therefore, the desired weight-preserving
bijection $\phi$ is given by mapping the path $P_{i,j}(\mathfrak{g}_{1})$ to
$P_{j-k_{1}-k_{2},i-k_{1}-k_{2}}(\mathfrak{g}_{2})$, for all distinct $i,j\in
T.$ The result follows. ∎

######  Remark 25.

In fact, for $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ as in Lemma 24, the
block corresponding to rows $\\{k_{1}+k_{2}+1,\ldots,2k_{1}+k_{2}\\}$ and
columns $\\{k_{1}+k_{2}+1,\ldots,2k_{1}+k_{2}\\}$ of
$\Sigma(\mathfrak{g}_{1})$ is the same as
$\widehat{\Sigma}(\mathfrak{g}_{2})^{t}$.

Now, in Lemma 27 below, we extend Lemma 24 so that it applies to the bottom
right $b\times b$ block of $\Sigma(\mathfrak{g})$ when
$\lfloor\frac{a}{b}\rfloor>1$. The following example provides an illustration
of Lemma 27.

######  Example 26.

Let $\mathfrak{g}=\mathfrak{p}^{A}\frac{5|2}{7}$ and
$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{1|1}{2}$. See Figure 4 above for
(a) $\Sigma(\mathfrak{g})$ and (b) $M(\mathfrak{g})$ and Figure 8 for (a)
$\widehat{\Sigma}(\mathfrak{g}^{\prime})$ and (b) $M(\mathfrak{g}^{\prime})$.
Note that the bottom right $2\times 2$ block of $\Sigma(\mathfrak{g})$
consists of the same multiset of values as
$\widehat{\Sigma}(\mathfrak{g}^{\prime})$. Moreover, note that there is a copy
of $M(\mathfrak{g}^{\prime})$ inside of $M(\mathfrak{g})$. In particular,
identifying vertices connected by all edges except $\\{v_{6},v_{7}\\}$ and
then rotating the resulting graph by 180 degrees yields
$M(\mathfrak{g}^{\prime})$.

01-10(a) $v_{1}$$v_{2}$(b) Figure 8: (a)
$\widehat{\Sigma}(\mathfrak{g}^{\prime})$ and (b) $M(\mathfrak{g}^{\prime})$

######  Lemma 27.

Let $k_{1},k_{2}\in\mathbb{Z}_{>0}$ satisfy $k_{1}>k_{2}$ and
$\gcd(k_{1},k_{2})=1$. If
$\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{mk_{1}+k_{2}|k_{1}}{(m+1)k_{1}+k_{2}}$
and
$\mathfrak{g}_{2}=\mathfrak{p}^{A}\frac{k_{1}-k_{2}|k_{2}}{k_{1}}$, then the
block corresponding to rows $\\{mk_{1}+k_{2}+1,\ldots,(m+1)k_{1}+k_{2}\\}$ and
columns $\\{mk_{1}+k_{2}+1,\ldots,(m+1)k_{1}+k_{2}\\}$ of
$\Sigma(\mathfrak{g}_{1})$ consists of the same values as
$\widehat{\Sigma}(\mathfrak{g}_{2})$.

###### Proof.

Note that if $m=1$, then the result follows by Lemma 24. So, assume that
$m>1$. Define the sets of vertices

$H=\\{1,\ldots,mk_{1}+k_{2}\\},\quad H_{1}=\\{1,\ldots,k_{1}\\},\quad
T_{1}=\\{mk_{1}+k_{2}+1,\ldots,(m+1)k_{1}+k_{2}\\},$

and

$T_{2}=\\{(m-1)k_{1}+k_{2}+1,\ldots,mk_{1}+k_{2}\\}.$

Let
$\mathfrak{g}_{3}=\mathfrak{p}^{A}\frac{(m-1)k_{1}+k_{2}|k_{1}}{mk_{1}+k_{2}}$.
First, we show that the block $\Delta_{1}$ of $\Sigma(\mathfrak{g}_{1})$
corresponding to rows $r\in T_{1}$ and columns $c\in T_{1}$ contains the same
values as the block $\Delta_{2}$ of $\Sigma(\mathfrak{g}_{3})$ corresponding
to rows $r\in T_{2}$ and columns $c\in T_{2}$. The block $\Delta_{1}$ of
$\Sigma(\mathfrak{g}_{1})$ consists of the values
$w(P_{i,j}(\mathfrak{g}_{1}))$, for $i,j\in T_{1}$, and the block $\Delta_{2}$
of $\Sigma(\mathfrak{g}_{3})$ consists of the values
$w(P_{i,j}(\mathfrak{g}_{3}))$, for $i,j\in T_{2}$. Clearly, $\Delta_{1}$ and
$\Delta_{2}$ have the same number, i.e., $k_{1}$, of 0’s on their respective
diagonals. Thus, we need to show that $\Delta_{1}$ and $\Delta_{2}$ have the
same multisets of values corresponding to off-diagonal entries. To do so, we
define a weight-preserving bijection

$\phi:~{}\\{P_{i,j}(\mathfrak{g}_{1})~{}|~{}i,j\in T_{1},~{}i\neq
j\\}\to\\{P_{i,j}(\mathfrak{g}_{3})~{}|~{}i,j\in T_{2},~{}i\neq j\\}.$

To aid in defining $\phi$, we make use of a transformation $\mathcal{S}$ of
$\overrightarrow{M}(\mathfrak{g}_{1})$ which is defined as follows. Note that
in $M(\mathfrak{g}_{1})$ (resp., $M(\mathfrak{g}_{3})$), each vertex of
$u\in\\{v_{i}~{}|~{}i\in H_{1}\\}$ is adjacent to a unique vertex of
$v\in\\{v_{i}~{}|~{}i\in T_{1}\\}$ (resp., $v\in\\{v_{i}~{}|~{}i\in T_{2}\\}$)
via an arc below. If $u\in\\{v_{i}~{}|~{}i\in H_{1}\\}$ is adjacent to
$v\in\\{v_{i}~{}|~{}i\in T_{1}\\},$ then $\mathcal{S}$ identifies $u$ and $v$
while removing any edges between them. Let
$\mathfrak{g}_{4}=\mathfrak{p}^{A}\frac{mk_{1}+k_{2}}{k_{1}|(m-1)k_{1}+k_{2}}$.
Note that
$\mathcal{S}(\overrightarrow{M}(\mathfrak{g}_{1}))=\overrightarrow{M}(\mathfrak{g}_{4})$
with $\mathcal{S}$ mapping vertices $v_{i}$ and $v_{(m+1)k_{1}+k_{2}-i+1}$ in
$\overrightarrow{M}(\mathfrak{g}_{1})$ to $v_{i}$ in
$\overrightarrow{M}(\mathfrak{g}_{4})$, for $1\leq i\leq k_{1}$, and vertices
$v_{i}$ in $\overrightarrow{M}(\mathfrak{g}_{1})$ to $v_{i}$ in
$\overrightarrow{M}(\mathfrak{g}_{4})$, for $k_{1}+1\leq i\leq mk_{1}+k_{2}$.
We claim that
$w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{(m+1)k_{1}+k_{2}-i+1,(m+1)k_{1}+k_{2}-j+1}(\mathfrak{g}_{4}))$,
for distinct $i,j\in T_{1}$.

To establish the claim, take distinct $i,j\in T_{1}$. If $v_{i}$ and $v_{j}$
are adjacent in $M(\mathfrak{g}_{1})$, then $\mathcal{S}$ maps
$P_{i,j}(\mathfrak{g}_{1})$ along with its orientation in
$\overrightarrow{M}(\mathfrak{g}_{1})$ to
$P_{(m+1)k_{1}+k_{2}-i+1,(m+1)k_{1}+k_{2}-j+1}(\mathfrak{g}_{4})$ along with
its orientation in $\overrightarrow{M}(\mathfrak{g}_{4})$; that is,
$w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{i,j}(\mathfrak{g}_{4}))$. On the other
hand, if $v_{i}$ and $v_{j}$ are not adjacent in $M(\mathfrak{g}_{1})$, then
$P_{i,j}(\mathfrak{g}_{1})$ contains subpaths of the form
$v_{t_{1}},v_{h_{1}},\ldots,v_{h_{\ell}},v_{t_{2}}$ with $t_{1},t_{2}\in
T_{1}$ and $h_{1},\ldots,h_{\ell}\in H$. To establish the claim, it suffices
to show that the weights of such subpaths $P_{t_{1},t_{2}}(\mathfrak{g}_{1})$
are invariant under $\mathcal{S}.$ Since

$\displaystyle w(P_{t_{1},t_{2}}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{t_{1},h_{1}}(\mathfrak{g}_{1}))+w(P_{h_{1},h_{\ell}}(\mathfrak{g}_{1}))+w(P_{h_{\ell},t_{2}}(\mathfrak{g}_{1}))$
$\displaystyle=-1+w(P_{h_{1},h_{\ell}}(\mathfrak{g}_{1}))+1$
$\displaystyle=w(P_{h_{1},h_{\ell}}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h_{1},h_{\ell}}(\mathfrak{g}_{4}))$
$\displaystyle=w(\mathcal{S}(P_{t_{1},t_{2}}(\mathfrak{g}_{1}))),$

the claim follows.

Now, applying Lemmas 12 and 15, it follows that
$w(P_{i,j}(\mathfrak{g}_{1}))=w(P_{i-k_{1},j-k_{1}}(\mathfrak{g}_{3}))$, for
all distinct $i,j\in T_{1}$. Consequently, our desired weight-preserving
bijection sends the path $P_{i,j}(\mathfrak{g}_{1})$ to
$P_{i-k_{1},j-k_{1}}(\mathfrak{g}_{3})$, for all distinct $i,j\in T_{1}$. As
this argument can be repeated until
$\mathfrak{g}_{3}=\mathfrak{p}^{A}\frac{k_{1}+k_{2}|k_{1}}{2k_{1}+k_{2}}$,
i.e., the case $m=1$, the result follows from Lemma 24. ∎

Having now addressed the top left $a\times a$ block and bottom right $b\times
b$ block of $\Sigma(\mathfrak{g}),$ where $a>b$ and
$\mathfrak{g}=\mathfrak{p}^{A}\frac{a|b}{n}$ is Frobenius, we proceed to the
top right $a\times b$ block of $\Sigma(\mathfrak{g}).$ As was the case with
the bottom right $b\times b$ block, we have two cases: one corresponding to
$\lfloor\frac{a}{b}\rfloor=1$ and another corresponding to
$\lfloor\frac{a}{b}\rfloor>1.$ The former of the two cases is covered by Lemma
29, and the latter by Lemma 32. Examples 28 and 31 illustrate each result
respectively.

######  Example 28.

Let $\mathfrak{g}=\mathfrak{p}^{A}\frac{3|2}{5}$ and
$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{2|1}{3}$. See Figure 9 for (a)
$\Sigma(\mathfrak{g})$ and (b) $M(\mathfrak{g})$, and see Figure 10 for (a)
$\Sigma(\mathfrak{g}^{\prime})$ and (b) $M(\mathfrak{g}^{\prime})$. Note that
the top right $3\times 2$ block (outlined by dashed lines) of
$\Sigma(\mathfrak{g})$ consists of the same multiset of values as the top
$2\times 3$ block of $\Sigma(\mathfrak{g}^{\prime})$ with each value
incremented by 1. Moreover, note that there is a copy of
$M(\mathfrak{g}^{\prime})$ inside of $M(\mathfrak{g})$. In particular,
identifying the vertices connected by dashed edges in Figure 9 (b) and then
reflecting the resulting graph across the horizontal line through the vertices
yields $M(\mathfrak{g}^{\prime})$.

01-121-10-210120320-110(a) $v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$(b) Figure 9:
(a) $\Sigma(\mathfrak{g})$ and (b) $M(\mathfrak{g})$ 01020-11(a)
$v_{1}$$v_{2}$$v_{3}$(b) Figure 10: (a) $\Sigma(\mathfrak{g}^{\prime})$ and
(b) $M(\mathfrak{g}^{\prime})$

######  Lemma 29.

Let $k_{1},k_{2}\in\mathbb{Z}_{>0}$ satisfy $k_{1}>k_{2}$ and
$\gcd(k_{1},k_{2})=1$. If
$\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{k_{1}+k_{2}|k_{1}}{2k_{1}+k_{2}}$ and
$\mathfrak{g}_{2}=\mathfrak{p}^{A}\frac{k_{1}|k_{2}}{k_{1}+k_{2}}$, then the
block corresponding to rows $\\{1,\ldots,k_{1}+k_{2}\\}$ and columns
$\\{k_{1}+k_{2}+1,\ldots,2k_{1}+k_{2}\\}$ of $\Sigma(\mathfrak{g}_{1})$
consists of the same multiset of values of entries as the block corresponding
to rows $\\{1,\ldots,k_{1}\\}$ and columns $\\{1,\ldots,k_{1}+k_{2}\\}$ of
$\Sigma(\mathfrak{g}_{2})$ with each value incremented by 1.

###### Proof.

Define the sets

$H_{1}=\\{1,\ldots,k_{1}+k_{2}\\},\quad
H_{2}=\\{1,\ldots,k_{1}\\},\quad\text{and}\quad
T=\\{k_{1}+k_{2}+1,\ldots,2k_{1}+k_{2}\\}.$

Note that $\\{v_{i}~{}|~{}i\in H_{1}\\}$ and $\\{v_{i}~{}|~{}i\in T\\}$ form a
partition of the vertices of $\overrightarrow{M}(\mathfrak{g}_{1})$. Also,
note that $H_{1}$, $H_{2}$, and $T$ are defined in such a way that top edges
in $M(\mathfrak{g}_{1})$ only connect pairs of vertices $u$ and $v$ with

  * •

$u,v\in\\{v_{i}~{}|~{}i\in H_{1}\\}$ or

  * •

$u,v\in\\{v_{i}~{}|~{}i\in T\\}$

and bottom edges in $M(\mathfrak{g}_{1})$ only connect pairs of vertices $u$
and $v$ with

  * •

$u\in\\{v_{i}~{}|~{}i\in H_{2}\\}$ and $v\in\\{v_{i}~{}|~{}i\in T\\}$ or

  * •

$u,v\in\\{v_{i}~{}|~{}i\in H_{1}\backslash H_{2}\\}$.

Now, the block of $\Sigma(\mathfrak{g}_{1})$ corresponding to rows $r\in
H_{1}$ and columns $c\in T$ consists of the values
$w(P_{i,j}(\mathfrak{g}_{1}))$, for $i\in H_{1}$ and $j\in T$, and the block
of $\Sigma(\mathfrak{g}_{2})$ corresponding to rows $r\in H_{2}$ and columns
$c\in H_{1}$ consists of the values $w(P_{i,j}(\mathfrak{g}_{2}))$, for $i\in
H_{2}$ and $j\in H_{1}$. To establish the result, we define a bijection

$\phi:\\{P_{i,j}(\mathfrak{g}_{1})~{}|~{}i\in H_{1},~{}j\in
T\\}\to\\{P_{i,j}(\mathfrak{g}_{2})~{}|~{}i\in H_{2},~{}j\in H_{1}\\}$

such that $w(P_{i,j}(\mathfrak{g}_{1}))=w(\phi(P_{i,j}(\mathfrak{g}_{1})))+1$.
To aid in defining $\phi$, we make use of a transformation $\mathcal{S}$ of
$\overrightarrow{M}(\mathfrak{g}_{1})$ which is defined as follows. If
$u\in\\{v_{i}~{}|~{}i\in H_{2}\\}$ is adjacent to $v\in\\{v_{i}~{}|~{}i\in
T\\}$, then identify $u$ and $v$ while removing edges between them. Let
$\mathfrak{g}_{3}=\mathfrak{p}^{A}\frac{k_{1}+k_{2}}{k_{1}|k_{2}}$. Note that
$\mathcal{S}(\overrightarrow{M}(\mathfrak{g}_{1}))=\overrightarrow{M}(\mathfrak{g}_{3})$
with $\mathcal{S}$ mapping vertices $v_{i}$ and $v_{2k_{1}+k_{2}-i+1}$ in
$\overrightarrow{M}(\mathfrak{g}_{1})$ to $v_{i}$ in
$\overrightarrow{M}(\mathfrak{g}_{3})$, for $i\in H_{2}$, and vertices $v_{i}$
in $\overrightarrow{M}(\mathfrak{g}_{1})$ to $v_{i}$ in
$\overrightarrow{M}(\mathfrak{g}_{3})$, for $i\in H_{1}\backslash H_{2}$. We
claim that
$w(P_{h,t}(\mathfrak{g}_{1}))=w(\mathcal{S}(P_{h,t}(\mathfrak{g}_{1})))+1$,
for all $h\in H_{1}$ and $t\in T$.

To establish the claim, take $h\in H_{1}$ and $t\in T$. There must exist a
unique $h^{\prime}\in H_{1}$ such that $(v_{h^{\prime}},v_{t})$ is an edge of
$\overrightarrow{M}(\mathfrak{g}_{1})$ and
$\mathcal{S}(P_{h,t}(\mathfrak{g}_{1}))=P_{h,h^{\prime}}(\mathfrak{g}_{3})$.
Recall that, in the proof of Lemma 21, it is shown that
$w(P_{h,h^{\prime}}(\mathfrak{g}_{1}))=w(P_{h,h^{\prime}}(\mathfrak{g}_{3}))$.
Now, if $(v_{h^{\prime}},v_{t})$ belongs to $P_{h,t}(\mathfrak{g}_{1})$, then

$\displaystyle w(P_{h,t}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h,h^{\prime}}(\mathfrak{g}_{1}))+w(P_{h^{\prime},t}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h,h^{\prime}}(\mathfrak{g}_{1}))+1$
$\displaystyle=w(P_{h,h^{\prime}}(\mathfrak{g}_{3}))+1.$

Otherwise, $P_{h,h^{\prime}}(\mathfrak{g}_{1})$ consists of
$P_{h,t}(\mathfrak{g}_{1})$ along with the edge $(v_{h^{\prime}},v_{t})$.
Consequently,

$\displaystyle w(P_{h,t}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h,h^{\prime}}(\mathfrak{g}_{1}))-w(P_{t,h^{\prime}}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h,h^{\prime}}(\mathfrak{g}_{1}))+1$
$\displaystyle=w(P_{h,h^{\prime}}(\mathfrak{g}_{3}))+1.$

Thus, in either case,

$w(P_{h,t}(\mathfrak{g}_{1}))=w(\mathcal{S}(P_{h,t}(\mathfrak{g}_{1})))+1=w(P_{h,h^{\prime}}(\mathfrak{g}_{3}))+1,$

establishing the claim.

Now, applying Lemma 12, we have
$w(P_{h,t}(\mathfrak{g}_{1}))=w(P_{h^{\prime},h}(\mathfrak{g}_{2}))+1$.
Therefore, the desired bijection $\phi$ is given by mapping the path
$P_{i,j}(\mathfrak{g}_{1})$ to $P_{2k_{1}+k_{2}-j+1,i}(\mathfrak{g}_{2}),$ for
all $i\in H_{1}$ and $j\in T.$ The result follows. ∎

######  Remark 30.

In fact, for $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ as in Lemma 29, the
block corresponding to rows $\\{1,\ldots,k_{1}+k_{2}\\}$ and columns
$\\{k_{1}+k_{2}+1,\ldots,2k_{1}+k_{2}\\}$ of $\Sigma(\mathfrak{g}_{1})$ is the
same as the block corresponding to rows $\\{1,\ldots,k_{1}\\}$ and columns
$\\{1,\ldots,k_{1}+k_{2}\\}$ of $\Sigma(\mathfrak{g}_{2})$ with every value
incremented by 1 and rotated 90 degrees clockwise.

######  Example 31.

Let $\mathfrak{g}=\mathfrak{p}^{A}\frac{5|2}{7}$ and
$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{3|2}{5}$. See Figure 11 for (a)
$\Sigma(\mathfrak{g})$ and (b) $M(\mathfrak{g})$, and see Figure 12 for (a)
$\Sigma(\mathfrak{g}^{\prime})$ and (b) $M(\mathfrak{g}^{\prime})$. Note that
the top right $5\times 2$ block (outlined by dashed lines) of
$\Sigma(\mathfrak{g})$ consists of the same multiset of values as the
rightmost $5\times 2$ block of $\Sigma(\mathfrak{g}^{\prime})$ with each value
incremented by 1. Moreover, note that there is a copy of
$M(\mathfrak{g}^{\prime})$ inside of $M(\mathfrak{g})$. In particular,
identifying the vertices connected by dashed edges in Figure 11 (b) and then
rotating the resulting graph by 180 degrees yields $M(\mathfrak{g}^{\prime})$.

01-20-121-10-3-1-210230214301-20-12112-110320-110(a)
$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$v_{7}$(b) Figure 11: (a)
$\Sigma(\mathfrak{g})$ and (b) $M(\mathfrak{g})$ 01-121-10-210120320-110(a)
$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$(b) Figure 12: (a)
$\Sigma(\mathfrak{g}^{\prime})$ and (b) $M(\mathfrak{g}^{\prime})$

######  Lemma 32.

Let $k_{1},k_{2}\in\mathbb{Z}_{>0}$ satisfy $k_{1}>k_{2}$ and
$\gcd(k_{1},k_{2})=1$. If
$\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{mk_{1}+k_{2}|k_{1}}{(m+1)k_{1}+k_{2}}$
and
$\mathfrak{g}_{2}=\mathfrak{p}^{A}\frac{(m-1)k_{1}+k_{2}|k_{1}}{mk_{1}+k_{2}}$,
then the block corresponding to rows $\\{1,\ldots,mk_{1}+k_{2}\\}$ and columns
$\\{mk_{1}+k_{2}+1,\ldots,(m+1)k_{1}+k_{2}\\}$ of $\Sigma(\mathfrak{g}_{1})$
consists of the same values as the block corresponding to rows
$\\{1,\ldots,mk_{1}+k_{2}\\}$ and columns
$\\{(m-1)k_{1}+k_{2}+1,\ldots,mk_{1}+k_{2}\\}$ of $\Sigma(\mathfrak{g}_{2})$
with each value incremented by 1.

###### Proof.

Define the sets

$H_{1}=\\{1,\ldots,k_{1}\\},\quad
H_{2}=\\{k_{1}+1,\ldots,mk_{1}+k_{2}\\},\quad
T_{1}=\\{mk_{1}+k_{2}+1,\ldots,(m+1)k_{1}+k_{2}\\},$

and

$T_{2}=\\{(m-1)k_{1}+k_{2}+1,\ldots,mk_{1}+k_{2}\\}.$

Note that $\\{v_{i}~{}|~{}i\in H_{1}\\},\\{v_{i}~{}|~{}i\in H_{2}\\},$ and
$\\{v_{i}~{}|~{}i\in T_{1}\\}$ form a partition of the vertices of
$M(\mathfrak{g}_{1})$ and are defined in such a way that top edges in
$M(\mathfrak{g}_{1})$ only connect pairs of vertices $u$ and $v$ with

  * •

$u,v\in\\{v_{i}~{}|~{}i\in H_{1}\cup H_{2}\\}$ or

  * •

$u,v\in\\{v_{i}~{}|~{}i\in T_{1}\\}$

and bottom edges in $M(\mathfrak{g}_{1})$ only connect pairs of vertices $u$
and $v$ with

  * •

$u\in\\{v_{i}~{}|~{}i\in H_{1}\\}$ and $v\in\\{v_{i}~{}|~{}i\in T_{1}\\}$ or

  * •

$u,v\in\\{v_{i}~{}|~{}i\in H_{2}\\}$.

Now, the block of $\Sigma(\mathfrak{g}_{1})$ corresponding to rows $r\in
H_{1}\cup H_{2}$ and columns $c\in T_{1}$ consists of the values
$w(P_{i,j}(\mathfrak{g}_{1}))$, for $i\in H_{1}\cup H_{2}$ and $j\in T_{1}$,
and the block of $\Sigma(\mathfrak{g}_{2})$ corresponding to rows $r\in
H_{1}\cup H_{2}$ and columns $c\in T_{2}$ consists of the values
$w(P_{i,j}(\mathfrak{g}_{2}))$, for $i\in H_{1}\cup H_{2}$ and $j\in T_{2}$.
To establish the result, we define a bijection

$\phi:\\{P_{i,j}(\mathfrak{g}_{1})~{}|~{}i\in H_{1}\cup H_{2},~{}j\in
T_{1}\\}\to\\{P_{i,j}(\mathfrak{g}_{2})~{}|~{}i\in H_{1}\cup H_{2},~{}j\in
T_{2}\\}$

such that $w(P_{i,j}(\mathfrak{g}_{1}))=w(\phi(P_{i,j}(\mathfrak{g}_{1})))+1$.
To aid in defining $\phi$, we make use of a transformation $\mathcal{S}$ of
$\overrightarrow{M}(\mathfrak{g}_{1})$ which is defined as follows. If
$u\in\\{v_{i}~{}|~{}i\in H_{1}\\}$ is adjacent to $v\in\\{v_{i}~{}|~{}i\in
T_{1}\\}$, then identify $u$ and $v$ while removing edges between them. Let
$\mathfrak{g}_{3}=\mathfrak{p}^{A}\frac{mk_{1}+k_{2}}{k_{1}|(m-1)k_{1}+k_{2}}$.
Note that
$\mathcal{S}(\overrightarrow{M}(\mathfrak{g}_{1}))=\overrightarrow{M}(\mathfrak{g}_{3})$
with $\mathcal{S}$ mapping vertices $v_{i}$ and $v_{(m+1)k_{1}+k_{2}-i+1}$ in
$\overrightarrow{M}(\mathfrak{g}_{1})$ to $v_{i}$ in
$\overrightarrow{M}(\mathfrak{g}_{3})$, for $i\in H_{1}$, and vertices $v_{i}$
in $\overrightarrow{M}(\mathfrak{g}_{1})$ to $v_{i}$ in
$\overrightarrow{M}(\mathfrak{g}_{3})$, for $i\in H_{2}$. We claim that
$w(P_{h,t}(\mathfrak{g}_{1}))=w(\mathcal{S}(P_{h,t}(\mathfrak{g}_{1})))+1$,
for $h\in H_{1}\cup H_{2}$ and $t\in T_{1}$.

To establish the claim, take $h\in H_{1}\cup H_{2}$ and $t\in T_{1}$. There
must exist a unique $h^{\prime}\in H_{1}$ such that $(v_{h^{\prime}},v_{t})$
is an edge of $\overrightarrow{M}(\mathfrak{g}_{1})$ and
$\mathcal{S}(P_{h,t}(\mathfrak{g}_{1}))=P_{h,h^{\prime}}(\mathfrak{g}_{3})$.
Note that in the proof of Lemma 21 it is shown that
$w(P_{h,h^{\prime}}(\mathfrak{g}_{1}))=w(P_{h,h^{\prime}}(\mathfrak{g}_{3}))$.
Now, if $(v_{h^{\prime}},v_{t})$ belongs to $P_{h,t}(\mathfrak{g}_{1})$, then

$\displaystyle w(P_{h,t}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h,h^{\prime}}(\mathfrak{g}_{1}))+w(P_{h^{\prime},t}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h,h^{\prime}}(\mathfrak{g}_{1}))+1$
$\displaystyle=w(P_{h,h^{\prime}}(\mathfrak{g}_{3}))+1.$

Otherwise, $P_{h,h^{\prime}}(\mathfrak{g}_{1})$ consists of
$P_{h,t}(\mathfrak{g}_{1})$ along with the edge $(v_{h^{\prime}},v_{t})$.
Consequently,

$\displaystyle w(P_{h,t}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h,h^{\prime}}(\mathfrak{g}_{1}))-w(P_{t,h^{\prime}}(\mathfrak{g}_{1}))$
$\displaystyle=w(P_{h,h^{\prime}}(\mathfrak{g}_{1}))+1$
$\displaystyle=w(P_{h,h^{\prime}}(\mathfrak{g}_{3}))+1.$

Thus, in either case,

$w(P_{h,t}(\mathfrak{g}_{1}))=w(\mathcal{S}(P_{h,t}(\mathfrak{g}_{1})))+1=w(P_{h,h^{\prime}}(\mathfrak{g}_{3}))+1,$

establishing the claim.

Now, applying Lemmas 12 and 15, we have

$w(P_{h,t}(\mathfrak{g}_{1}))=w(P_{mk_{1}+k_{2}-h+1,mk_{1}+k_{2}-h^{\prime}+1}(\mathfrak{g}_{2}))+1.$

Therefore, the desired bijection $\phi$ is given by mapping the path
$P_{i,j}(\mathfrak{g}_{1})$ to
$P_{mk_{1}+k_{2}-i+1,j-k_{1}}(\mathfrak{g}_{2}),$ for all $i\in H_{1}\cup
H_{2}$ and $j\in T_{1}.$ The result follows. ∎

In the following subsections, we utilize the structural lemmas above to
inductively determine closed formulas for the spectra of certain families of
Frobenius, maximal parabolic, type-A seaweeds.

###  3.1 $\mathfrak{p}^{A}\frac{k|1}{k+1}$

In this subsection, we compute the (extended) spectra of Frobenius, type-A
seaweed algebras of the form $\mathfrak{p}^{A}\frac{k|1}{k+1}$, for $k\geq 1$.
See Figure 13 for illustrations of the oriented meanders corresponding to
$\mathfrak{p}^{A}\frac{k|1}{k+1}$ for $k=1,2,$ and 3.

$v_{1}$$v_{2}$(a)

$v_{1}$$v_{2}$$v_{3}$(b)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$(c)

Figure 13: Oriented meanders of (a) $\mathfrak{p}^{A}\frac{1|1}{2}$, (b)
$\mathfrak{p}^{A}\frac{2|1}{3}$, and (c) $\mathfrak{p}^{A}\frac{3|1}{4}$

To establish our result, we require the following lemma which determines the
multiset of values contained in the rightmost $(k+1)\times 1$ block of
$\Sigma(\mathfrak{g})$, for $\mathfrak{g}=\mathfrak{p}^{A}\frac{k|1}{k+1}$.

######  Lemma 33.

Let $\mathfrak{g}_{k}=\mathfrak{p}^{A}\frac{k|1}{k+1}$, for $k\geq 1$. Then

$\\{w(P_{i,k+1}(\mathfrak{g}_{k}))~{}|~{}1\leq i\leq
k+1\\}=\\{0,1,\ldots,k\\}$

as multisets.

###### Proof.

By induction on $k$. For $k=1$, using Figure 13 (a) we compute that

$\\{w(P_{i,k+1}(\mathfrak{g}_{k}))~{}|~{}1\leq i\leq
k+1\\}=\\{w(P_{1,2}(\mathfrak{g}_{1})),w(P_{2,2}(\mathfrak{g}_{1}))\\}=\\{0,1\\}.$

Assume the result holds for $k-1\geq 1$. Evidently,
$w(P_{k+1,k+1}(\mathfrak{g}_{k}))=0$. Note that the path
$P_{i,k+1}(\mathfrak{g}_{1})$ in $\overrightarrow{M}(\mathfrak{g}_{k})$, for
$1\leq i<k+1$, must contain the edge $(v_{1},v_{k+1})$. Thus,

$w(P_{i,k+1}(\mathfrak{g}_{k}))=w(P_{i,1}(\mathfrak{g}_{k}))+w(P_{1,k+1}(\mathfrak{g}_{k}))=w(P_{i,1}(\mathfrak{g}_{k}))+1.$

Consequently, it suffices to compute the multiset

$\\{w(P_{i,1}(\mathfrak{g}_{k}))~{}|~{}1\leq i<k+1\\}.$

Let $\mathfrak{g}=\mathfrak{p}^{A}\frac{k}{1|k-1}$. Note that removing the
edge $(v_{1},v_{k+1})$ and vertex $v_{k+1}$ from
$\overrightarrow{M}(\mathfrak{g}_{k})$ results in
$\overrightarrow{M}(\mathfrak{g})$. Therefore, since no path
$P_{i,1}(\mathfrak{g}_{k})$, for $1\leq i<k+1$, contains the edge
$(v_{1},v_{k}+1)$, it follows that

$\\{w(P_{i,1}(\mathfrak{g}_{k}))~{}|~{}1\leq
i<k+1\\}=\\{w(P_{i,1}(\mathfrak{g}))~{}|~{}1\leq i<k+1\\}.$

Applying Lemmas 12 and 15 along with our induction hypothesis, we find that

$\\{w(P_{i,1}(\mathfrak{g}))~{}|~{}1\leq
i<k+1\\}=\\{w(P_{k-i+1,k}(\mathfrak{g}_{k-1}))~{}|~{}1\leq
i<k+1\\}=\\{0,1,\ldots,k-1\\};$

that is,

$\\{w(P_{i,k+1}(\mathfrak{g}_{k}))~{}|~{}1\leq i<k+1\\}=\\{1,2,\ldots,k\\}.$

The result follows. ∎

######  Theorem 34.

Let $\mathfrak{g}_{k}=\mathfrak{p}^{A}\frac{k|1}{k+1}$, for $k\geq 1$. The
spectrum of $\mathfrak{g}_{k}$ is equal to

$\bigcup_{i=1}^{k}\left\\{(-k+i)^{i},(k-i+1)^{i}\right\\}$

and the extended spectrum is equal to

$\\{0^{k}\\}\cup\bigcup_{i=0}^{k-1}\left\\{(-k+i)^{i+1},(k-i)^{i+1}\right\\}.$

###### Proof.

By induction on $k.$ Let $\mathfrak{g}_{k}=\mathfrak{p}^{A}\frac{k|1}{k+1}$.
For the base case, constructing the (extended) spectrum matrix of the seaweed
algebra $\mathfrak{g}_{1}=\mathfrak{p}^{A}\frac{1|1}{2}$ directly from
$\overrightarrow{M}(\mathfrak{g}_{1})$ (see Figure 13 (a)), we find

$\Sigma(\mathfrak{g}_{1})=\begin{bmatrix}0&1\\\
&0\end{bmatrix}\quad\quad\text{and}\quad\quad\widehat{\Sigma}(\mathfrak{g}_{1})=\begin{bmatrix}0&1\\\
-1&0\end{bmatrix}.$

Thus, the spectrum of $\mathfrak{g}_{1}$ is equal to $\\{0,1\\}$ and the
extended spectrum is equal to $\\{-1,0,1\\}$.

Now, assume the result holds for $k-1\geq 1$. Note that

$\Sigma(\mathfrak{g}_{k})=\begin{bmatrix}B_{1}&B_{2}\\\ &0\end{bmatrix},$

where

  * •

$B_{1}$ is the $k\times k$ block consisting of the values
$w(P_{i,j}(\mathfrak{g}_{k}))$, for $1\leq i,j\leq k$, and

  * •

$B_{2}$ is the $k\times 1$ block consisting of the values
$w(P_{i,k+1}(\mathfrak{g}_{k}))$, for $1\leq i\leq k$.

Consequently, considering Remark 11, it follows that

$\widehat{\Sigma}(\mathfrak{g}_{k})=\begin{bmatrix}B_{1}&B_{2}\\\
-B_{2}^{t}&0\end{bmatrix}.$

Now, by Lemma 21, $B_{1}$ contains the same multiset of values as
$\widehat{\Sigma}(\mathfrak{g}_{k-1})$. Thus, applying our inductive
hypothesis – and keeping Remark 7 in mind – it follows that $B_{1}$
contributes

$\\{0^{k}\\}\cup\bigcup_{i=1}^{k-1}\\{(-k+i)^{i},(k-i)^{i}\\}$

to the multiset of values contained in the (extended) spectrum matrix of
$\mathfrak{g}_{k}$. It then follows from Lemma 33 that $B_{2}$ contributes
$\\{1,\ldots,k\\}$ to the multiset of values contained in the (extended)
spectrum of $\mathfrak{g}_{k}$ and $-B_{2}^{t}$ contributes
$\\{-k,\ldots,-1\\}$ to the multiset of values contained in the extended
spectrum of $\mathfrak{g}_{k}$. Putting these contributions together, the
result follows. ∎

Considering the spectrum formula of Theorem 34, we are immediately led to the
following.

######  Corollary 35.

For $k\geq 1$, if $\mathfrak{g}=\mathfrak{p}^{A}\frac{k|1}{k+1}$, then
$\mathfrak{g}$ has the log-concave spectrum property.

######  Remark 36.

Combining Theorem 34 and Corollary 35 with Corollaries 14, 17, and 18, we
obtain similar results to Theorem 34 and Corollary 35 for related families of
Frobenius, type-A seaweeds. In particular, the (extended) spectrum of each of
$\mathfrak{p}^{A}\frac{k+1}{k|1}$, $\mathfrak{p}^{A}\frac{1|k}{k+1}$, and
$\mathfrak{p}^{A}\frac{k+1}{1|k},$ for $k\geq 1,$ is given in Theorem 34, and
each algebra possesses the log-concave spectrum property.

###  3.2 $\mathfrak{p}^{A}\frac{k|2}{k+2}$

In this section, we consider the (extended) spectra of Frobenius, type-A
seaweed algebras of the form $\mathfrak{p}^{A}\frac{k|2}{k+2}$, for $k>1$ odd.
See Figure 14 for illustrations of the oriented meanders of
$\mathfrak{p}^{A}\frac{k|2}{k+2}$, for $k=3$ and 5.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$(a)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{5}$$v_{5}$(b)

Figure 14: Oriented meanders of (a) $\mathfrak{p}^{A}\frac{3|2}{5}$ and (b)
$\mathfrak{p}^{A}\frac{5|2}{7}$

Analogous to the case of type-A seaweeds of the form
$\mathfrak{p}^{A}\frac{k|1}{k+1}$, we require the following lemma, which
determines the multiset of values contained in the top right $(k+2)\times 2$
block of $\Sigma(\mathfrak{g})$, for
$\mathfrak{g}=\mathfrak{p}^{A}\frac{k|2}{k+2}$.

######  Lemma 37.

Let $\mathfrak{g}_{k}=\mathfrak{p}^{A}\frac{k|2}{k+2}$ for $k=2m-1>5$. Then
the multiset of values contained in the block of $\Sigma(\mathfrak{g}_{k})$
corresponding to rows $\\{1,\ldots,k\\}$ and columns $\\{k+1,k+2\\}$ is equal
to

$\\{0,1^{3},(m-1)^{3},m^{2},m+1\\}\cup\bigcup_{i=2}^{m-2}\\{i^{4}\\}.$

Moreover, the multiset of values contained in the block of
$\Sigma(\mathfrak{g}_{k})$ corresponding to rows $\\{1,\ldots,k+2\\}$ and
columns $\\{k+1,k+2\\}$ is equal to

$\\{-1,0^{3},(m-1)^{3},m^{2},m+1\\}\cup\bigcup_{i=1}^{m-2}\\{i^{4}\\}.$

###### Proof.

By induction on $m$. For $m=4$ (or $k=7$) one can compute directly that the
multiset of values of $\Sigma(\mathfrak{g}_{7})$ contained in rows
$\\{1,2,3,4,5,6,7\\}$ and columns $\\{8,9\\}$ is

$\\{0,1^{3},2^{4},3^{3},4^{2},5\\}$

and the multiset of values of $\Sigma(\mathfrak{g}_{7})$ contained in rows
$\\{1,2,3,4,5,6,7,8,9\\}$ and columns $\\{8,9\\}$ is

$\\{-1,0^{3},1^{4},2^{4},3^{3},4^{2},5\\}.$

Now, assume the result holds for $m-1\geq 4$ (or $k-2=2(m-1)-1=2m-3\geq 7$).
By Lemma 32, the multiset of values of $\Sigma(\mathfrak{g}_{k})$ contained in
rows $\\{1,\ldots,k\\}$ and columns $\\{k+1,k+2\\}$ is equal to the multiset
of values of $\Sigma(\mathfrak{g}_{k-2})$ contained in rows $\\{1,\ldots,k\\}$
and columns $\\{k-1,k\\}$ with all values incremented by 1; that is, by the
inductive hypothesis, the multiset of values of $\Sigma(\mathfrak{g}_{k})$
contained in rows $\\{1,\ldots,k\\}$ and columns $\\{k+1,k+2\\}$ is equal to

$\displaystyle M$
$\displaystyle=\\{-1+1,(0+1)^{3},(m-2+1)^{3},(m-1+1)^{2},m+1\\}\cup\bigcup_{i=1}^{m-3}\\{(i+1)^{4}\\}$
$\displaystyle=\\{0,1^{3},(m-1)^{3},m^{2},m+1\\}\cup\bigcup_{i=2}^{m-2}\\{i^{4}\\},$

as desired. As for the multiset of values contained in rows
$\\{1,\ldots,k+2\\}$ and columns $\\{k+1,k+2\\}$ of
$\Sigma(\mathfrak{g}_{k})$, we simply need to add the values contained in rows
$\\{k+1,k+2\\}$ and columns $\\{k+1,k+2\\}$ to $M$. By Lemma 27, the multiset
of values contained in rows $\\{k+1,k+2\\}$ and columns $\\{k+1,k+2\\}$ of
$\Sigma(\mathfrak{g}_{k})$ is equal to the extended spectrum of
$\mathfrak{p}^{A}\frac{1|1}{2}$, i.e., $\\{-1,0^{2},1\\}$. The result follows
by induction. ∎

######  Theorem 38.

Let $\mathfrak{g}_{k}=\mathfrak{p}^{A}\frac{k|2}{k+2}$, for $k=2m-1>1$. Then
the spectrum of $\mathfrak{g}_{k}$ is equal to
$\\{-2,-1^{3},0^{5},1^{5},2^{3},3\\}$ if $m=2$ (or $k=3$) and is equal to

$\\{-m,(-m+1)^{3},0^{2k-1},1^{2k-1},m^{3},m+1\\}\cup\bigcup_{i=2}^{m-1}\\{(-m+i)^{4i-2},(m-i+1)^{4i-2}\\},$

for $m>2$ (or $k>3$). Furthermore, the extended spectrum of $\mathfrak{g}_{k}$
is equal to $\\{-3,-2^{3},-1^{5},0^{6},1^{5},2^{3},3\\}$ if $m=2$ (or $k=3$)
and is equal to

$\\{-m-1,-m^{3},-1^{2k-1},0^{2k},1^{2k-1},m^{3},m+1\\}\cup\bigcup_{i=1}^{m-2}\\{(-m+i)^{4i+2},(m-i)^{4i+2}\\},$

for $m>2$ (or $k>3$).

###### Proof.

By induction on $m$. For $m=2$ (or $k=3$), computing directly we find that the
spectrum of $\mathfrak{g}_{3}$ is equal to

$\\{-2,-1^{3},0^{5},1^{5},2^{3},3\\},$

while the extended spectrum is equal to

$\\{-3,-2^{3},-1^{5},0^{6},1^{5},2^{3},3\\}.$

Similarly, for $m=3$ (or $k=5$), we find that the spectrum of
$\mathfrak{g}_{5}$ is equal to

$\\{-3,-2^{3},-1^{6},0^{9},1^{9},2^{6},3^{3},4\\},$

while the extended spectrum is equal to

$\\{-4,-3^{3},-2^{6},-1^{9},0^{10},1^{9},2^{6},3^{3},4\\}.$

Now, assume the result holds for $m-1\geq 3$ (or $k-2=2(m-1)-1=2m-3\geq 5$).
Note that

$\Sigma(\mathfrak{g}_{k})=\begin{bmatrix}B_{1}&B_{2}\\\ &B_{3}\end{bmatrix},$

where

  * •

$B_{1}$ is the $k\times k$ block consisting of the values
$w(P_{i,j}(\mathfrak{g}_{k}))$, for $1\leq i,j\leq k$,

  * •

$B_{2}$ is the $k\times 2$ block consisting of the values
$w(P_{i,k+1}(\mathfrak{g}_{k}))$ and $w(P_{i,k+2}(\mathfrak{g}_{k}))$, for
$1\leq i\leq k$,

  * •

and $B_{3}$ is the $2\times 2$ block consisting of the values
$w(P_{k+1,k+1}(\mathfrak{g}_{k}))$, $w(P_{k+1,k+2}(\mathfrak{g}_{k}))$,
$w(P_{k+2,k+1}(\mathfrak{g}_{k}))$, and $w(P_{k+2,k+2}(\mathfrak{g}_{k}))$.

Consequently, considering Remark 11, it follows that

$\widehat{\Sigma}(\mathfrak{g}_{k})=\begin{bmatrix}B_{1}&B_{2}\\\
-B_{2}^{t}&B_{3}\end{bmatrix}.$

Now, by Lemma 21, $B_{1}$ contains the same multiset of values as
$\widehat{\Sigma}(\mathfrak{g}_{k-2})$. Thus, applying our induction
hypothesis – and keeping Remark 7 in mind – it follows that $B_{1}$
contributes

$\\{-m,(-m+1)^{3},-1^{2k-5},0^{2k-3},1^{2k-5},(m-1)^{3},m\\}\cup\bigcup_{i=1}^{m-3}\\{(-m+i+1)^{4i+2},(m-i-1)^{4i+2}\\}$
(1)

to the multiset of values contained in the (extended) spectrum matrix of
$\mathfrak{g}_{k}$. It then follows from Lemma 37 that $B_{2}$ and $B_{3}$
together contribute

$\\{-1,0^{3},(m-1)^{3},m^{2},m+1\\}\cup\bigcup_{i=1}^{m-2}\\{i^{4}\\}$ (2)

to the multiset of values contained in the (extended) spectrum matrix of
$\mathfrak{g}_{k}$. Thus, combining the contributions (1) and (2), the
multiset of values contained in $\Sigma(\mathfrak{g}_{k})$ is equal to

$\\{-m,(-m+1)^{3},0^{2k},1^{2k-1},m^{3},m+1\\}\cup\bigcup_{i=2}^{m-1}\\{(-m+i)^{4i-2},(m-i+1)^{4i-2}\\}.$

Considering Remark 7, we have established the claimed form of the spectrum of
$\mathfrak{g}_{k}$.

As for the extended spectrum, it remains to add the multiset of values
contained in $-B_{2}^{t}$ to the spectrum of $\mathfrak{g}_{k}$. Applying
Lemma 37, it follows that $-B_{2}^{t}$ contains the multiset of values

$\\{-m-1,-m^{2},(-m+1)^{3},-1^{3},0\\}\cup\bigcup_{i=2}^{m-2}\\{-i^{4}\\};$

that is, the extended spectrum of $\mathfrak{g}_{k}$ is equal to

$\\{-m-1,-m^{3},-1^{2k-1},0^{2k},1^{2k-1},m^{3},m+1\\}\cup\bigcup_{i=1}^{m-2}\\{(-m+i)^{4i+2},(m-i)^{4i+2}\\},$

as desired. The result follows by induction. ∎

Considering the spectrum formula of Theorem 38, we are immediately led to the
following.

######  Corollary 39.

For $k\geq 1$ odd, if $\mathfrak{g}=\mathfrak{p}^{A}\frac{k|2}{k+2}$, then
$\mathfrak{g}$ has the log-concave spectrum property.

######  Remark 40.

Combining Theorem 38 and Corollary 39 with Corollaries 14, 17, and 18, we
obtain similar results to Theorem 38 and Corollary 39 for related families of
Frobenius, type-A seaweeds. In particular, the (extended) spectrum of each of
$\mathfrak{p}^{A}\frac{k+2}{k|2}$, $\mathfrak{p}^{A}\frac{2|k}{k+2}$, and
$\mathfrak{p}^{A}\frac{k+2}{2|k}$, for $k\geq 1$ odd, is given in Theorem 38,
and each algebra possesses the log-concave spectrum property.

As displayed by Theorems 34 and 38, the structural lemmas established at the
beginning of this section allow for the inductive determination of formulas
for the spectra of Frobenius, maximal parabolic, type-A seaweeds
$\mathfrak{p}^{A}\frac{a|b}{n}$ with $a>b$ for fixed $b$. Unfortunately, each
value of $b$ seems to require its own supporting lemma analogous to Lemma 37;
that is, it does not seem possible to extend the inductive procedure above to
obtain explicit formulae – or establish log-concavity or unimodality – for the
spectra of all Frobenius, maximal parabolic, type-A seaweeds. Consequently, we
do not proceed by finding the spectra of Frobenius, type-A seaweeds of the
form $\mathfrak{p}^{A}\frac{k|3}{k+3}$.

Instead, to finish this section, we use Theorems 34 and 38 to determine the
spectra of Frobenius, type-A seaweeds of the form
$\mathfrak{p}^{A}\frac{a|b}{n}=\mathfrak{p}^{A}\frac{k+1|k}{2k+1}$ and
$\mathfrak{p}^{A}\frac{k+2|k}{2k+2}$, where, unlike in our previous examples,
neither $a$ nor $b$ is fixed.

###  3.3 $\mathfrak{p}^{A}\frac{k+1|k}{2k+1}$

In this subsection, we compute the spectra of Frobenius, type-A seaweed
algebras of the form $\mathfrak{p}^{A}\frac{k+1|k}{2k+1}$, for $k\geq 1$. See
Figure 15 for illustrations of the oriented meanders corresponding to
$\mathfrak{p}^{A}\frac{k+1|k}{2k+1}$ for $k=1,2,$ and 3.

$v_{1}$$v_{2}$$v_{3}$(a)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$(b)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{5}$$v_{5}$(c)

Figure 15: Oriented meanders of (a) $\mathfrak{p}^{A}\frac{2|1}{3}$, (b)
$\mathfrak{p}^{A}\frac{3|2}{5}$, and (c) $\mathfrak{p}^{A}\frac{4|3}{7}$

######  Remark 41.

In the preceding two sections, to compute the spectra of the algebras of
interest it was necessary to also compute the extended spectra. This is not
true of the seaweeds considered for the remainder of this paper, so we omit
discussion about extended spectra ongoing.

######  Theorem 42.

Let $\mathfrak{g}_{k}=\mathfrak{p}^{A}\frac{k+1|k}{2k+1}$, for $k\geq 1$. The
spectrum of $\mathfrak{g}_{k}$ is equal to $\\{-1,0^{2},1^{2},2\\}$ if $k=1$
and is equal to

$\\{-k,0^{3k-1},1^{3k-1},k+1\\}\cup\bigcup_{i=1}^{k-1}\\{(-k+i)^{3i},(k-i+1)^{3i}\\},$

for $k>1$.

###### Proof.

By induction on $k$. The cases $k=1$ and 2 were considered in Theorems 34 and
38, respectively. So, assume the result holds for $k-1\geq 2$. Note that

$\Sigma(\mathfrak{g}_{k})=\begin{bmatrix}B_{1}&B_{2}\\\ &B_{3}\end{bmatrix},$

where

  * •

$B_{1}$ is the $(k+1)\times(k+1)$ block consisting of the values
$w(P_{i,j}(\mathfrak{g}_{k}))$, for $1\leq i,j\leq k+1$,

  * •

$B_{2}$ is the $(k+1)\times k$ block consisting of the values
$w(P_{i,j}(\mathfrak{g}_{k}))$, for $1\leq i\leq k+1$ and $k+2\leq j\leq
2k+1$, and

  * •

$B_{3}$ is the $k\times k$ block consisting of the values
$w(P_{i,j}(\mathfrak{g}_{k}))$, for $k+2\leq i,j\leq 2k+1$.

Now, if $\mathfrak{p}_{k}=\mathfrak{p}^{A}\frac{k|1}{k+1}$, then by Lemma 21
combined with Corollary 17, $B_{1}$ contains the same multiset of values as
$\widehat{\Sigma}(\mathfrak{p}_{k})$. Thus, applying Theorem 34 – and keeping
Remark 7 in mind – it follows that $B_{1}$ contributes

$\\{0^{k+1}\\}\cup\bigcup_{i=0}^{k-1}\left\\{(-k+i)^{i+1},(k-i)^{i+1}\right\\}$
(3)

to the multiset of values contained in $\Sigma(\mathfrak{g}_{k})$. It then
follows from Lemma 29 that the multiset of values contained in $B_{2}$ is
equal to that of $\Sigma(\mathfrak{p}_{k})$ with each value increased by 1;
that is, applying Theorem 34, $B_{2}$ contributes

$\bigcup_{i=2}^{k+1}\left\\{(-k+i)^{i-1},(k-i+3)^{i-1}\right\\}$ (4)

to the multiset of values contained in $\Sigma(\mathfrak{g}_{k})$. Finally, by
Lemma 24, it follows that the multiset of values contained in $B_{3}$ is equal
to that of $\widehat{\Sigma}(\mathfrak{p}_{k-1})$; that is, applying Theorem
34 – and keeping Remark 7 in mind – $B_{3}$ contributes

$\\{0^{k}\\}\cup\bigcup_{i=0}^{k-2}\left\\{(-k+i+1)^{i+1},(k-i-1)^{i+1}\right\\}$
(5)

to the multiset of values contained in $\Sigma(\mathfrak{g}_{k})$. Therefore,
putting contributions (3), (4), and (5) together – keeping Remark 7 in mind –
we find that the spectrum of $\mathfrak{g}_{k}$ is equal to

$\\{-k,0^{3k-1},1^{3k-1},k+1\\}\cup\bigcup_{i=1}^{k-1}\\{(-k+i)^{3i},(k-i+1)^{3i}\\},$

as desired. ∎

Considering the spectrum formula of Theorem 42, we are immediately led to the
following.

######  Corollary 43.

For $k\geq 1$, if $\mathfrak{g}=\mathfrak{p}^{A}\frac{k+1|k}{2k+1}$, then
$\mathfrak{g}$ has the log-concave spectrum property.

######  Remark 44.

Combining Theorem 42 and Corollary 43 with Corollaries 14, 17, and 18, we
obtain similar results to Theorem 42 and Corollary 43 for related families of
Frobenius, type-A seaweeds. In particular, the spectrum of each of
$\mathfrak{p}^{A}\frac{2k+1}{k+1|k}$, $\mathfrak{p}^{A}\frac{k|k+1}{2k+1}$,
and $\mathfrak{p}^{A}\frac{2k+1}{k|k+1}$, for $k\geq 1$, is given in Theorem
42, and each algebra possesses the log-concave spectrum property.

###  3.4 $\mathfrak{p}^{A}\frac{k+2|k}{2k+2}$

In this subsection, we compute the spectra of Frobenius, type-A seaweed
algebras of the form $\mathfrak{p}^{A}\frac{k+2|k}{2k+2}$, for $k\geq 1$ odd.
See Figure 16 for illustrations of the oriented menaders corresponding to
$\mathfrak{p}^{A}\frac{k+2|k}{2k+2}$ for $k=1$ and $3.$

$v_{1}$$v_{2}$$v_{3}$$v_{4}$(a)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$v_{7}$$v_{8}$(b)

Figure 16: Oriented meanders of (a) $\mathfrak{p}^{A}\frac{3|1}{4}$ and (b)
$\mathfrak{p}^{A}\frac{5|3}{8}$

######  Theorem 45.

Let $\mathfrak{g}_{k}=\mathfrak{p}^{A}\frac{k+2|k}{2k+2},$ for $k=2m-1\geq 1.$
The spectrum of $\mathfrak{g}_{k}$ is equal to

$\\{-2,-1^{2},0^{3},1^{3},2^{2},3\\}$

if $k=1$,

$\\{-3,-2^{4},-1^{8},0^{11},1^{11},2^{8},3^{4},4\\}$

if $k=3$,

$\\{-4,-3^{4},-2^{10},-1^{17},0^{22},1^{22},2^{17},3^{10},4^{4},5\\}$

if $k=5$,

$\\{-5,-4^{4},-3^{10},-2^{19},-1^{28},0^{34},1^{34},2^{28},3^{19},4^{10},5^{4},6\\}$

if $k=7$, and

$\\{-m-1,-m^{4},(-m+1)^{10},(-m+2)^{19},-1^{6k-14},0^{6k-8},1^{6k-8},2^{6k-14},(m-1)^{19},m^{10},(m+1)^{4},m+2\\}$
$\cup\bigcup_{i=1}^{m-4}\\{(-m+i+2)^{12i+18},(m-i-1)^{12i+18}\\}$

if $k>7$.

###### Proof.

By induction on $m.$ The spectra of $\mathfrak{g}_{1},$ $\mathfrak{g}_{3},$
$\mathfrak{g}_{5},$ and $\mathfrak{g}_{7}$ can be easily verified by direct
calculation. For $m=5$ (or $k=9$), computing directly we find that the
spectrum of $\mathfrak{g}_{9}$ is equal to

$\\{-6,-5^{4},-4^{10},-3^{19},-2^{30},-1^{40},0^{46},1^{46},2^{40},3^{30},4^{19},5^{10},6^{4},7\\}.$

So, assume the result holds for $m-1\geq 5.$ Note that

$\Sigma(\mathfrak{g}_{k})=\begin{bmatrix}B_{1}&B_{2}\\\ &B_{3}\end{bmatrix},$

where

  * •

$B_{1}$ is the $(k+2)\times(k+2)$ block consisting of the values
$w(P_{i,j}(\mathfrak{g}_{k})),$ for $1\leq i,j\leq k+2,$

  * •

$B_{2}$ is the $(k+2)\times k$ block consisting of the values
$w(P_{i,j}(\mathfrak{g}_{k})),$ for $1\leq i\leq k+2$ and $k+3\leq j\leq
2k+2,$ and

  * •

$B_{3}$ is the $k\times k$ block consisting of the values
$w(P_{i,j}(\mathfrak{g}_{k})),$ for $k+3\leq i,j\leq 2k+2.$

Now, if $\mathfrak{p}_{k}=\mathfrak{p}^{A}\frac{k|2}{k+2},$ then by Lemma 21
combined with Corollary 17, $B_{1}$ contains the same multiset of values of
entries as $\widehat{\Sigma}(\mathfrak{p}_{k}).$ Thus, applying Theorem 38 –
and keeping Remark 7 in mind – it follows that $B_{1}$ contributes

$\\{-m-1,-m^{3},-1^{2k-1},0^{2k+1},1^{2k-1},m^{3},m+1\\}\cup\bigcup_{i=1}^{m-2}\\{(-m+i)^{4i+2},(m-i)^{4i+2}\\}$
(6)

to the multiset of values contained in $\Sigma(\mathfrak{g}_{k}).$ It then
follows from Lemma 29 that the multiset of values contained in $B_{2}$ is
equal to that of rows $\\{1,\dots,k\\}$ and columns $\\{1,\dots,k+2\\}$ in
$\Sigma(\mathfrak{p}_{k})$ with each value increased by 1. By the proof of
Theorem 38, the multiset of values contained in rows $\\{1,\dots,k\\}$ and
columns $\\{1,\dots,k\\}$ of $\mathfrak{p}_{k}$ is

$\\{-m,(-m+1)^{3},-1^{2k-5},0^{2k-3},1^{2k-5},(m-1)^{3},m\\}\cup\bigcup_{i=1}^{m-3}\\{(-m+i+1)^{4i+2},(m-i-1)^{4i+2}\\},$

and by Lemma 37, the multiset of values contained in rows $\\{1,\dots,k\\}$
and columns $\\{k+1,k+2\\}$ of $\mathfrak{p}_{k}$ is

$\\{0,1^{3},(m-1)^{3},m^{2},m+1\\}\cup\bigcup_{i=2}^{m-2}\\{i^{4}\\}.$

Thus, $B_{2}$ contributes

$\displaystyle\\{-m+1$
$\displaystyle,(-m+2)^{3},0^{2k-5},1^{2k-2},2^{2k-2},m^{6},(m+1)^{3},m+2\\}$
(7) $\displaystyle\cup\bigcup_{i=1}^{m-3}\\{(-m+i+2)^{4i+2},(m-i)^{4i+6}\\}$

to the multiset of values contained in $\Sigma(\mathfrak{g}_{k}).$ Finally, by
Lemma 24, it follows that the multiset of values contained in $B_{3}$ is equal
to that of $\widehat{\Sigma}(\mathfrak{p}_{k-2});$ that is, applying Theorem
38, $B_{3}$ contributes

$\\{-m,(-m+1)^{3},-1^{2k-5},0^{2k-3},1^{2k-5},(m-1)^{3},m\\}\cup\bigcup_{i=1}^{m-3}\\{(-m+i+1)^{4i+2},(m-i-1)^{4i+2}\\}$
(8)

to the multiset of values contained in $\Sigma(\mathfrak{g}_{k}).$ Therefore,
putting contributions (6), (7), and (8) together – keeping Remark 7 in mind –
we find that the spectrum of $\mathfrak{g}_{k}$ is equal to

$\\{-m-1,-m^{4},(-m+1)^{10},(-m+2)^{19},-1^{6k-14},0^{6k-8},1^{6k-8},2^{6k-14},(m-1)^{19},m^{10},(m+1)^{4},m+2\\}$
$\cup\bigcup_{i=1}^{m-4}\\{(-m+i+2)^{12i+18},(m-i-1)^{12i+18}\\}$

as desired. ∎

Considering the spectrum formula of Theorem 45, we are immediately led to the
following.

######  Corollary 46.

For $k\geq 1$, if $\mathfrak{g}=\mathfrak{p}^{A}\frac{k+2|k}{2k+2}$, then
$\mathfrak{g}$ has the log-concave spectrum property.

######  Remark 47.

Combining Theorem 45 and Corollary 46 with Corollaries 14, 17, and 18, we
obtain similar results to Theorem 45 and Corollary 46 for related families of
Frobenius, type-A seaweeds. In particular, the specctrum of each of
$\mathfrak{p}^{A}\frac{2k+2}{k+2|k}$, $\mathfrak{p}^{A}\frac{k|k+2}{2k+2}$,
and $\mathfrak{p}^{A}\frac{2k+2}{k|k+2}$, for $k\geq 1$ odd, is given in
Theorem 45, and each algebra possesses the log-concave spectrum property.

One quality that makes the family of maximal parabolic, type-A seaweeds a
desirable candidate family in our investigations is Elashvili’s ([9], 1990)
simple closed-form index formula

${\rm ind\hskip 2.84544pt}\mathfrak{p}^{A}\frac{a|b}{n}=\gcd(a,b)-1,$

allowing for quick identification of Frobenius such algebras. Outside of this
family, there are two others known with similar index formulas: the families
of type-A seaweeds of the form $\mathfrak{p}^{A}\frac{a|b|c}{a+b+c}$ and of
the form $\mathfrak{p}^{A}\frac{a|b}{c|d}$ which both have index given by
$\gcd(a+b,b+c)-1$ (see [2]). For the sake of comparison, we provide the
formulas for the spectra of two such families of seaweeds below. The proofs
are left to the interested reader.

######  Theorem 48.

Let $\mathfrak{g}_{k}=\mathfrak{p}^{A}\frac{2k|1}{1|2k}$, for $k\geq 1$. Then
the spectrum of $\mathfrak{g}_{k}$ is equal to

$\bigcup_{i=1}^{k}\\{(-k+i)^{4i-2},(k-i+1)^{4i-2}\\}.$

Moreover, $\mathfrak{g}_{k}$ has the log-concave spectrum property.

######  Theorem 49.

Let $\mathfrak{g}_{k}=\mathfrak{p}^{A}\frac{2k|1|1}{2k+2}$, for $k\geq 1$.
Then the spectrum of $\mathfrak{g}_{k}$ is equal to

$\\{-k,k+1\\}\cup\bigcup_{i=1}^{k}\\{(-k+i)^{4i},(k-i+1)^{4i}\\}.$

Moreover, $\mathfrak{g}_{k}$ has the log-concave spectrum property.

Note that all families of Frobenius, type-A seaweeds considered above have
been defined by compositions with a fixed number of parts, while the sizes of
the parts vary. In the next section, we consider families of Frobenius, type-A
seaweeds parametrized by the number of parts of a fixed size in their defining
compositions. In contrast to the spectra of this section – where both the set
of distinct eigenvalues and the sequence of multiplicities varied with the
parameter – the seaweeds discussed in Section 4 have spectra whose sets of
distinct eigenvalues exhibit a suprising stability property.

##  4 Stability

In this section, we consider families of Frobenius, type-A seaweeds that are
parametrized by the number of 2’s (Section 4.1) and by the number of 4’s
(Section 4.2) in the defining compositions. For such families, we find that
the sets of distinct eigenvalues in the spectra stabilize. At the end of
Section 4.2, we conjecture that such stabilization occurs among more general
families as well. Note that this behavior stands in sharp contrast to that of
the Frobenius seaweeds $\mathfrak{p}^{A}\frac{k|1}{k+1}$,
$\mathfrak{p}^{A}\frac{k|2}{k+2}$, $\mathfrak{p}^{A}\frac{k+1|k}{2k+1}$,
$\mathfrak{p}^{A}\frac{k+2|k}{2k+2}$, $\mathfrak{p}^{A}\frac{2k|1}{1|2k}$, and
$\mathfrak{p}^{A}\frac{2k|1|1}{2k+2}$, where the number of eigenvalues
contained in the spectrum strictly increased with $k$.

######  Remark 50.

We do not consider families of Frobenius, type-A seaweeds parametrized by the
number of occurrences of an odd integer in the defining compositions because
such families do not exist. For a given type-A seaweed $\mathfrak{g}$, each
odd part in the defining composition of $\mathfrak{g}$ contributes a vertex of
degree 1 to $M(\mathfrak{g})$. Consequently, if $\mathfrak{g}$ is to be
Frobenius, i.e., if $M(\mathfrak{g})$ consists of a single path, then
$\mathfrak{g}$ can have at most two odd parts in its defining composition.

###  4.1 Parameterized by number of 2’s

In this subsection, we consider families of Frobenius, type-A seaweeds which
are parameterized by the number of 2’s in the defining compositions. More
specifically, we determine the spectra of type-A seaweeds of the form

$\mathfrak{p}^{A}\frac{k|2|\cdots|2}{k+1|2|\cdots|2|1},\quad\quad\mathfrak{p}^{A}\frac{k|2|\cdots|2|1}{k+1|2|\cdots|2},\quad\quad\text{and}\quad\quad\mathfrak{p}^{A}\frac{2|\cdots|2|1}{2r+1}.$

To start, we establish a general result concerning the relationship between
the spectra of the Frobenius, type-A seaweeds

$\mathfrak{g}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|1}{b_{1}|\ldots|b_{t}},\quad\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{2|\cdots|2}^{r}}{b_{1}|\ldots|b_{t}|\underbrace{2|\cdots|2}_{r-1}|1},\quad\text{and}\quad\mathfrak{g}^{\prime\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{2|\cdots|2}^{r}|1}{b_{1}|\ldots|b_{t}|\underbrace{2|\cdots|2}_{r}},$

for $r\geq 1$. For example, taking $\mathfrak{g}=\frac{3|1}{4}$ and $r=1$, we
have $\mathfrak{g}^{\prime}=\frac{3|2}{4|1}$ and
$\mathfrak{g}^{\prime\prime}=\frac{3|2|1}{4|2}$. The oriented meanders of
these type-A seaweeds are illustrated in Figure 17.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$(a)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$(b)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$(c)

Figure 17: Oriented meanders of (a) $\mathfrak{p}^{A}\frac{3|1}{4}$, (b)
$\mathfrak{p}^{A}\frac{3|2}{4|1}$, and (c) $\mathfrak{p}^{A}\frac{3|2|1}{4|2}$

######  Lemma 51.

If
$\mathfrak{g}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|1}{b_{1}|\ldots|b_{t}}$
is a Frobenius, type-A seaweed with spectrum $S$, then
$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|2}{b_{1}|\ldots|b_{t}|1}$
is a Frobenius, type-A seaweed with spectrum $S\cup\\{0,1\\}$.

###### Proof.

Let $n=1+\sum_{i=1}^{m}a_{i}=\sum_{j=1}^{t}b_{j}$. Evidently,
$\overrightarrow{M}(\mathfrak{g}^{\prime})$ is equal to
$\overrightarrow{M}(\mathfrak{g})$ with the addition of a vertex $v_{n+1}$ and
a directed edge $(v_{n+1},v_{n})$. Considering Corollary 3, it follows that
$\mathfrak{g}^{\prime}$ is Frobenius. Now, setting $a_{0}=b_{0}=0$, note that
the spectrum of $\mathfrak{g}$ is the multiset

$\displaystyle S=\bigcup_{k=1}^{m}$
$\displaystyle\left\\{w(P_{i,j}(\mathfrak{g}))~{}|~{}1+\sum_{l=0}^{k-1}a_{l}\leq
j<i\leq\sum_{l=1}^{k}a_{l}\right\\}$
$\displaystyle\cup\bigcup_{k=1}^{t}\left\\{w(P_{i,j}(\mathfrak{g}))~{}|~{}1+\sum_{l=0}^{k-1}b_{l}\leq
i<j\leq\sum_{l=1}^{k}b_{l}\right\\}\cup\\{0^{n-1}\\},$

and the spectrum of $\mathfrak{g}^{\prime}$ is the multiset

$\displaystyle S^{\prime}=\bigcup_{k=1}^{m}$
$\displaystyle\left\\{w(P_{i,j}(\mathfrak{g}^{\prime}))~{}|~{}1+\sum_{l=0}^{k-1}a_{l}\leq
j<i\leq\sum_{l=1}^{k}a_{l}\right\\}$
$\displaystyle\cup\bigcup_{k=1}^{t}\left\\{w(P_{i,j}(\mathfrak{g}^{\prime}))~{}|~{}1+\sum_{l=0}^{k-1}b_{l}\leq
i<j\leq\sum_{l=1}^{k}b_{l}\right\\}\cup\\{0^{n},w(P_{n+1,n}(\mathfrak{g}^{\prime}))\\}.$

Given the relationship between $\overrightarrow{M}(\mathfrak{g}^{\prime})$ and
$\overrightarrow{M}(\mathfrak{g})$ outlined above, it follows that

$S^{\prime}=S\cup\\{0,w(P_{n+1,n}(\mathfrak{g}^{\prime}))\\}=S\cup\\{0,1\\}.$

∎

######  Theorem 52.

If
$\mathfrak{g}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|1}{b_{1}|\ldots|b_{t}}$
is a Frobenius, type-A seaweed with spectrum $S$, then

  1. (1)

$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{2|\cdots|2}^{r}}{b_{1}|\ldots|b_{t}|\underbrace{2|\cdots|2}_{r-1}|1},$

for $r\geq 1$, is a Frobenius, type-A seaweed with spectrum
$S\cup\\{0^{2r-1},1^{2r-1}\\}$; and

  2. (2)

$\mathfrak{g}^{\prime\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{2|\cdots|2}^{r}|1}{b_{1}|\ldots|b_{t}|\underbrace{2|\cdots|2}_{r}},$

for $r\geq 1$, is a Frobenius, type-A seaweed with spectrum
$S\cup\\{0^{2r},1^{2r}\\}$.

###### Proof.

By induction on $r$. An application of Lemma 51 establishes the result for
$r=1$ in (1). Then applying Corollary 14, followed by Lemma 51, and then
Corollary 14 again, to

$\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|2}{b_{1}|\ldots|b_{t}|1}$

establishes the case $r=1$ for (2). Assume the result holds for $r-1\geq 1$.
In particular, assume that the algebra

$\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{2|\cdots|2}^{r-1}|1}{b_{1}|\ldots|b_{t}|\underbrace{2|\cdots|2}_{r-1}}$

is Frobenius with spectrum equal to $S\cup\\{0^{2r-2},1^{2r-2}\\}$. Applying
Lemma 51, we find that the algebra

$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{2|\cdots|2}^{r}}{b_{1}|\ldots|b_{t}|\underbrace{2|\cdots|2}_{r-1}|1}$

is Frobenius with spectrum equal to $S\cup\\{0^{2r-1},1^{2r-1}\\}$, as
desired. Now, applying Corollary 14 followed by Lemma 51 and then a second
application of Corollary 14 to $\mathfrak{g}^{\prime}$, we find that the
algebra

$\mathfrak{g}^{\prime\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{2|\cdots|2}^{r}|1}{b_{1}|\ldots|b_{t}|\underbrace{2|\cdots|2}_{r}}$

is Frobenius with spectrum equal to $S\cup\\{0^{2r},1^{2r}\\}$, as desired.
The result follows by induction. ∎

Considering the spectrum formulas of Theorem 52, we are immediately led to the
following.

######  Corollary 53.

Let
$\mathfrak{g}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|1}{b_{1}|\ldots|b_{t}}$
be a Frobenius, type-A seaweed with the unimodal spectrum property. If

$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{2|\cdots|2}^{r}}{b_{1}|\ldots|b_{t}|\underbrace{2|\cdots|2}_{r-1}|1}\quad\text{and}\quad\mathfrak{g}^{\prime\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{2|\cdots|2}^{r}|1}{b_{1}|\ldots|b_{t}|\underbrace{2|\cdots|2}_{r}},$

then $\mathfrak{g}^{\prime}$ and $\mathfrak{g}^{\prime\prime}$ have the
unimodal spectrum property.

It is important to note that “unimodal” cannot be strengthened to “log-
concave” in the conclusion of Corollary 53. A counterexample is provided by
the following theorem (see Remark 55) which is a corollary of Theorems 34 and
52.

######  Theorem 54.

  1. (1)

For $k\geq 1$, if

$\mathfrak{g}_{k}=\mathfrak{p}^{A}\frac{k|\overbrace{2|\cdots|2}^{r}}{k+1|\underbrace{2|\cdots|2}_{r-1}|1},$

then $\mathfrak{g}_{k}$ has the unimodal spectrum property with spectrum equal
to

$\\{0^{k+2r-1},1^{k+2r-1}\\}\cup\bigcup_{i=1}^{k-1}\\{(-k+i)^{i},(k-i)^{i}\\}.$

  2. (2)

For $k\geq 1$, if

$\mathfrak{g}_{k}=\mathfrak{p}^{A}\frac{k|\overbrace{2|\cdots|2}^{r}|1}{k+1|\underbrace{2|\cdots|2}_{r}},$

then $\mathfrak{g}_{k}$ has the unimodal spectrum property with spectrum equal
to

$\\{0^{k+2r},1^{k+2r}\\}\cup\bigcup_{i=1}^{k-1}\\{(-k+i)^{i},(k-i)^{i}\\}.$

######  Remark 55.

Utilizing Theorem 54, we can construct examples of Frobenius, type-A seaweeds
which do not have the log-concave spectrum property. For example, let
$\mathfrak{g}=\mathfrak{p}^{A}\frac{3|1}{4}$ and
$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{3|2|2}{4|2|1}.$ Recall from
Theorem 34 that the spectrum of $\mathfrak{g}$ is
$\\{-2,-1^{2},0^{3},1^{3},2^{2},3\\}$. On the other hand, by Theorem 52, we
find that the spectrum of $\mathfrak{g}^{\prime}$ is
$\\{-2,-1^{2},0^{6},1^{6},2^{2},3\\}.$ Clearly, $\mathfrak{g}^{\prime}$ does
not have the log-concave spectrum property.

######  Remark 56.

Note that for fixed $k$ and varying values of $r$, the collection of distinct
eigenvalues in the spectra of the Frobenius, type-A seaweeds considered in
Theorem 54 is fixed.

######  Remark 57.

Note that one can also apply Corollary 53 to the Frobenius, type-A seaweed
$\mathfrak{p}^{A}\frac{1|1}{2}$ to determine the spectrum of type-A seaweeds
of the form
$\mathfrak{g}_{r}=\mathfrak{p}^{A}\frac{1|2|\cdots|2}{2|\cdots|2|1}$, where
$r\geq 1$ is the number of $2$’s in the numerator and denominator. See Figure
18 for some example meanders of such type-A seaweeds. In particular, it is
straightforward to prove that the spectrum of $\mathfrak{g}_{r}$ is
$\\{0^{2r},1^{2r}\\}$. Now, one can show that type-A seaweeds of the form
$\mathfrak{p}^{A}\frac{1|2|\cdots|2}{2|\cdots|2|1}$ are related to algebras of
another combinatorially defined family of Lie subalgebras of
$\mathfrak{sl}(n)$. In particular,
$\mathfrak{p}^{A}\frac{1|2|\cdots|2}{2|\cdots|2|1}$ is isomorphic to a type-A
Lie poset algebra (see [3]). Type-A Lie poset algebras are subalgebras of
$\mathfrak{sl}(n)$ parametrized by posets on $\\{1,\ldots,n\\}$. It is known
that the spectra of Frobenius, type-A Lie poset algebras corresponding to
posets with chains of cardinality at most two must consist of an equal number
of 0’s and 1’s (see [4]). It is conjectured that this is true in general for
Frobenius Lie poset algebras (see [4, 5]).

$v_{1}$$v_{2}$$v_{3}$(a)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$(b)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{5}$$v_{5}$(c)

Figure 18: Oriented meanders of (a) $\mathfrak{p}^{A}\frac{1|2}{2|1}$, (b)
$\mathfrak{p}^{A}\frac{1|2|2}{2|2|1}$, and (c)
$\mathfrak{p}^{A}\frac{1|2|2|2}{2|2|2|1}$

######  Remark 58.

Additionally, one can apply Theorem 52 to the Frobenius, type-A seaweeds of
Theorems 48 and 49. We leave such applications as exercises for interested
readers.

We now consider Frobenius, type-A seaweeds of the form
$\mathfrak{p}^{A}\frac{2|\dots|2|1}{2r+1}.$ Although such algebras are
parametrized by the number of 2’s in their defining compositions, they are
parabolic and so descriptions of their spectra do not follow from Corollary
53. See Figure 19 for some example meanders of such type-A seaweeds.

$v_{1}$$v_{2}$$v_{3}$(a)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$(b)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{5}$$v_{5}$(c)

Figure 19: Oriented meanders of (a) $\mathfrak{p}^{A}\frac{2|1}{3}$, (b)
$\mathfrak{p}^{A}\frac{2|2|1}{5}$, and (c) $\mathfrak{p}^{A}\frac{2|2|2|1}{7}$

To determine the form of the spectrum of the Frobenius, type-A seaweed
$\mathfrak{p}^{A}\frac{2|\cdots|2|1}{2r+1}$, we require the following lemma.

######  Lemma 59.

Let $\mathfrak{g}_{r}=\mathfrak{p}^{A}\frac{2|\cdots|2|1}{2r+1}$, for $r\geq
1$. Then

$\\{w(P_{i,2r+1}(\mathfrak{g}_{r}))~{}|~{}1\leq
i<2r+1\\}=\\{2^{\lceil\frac{r}{2}\rceil},1^{r},0^{\lceil\frac{r-1}{2}\rceil}\\}.$

###### Proof.

By induction on $r$. For $r=1$, we can compute directly from
$\overrightarrow{M}(\mathfrak{g}_{1})$ (see Figure 19(a)) that

$w(P_{1,3}(\mathfrak{g}_{1}))=1\quad\text{and}\quad
w(P_{2,3}(\mathfrak{g}_{1}))=2.$

Thus,

$\\{w(P_{i,3}(\mathfrak{g}_{1}))~{}|~{}1\leq
i<3\\}=\\{2,1\\}=\\{2^{\lceil\frac{1}{2}\rceil},1^{1},0^{\lceil\frac{1-1}{2}\rceil}\\}.$

Now, assume that the result holds for $r-1\geq 1$. Let
$\mathfrak{p}_{r-1}=\mathfrak{p}^{A}\frac{1|2|\cdots|2}{2r-1}$. Note that
removing the edges $(v_{1},v_{2r+1})$ and $(v_{2},v_{1})$ and the vertices
$v_{1}$ and $v_{2r+1}$ from $\overrightarrow{M}(\mathfrak{g}_{r})$ yields
$\overrightarrow{M}(\mathfrak{p}_{r-1}),$ with $v_{i+1}$ in place of $v_{i}$,
for $1\leq i\leq 2r-1$. Thus,

$\displaystyle w(P_{i,2r+1}(\mathfrak{g}_{r}))$
$\displaystyle=w(P_{i,2}(\mathfrak{g}_{r}))+w(P_{2,1}(\mathfrak{g}_{r}))+w(P_{1,2r+1}(\mathfrak{g}_{r}))$
$\displaystyle=w(P_{i-1,1}(\mathfrak{p}_{r-1}))+2,$

for $2\leq i\leq 2r$. Applying Lemma 15, it follows that

$w(P_{i,2r+1}(\mathfrak{g}_{k}))=w(P_{i-1,1}(\mathfrak{p}_{r-1}))+2=-w(P_{2r-i+1,2r-1}(\mathfrak{g}_{r-1}))+2,$

for $2\leq i\leq 2r$. Therefore, applying the induction hypothesis,

$\\{w(P_{i,2r+1}(\mathfrak{g}_{r}))~{}|~{}3\leq i\leq
2r\\}=\\{-w(P_{i,2r-1}(\mathfrak{g}_{r-1}))+2~{}|~{}1\leq
i<2r-1\\}=\\{0^{\lceil\frac{r-1}{2}\rceil},1^{r-1},2^{\lceil\frac{r-2}{2}\rceil}\\}.$

As $w(P_{1,2r+1}(\mathfrak{g}_{r}))=1$ and
$w(P_{2,2r+1}(\mathfrak{g}_{r}))=2$, it follows that

$\\{w(P_{i,2r+1}(\mathfrak{g}_{r}))~{}|~{}1\leq
i<2r+1\\}=\\{0^{\lceil\frac{r-1}{2}\rceil},1^{r},2^{\lceil\frac{r}{2}\rceil}\\},$

as desired. ∎

######  Theorem 60.

Let $\mathfrak{g}_{r}=\mathfrak{p}^{A}\frac{2|\cdots|2|1}{2r+1}$, for $r\geq
1$. There exists positive integers $a_{r}>b_{r}$ such that the spectrum of
$\mathfrak{g}_{r}$ is given by

$\\{-1^{b_{r}},0^{a_{r}},1^{a_{r}},2^{b_{r}}\\}.$

###### Proof.

By induction on $r$. For $r=1$, calculating directly using
$\overrightarrow{M}(\mathfrak{g}_{1})$ (see Figure 19(a)) we find that the
spectrum of $\mathfrak{g}_{1}$ is

$\\{-1^{1},0^{2},1^{2},2^{1}\\}.$

Assume the result holds for $r-1\geq 1$. We break the spectrum of
$\mathfrak{g}_{r}$ into three groups:

$G_{1}=\\{w(P_{i,j}(\mathfrak{g}_{r}))~{}|~{}2\leq i\leq j\leq
2r\\}\cup\\{w(P_{2j,2j-1}(\mathfrak{g}_{r}))~{}|~{}1\leq j\leq r\\},$
$G_{2}=\\{w(P_{i,2r+1}(\mathfrak{g}_{r}))~{}|~{}1\leq i<2r+1\\},$

and

$G_{3}=\\{w(P_{1,i}(\mathfrak{g}_{r}))~{}|~{}1\leq i\leq 2r\\}.$

Let $\mathfrak{p}_{r-1}=\mathfrak{p}^{A}\frac{1|2|\cdots|2}{2r-1}$. Note that
removing the edges $(v_{1},v_{2r+1})$ and $(v_{2},v_{1})$ and the vertices
$v_{1}$ and $v_{2r+1}$ from $\overrightarrow{M}(\mathfrak{g}_{r})$ yields
$\overrightarrow{M}(\mathfrak{p}_{r-1})$ with $v_{i+1}$ in place of $v_{i}$,
for $1\leq i\leq 2r-1$. Thus,

$w(P_{i,j}(\mathfrak{g}_{r}))=w(P_{i-1,j-1}(\mathfrak{p}_{r-1})),$

for $2\leq i\leq j\leq 2r$, and

$w(P_{2j,2j-1}(\mathfrak{g}_{r}))=w(P_{2j-1,2j-2}(\mathfrak{p}_{r-1})),$

for $2\leq j\leq r$. Consequently, letting $S$ denote the spectrum of of
$\mathfrak{p}_{r-1}$, we have that

$G_{1}=S\cup\\{w(P_{2r,2r}(\mathfrak{g}_{r})),w(P_{2,1}(\mathfrak{g}_{r}))\\}=S\cup\\{0,1\\}.$

Applying Corollary 17, the spectrum of $\mathfrak{g}_{r-1}$ is also $S$; that
is, applying our induction hypothesis,

$G_{1}=\\{-1^{b_{r-1}},0^{a_{r-1}+1},1^{a_{r-1}+1},2^{b_{r-1}}\\}.$

Now, considering Lemma 59, we have that

$G_{2}=\\{2^{\lceil\frac{r}{2}\rceil},1^{r},0^{\lceil\frac{r-1}{2}\rceil}\\}.$

As for $G_{3}$, note that

$\displaystyle w(P_{1,i}(\mathfrak{g}_{r}))$
$\displaystyle=-w(P_{i,1}(\mathfrak{g}_{r}))$
$\displaystyle=-(w(P_{i,2r+1}(\mathfrak{g}_{r}))-w(P_{1,2r+1}(\mathfrak{g}_{r})))$
$\displaystyle=-w(P_{i,2r+1}(\mathfrak{g}_{r}))+1,$

for $1\leq i\leq 2r$. Thus, applying Lemma 59, it follows that

$G_{3}=\\{-1^{\lceil\frac{r}{2}\rceil},0^{r},1^{\lceil\frac{r-1}{2}\rceil}\\}.$

Putting everything together, we find that the spectrum of $\mathfrak{g}_{r}$
is given by

$\\{-1^{b_{r-1}+\lceil\frac{r}{2}\rceil},0^{a_{r-1}+r+\lceil\frac{r-1}{2}\rceil+1},1^{a_{r-1}+r+\lceil\frac{r-1}{2}\rceil+1},2^{b_{r-1}+\lceil\frac{r}{2}\rceil}\\}.$

The result follows. ∎

Considering the spectrum formula of Theorem 60, we are immediately led to the
following.

######  Corollary 61.

For $r\geq 1$, if
$\mathfrak{g}_{r}=\mathfrak{p}^{A}\frac{2|\cdots|2|1}{2r+1}$, then
$\mathfrak{g}_{r}$ has the log-concave spectrum property.

######  Remark 62.

Combining Theorems 52, 54, and 60 and Corollaries 61 and 53 with Corollaries
14, 17, and 18, we obtain similar results for related families of Frobenius,
type-A seaweeds.

###  4.2 Parameterized by number of 4’s

In this subsection, we consider families of Frobneius, type-A seaweeds which
are parameterized by the number of 4’s in the defining compositions. More
specifically, we determine the spectra of type-A seaweeds of the form

$\mathfrak{p}^{A}\frac{k|4|\cdots|4}{k+2|4|\cdots|4|2}\quad\quad\text{and}\quad\quad\mathfrak{p}^{A}\frac{k|4|\cdots|4|2}{k+2|4|\cdots|4}.$

Similar to Section 4.1, we start by establishing a general result concerning
the relationship between the spectra of the Frobenius, type-A seaweeds

$\mathfrak{g}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|2}{b_{1}|\ldots|b_{t}},\quad\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{4|\cdots|4}^{r}}{b_{1}|\ldots|b_{t}|\underbrace{4|\cdots|4}_{r-1}|2},\quad\text{and}\quad\mathfrak{g}^{\prime\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{4|\cdots|4}^{r}|2}{b_{1}|\ldots|b_{t}|\underbrace{4|\cdots|4}_{r}},$

for $r\geq 1$. For example, taking $\mathfrak{g}=\frac{1|2}{3}$ and $r=1$, we
have $\mathfrak{g}^{\prime}=\frac{1|4}{3|2}$ and
$\mathfrak{g}^{\prime\prime}=\frac{1|4|2}{3|4}$. The oriented meanders of
these type-A seaweeds are illustrated in Figure 20.

$v_{1}$$v_{2}$$v_{3}$(a)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$(b)

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$v_{7}$(c)

Figure 20: Oriented meanders of (a) $\mathfrak{p}^{A}\frac{1|2}{3}$, (b)
$\mathfrak{p}^{A}\frac{1|4}{3|2}$, and (c) $\mathfrak{p}^{A}\frac{1|4|2}{3|4}$

######  Lemma 63.

If
$\mathfrak{g}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|2}{b_{1}|\ldots|b_{t}}$
is a Frobenius, type-A seaweed with spectrum $S$, then
$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|4}{b_{1}|\ldots|b_{t}|2}$
is a Frobenius, type-A seaweed with spectrum $S\cup\\{-1,0^{3},1^{3},2\\}$.

###### Proof.

Let $n=2+\sum_{i=1}^{m}a_{i}=\sum_{j=1}^{t}b_{j}$. Evidently,
$\overrightarrow{M}(\mathfrak{g}^{\prime})$ is equal to
$\overrightarrow{M}(\mathfrak{g})$ with the removal of the directed edge
$(v_{n},v_{n-1})$, the addition of vertices $v_{n+1}$ and $v_{n+2}$, and the
addition of directed edges $(v_{n+1},v_{n})$, $(v_{n+2},v_{n-1})$, and
$(v_{n+1},v_{n+2})$. Considering Corollary 3, it follows that
$\mathfrak{g}^{\prime}$ is Frobenius. Now, setting $a_{0}=b_{0}=0$ and
$a_{m+1}=2$, note that the spectrum of $\mathfrak{g}^{\prime}$ is the multiset

$\displaystyle S^{\prime}=\bigcup_{k=1}^{m+1}$
$\displaystyle\left\\{w(P_{i,j}(\mathfrak{g}^{\prime}))~{}|~{}1+\sum_{l=0}^{k-1}a_{l}\leq
j<i\leq\sum_{l=1}^{k}a_{l}\right\\}$
$\displaystyle\cup\bigcup_{k=1}^{t}\left\\{w(P_{i,j}(\mathfrak{g}^{\prime}))~{}|~{}1+\sum_{l=0}^{k-1}b_{l}\leq
i<j\leq\sum_{l=1}^{k}b_{l}\right\\}$
$\displaystyle\cup\\{0^{n+1},w(P_{n+1,n-1}(\mathfrak{g}^{\prime})),w(P_{n+1,n}(\mathfrak{g}^{\prime})),w(P_{n+1,n+2}(\mathfrak{g}^{\prime}))\\}$
$\displaystyle\cup\\{w(P_{n+2,n-1}(\mathfrak{g}^{\prime})),w(P_{n+2,n}(\mathfrak{g}^{\prime})),w(P_{n+2,n+1}(\mathfrak{g}^{\prime}))\\};$

that is,

$\displaystyle S^{\prime}=\bigcup_{k=1}^{m+1}$
$\displaystyle\left\\{w(P_{i,j}(\mathfrak{g}^{\prime}))~{}|~{}1+\sum_{l=0}^{k-1}a_{l}\leq
j<i\leq\sum_{l=1}^{k}a_{l}\right\\}$
$\displaystyle\cup\bigcup_{k=1}^{t}\left\\{w(P_{i,j}(\mathfrak{g}^{\prime}))~{}|~{}1+\sum_{l=0}^{k-1}b_{l}\leq
i<j\leq\sum_{l=1}^{k}b_{l}\right\\}$
$\displaystyle\cup\\{-1,0^{n+2},1^{3},2\\}.$

We claim that $w(P_{i,j}(\mathfrak{g}^{\prime}))=w(P_{i,j}(\mathfrak{g}))$,
for $1\leq i\neq j\leq n$. To establish the claim, we break it into two cases.

Case 1: $\\{v_{n-1},v_{n}\\}$ is not an edge in $P_{i,j}(\mathfrak{g})$.
Considering the relationship between $\overrightarrow{M}(\mathfrak{g})$ and
$\overrightarrow{M}(\mathfrak{g}^{\prime})$ outlined above, we have that
$P_{i,j}(\mathfrak{g})=P_{i,j}(\mathfrak{g}^{\prime})$, and the claim follows.

Case 2: $\\{v_{n-1},v_{n}\\}$ is an edge in $P_{i,j}(\mathfrak{g})$. In this
case $P_{i,j}(\mathfrak{g})$ can be decomposed into three (possibly trivial)
subpaths as follows: either

  * •

$P_{i,j}(\mathfrak{g})$ is equal to the concatenation of the paths
$P_{i,n-1}(\mathfrak{g})$, $P_{n-1,n}(\mathfrak{g})$, and
$P_{n,j}(\mathfrak{g})$, or

  * •

$P_{i,j}(\mathfrak{g})$ is equal to the concatenation of the paths
$P_{i,n}(\mathfrak{g})$, $P_{n,n-1}(\mathfrak{g})$, and
$P_{n-1,j}(\mathfrak{g})$.

Assume that $P_{i,j}(\mathfrak{g})$ is equal to the concatenation of the paths
$P_{i,n-1}(\mathfrak{g})$, $P_{n-1,n}(\mathfrak{g})$, and
$P_{n,j}(\mathfrak{g})$; the other case follows via a similar argument.
Considering the relationship between $\overrightarrow{M}(\mathfrak{g})$ and
$\overrightarrow{M}(\mathfrak{g}^{\prime})$ outlined above, it follows that
$P_{i,j}(\mathfrak{g}^{\prime})$ is the concatenation of the paths
$P_{i,n-1}(\mathfrak{g}^{\prime})$, $P_{n-1,n+2}(\mathfrak{g}^{\prime})$,
$P_{n+2,n+1}(\mathfrak{g}^{\prime})$, $P_{n+1,n}(\mathfrak{g}^{\prime})$, and
$P_{n,j}(\mathfrak{g}^{\prime})$, where
$P_{i,n-1}(\mathfrak{g}^{\prime}))=P_{i,n-1}(\mathfrak{g}))$ and
$P_{n,j}(\mathfrak{g}^{\prime}))=P_{n,j}(\mathfrak{g}))$. Thus,

$\displaystyle w(P_{i,j}(\mathfrak{g}^{\prime}))$
$\displaystyle=w(P_{i,n-1}(\mathfrak{g}^{\prime}))+w(P_{n-1,n+2}(\mathfrak{g}^{\prime}))+w(P_{n+2,n+1}(\mathfrak{g}^{\prime}))+w(P_{n+1,n}(\mathfrak{g}^{\prime}))+w(P_{n,j}(\mathfrak{g}^{\prime}))$
$\displaystyle=w(P_{i,n-1}(\mathfrak{g}^{\prime}))-1-1+1+w(P_{n,j}(\mathfrak{g}^{\prime}))$
$\displaystyle=w(P_{i,n-1}(\mathfrak{g}))-1+w(P_{n,j}(\mathfrak{g}))$
$\displaystyle=w(P_{i,n-1}(\mathfrak{g}))+w(P_{n-1,n}(\mathfrak{g}))+w(P_{n,j}(\mathfrak{g}))$
$\displaystyle=w(P_{i,j}(\mathfrak{g})),$

establishing the claim.

Therefore,

$\displaystyle S^{\prime}=\bigcup_{k=1}^{m+1}$
$\displaystyle\left\\{w(P_{i,j}(\mathfrak{g}))~{}|~{}1+\sum_{l=0}^{k-1}a_{l}\leq
j<i\leq\sum_{l=1}^{k}a_{l}\right\\}$
$\displaystyle\cup\bigcup_{k=1}^{t}\left\\{w(P_{i,j}(\mathfrak{g}))~{}|~{}1+\sum_{l=0}^{k-1}b_{l}\leq
i<j\leq\sum_{l=1}^{k}b_{l}\right\\}$
$\displaystyle\cup\\{-1,0^{n+2},1^{3},2\\};$

that is,

$S^{\prime}=S\cup\\{-1,0^{3},1^{3},2\\}.$

∎

######  Theorem 64.

If
$\mathfrak{g}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|2}{b_{1}|\ldots|b_{t}}$
is a Frobenius, type-A seaweed with spectrum $S$, then

  1. (1)

$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{4|\cdots|4}^{r}}{b_{1}|\ldots|b_{t}|\underbrace{4|\cdots|4}_{r-1}|2},$

for $r\geq 1$, is a Frobenius, type-A seaweed with spectrum
$S\cup\\{-1^{2r-1},0^{6r-3},1^{6r-3},2^{2r-1}\\}$; and

  2. (2)

$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{4|\cdots|4}^{r}|2}{b_{1}|\ldots|b_{t}|\underbrace{4|\cdots|4}_{r}},$

for $r\geq 1$, is a Frobenius, type-A seaweed with spectrum
$S\cup\\{-1^{2r},0^{6r},1^{6r},2^{2r}\\}$.

###### Proof.

By induction on $r$. An application of Lemma 63 establishes the result for
$r=1$ in (1). Then applying Corollary 14, followed by Lemma 63, and then
Corollary 14 again, to

$\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|4}{b_{1}|\ldots|b_{t}|2}$

establishes the case $r=1$ for (2). Assume the result holds for $r-1\geq 1$.
In particular, assume that the algebra

$\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{4|\cdots|4}^{r-1}|2}{b_{1}|\ldots|b_{t}|\underbrace{4|\cdots|4}_{r-1}}$

is Frobenius with spectrum equal to
$S\cup\\{-1^{2r-2},0^{6r-6},1^{6r-6},2^{2r-2}\\}$. Applying Lemma 63, we find
that the algebra

$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{4|\cdots|4}^{r}}{b_{1}|\ldots|b_{t}|\underbrace{4|\cdots|4}_{r-1}|2}$

is Frobenius with spectrum equal to
$S\cup\\{-1^{2r-1},0^{6r-3},1^{6r-3},2^{2r-1}\\}$, as desired. Now, applying
Corollary 14, followed by Lemma 63, and then Corollary 14 again, to
$\mathfrak{g}^{\prime}$, we find that the algebra

$\mathfrak{g}^{\prime\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{4|\cdots|4}^{r}|2}{b_{1}|\ldots|b_{t}|\underbrace{4|\cdots|4}_{r}}$

is Frobenius with spectrum equal to $S\cup\\{-1^{2r},0^{6r},1^{6r},2^{2r}\\}$,
as desired. The result follows by induction. ∎

Considering the spectrum formula of Theorem 64, we are immediately led to the
following.

######  Corollary 65.

Let
$\mathfrak{g}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|2}{b_{1}|\ldots|b_{t}}$
be a Frobenius, type-A seaweed with the unimodal spectrum property. If

$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{4|\cdots|4}^{r}}{b_{1}|\ldots|b_{t}|\underbrace{4|\cdots|4}_{r-1}|2}\quad\text{and}\quad\mathfrak{g}^{\prime\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{4|\cdots|4}^{r}|2}{b_{1}|\ldots|b_{t}|\underbrace{4|\cdots|4}_{r}},$

then $\mathfrak{g}^{\prime}$ and $\mathfrak{g}^{\prime\prime}$ have the
unimodal spectrum property.

As in Corollary 53, “unimodal” cannot be strengthened to “log-concave” in; the
conclusion of Corollary 65. A counterexample is provided by the following
theorem (see Remark 67) which is a corollary of Theorems 38 and 64.

######  Theorem 66.

Let $k=2m-1\geq 1.$

  1. (1)

For $k=2m-1\geq 1$, if

$\mathfrak{g}_{k,r}=\mathfrak{p}^{A}\frac{k|\overbrace{4|\cdots|4}^{r}}{k+2|\underbrace{4|\cdots|4}_{r-1}|2},$

then $\mathfrak{g}_{k,r}$ has the unimodal spectrum property with spectrum
equal to $\\{-1^{2r},0^{6r-1},1^{6r-1},2^{2r}\\}$ if $k=1$,
$\\{-2,-1^{2r+2},0^{6r+2},1^{6r+2},2^{2r+2},3\\}$ if $k=3$, and

$\displaystyle\\{-m,(-m+1)^{3}$
$\displaystyle,-1^{2(k+r)-5},0^{2(k+3r)-4},1^{2(k+3r)-4},2^{2(k+r)-5},m^{3},m+1\\}$
$\displaystyle\cup\bigcup_{i=2}^{m-2}\\{(-m+i)^{4i-2},(m-i+1)^{4i-2}\\},$

if $k>3$.

  1. (2)

For $k=2m-1\geq 1$, if

$\mathfrak{g}_{k,r}=\mathfrak{p}^{A}\frac{k|\overbrace{4|\cdots|4}^{r}|2}{k+2|\underbrace{4|\cdots|4}_{r}},$

then $\mathfrak{g}_{k,r}$ has the unimodal spectrum property with spectrum
equal to $\\{-1^{2r+1},0^{6r+2},1^{6r+2},2^{2r+1}\\}$ if $k=1$,
$\\{-2,-1^{2r+3},0^{6r+5},1^{6r+5},2^{2r+3},3\\}$ if $k=3$, and

$\displaystyle\\{-m,(-m+1)^{3}$
$\displaystyle,-1^{2(k+r)-4},0^{2(k+3r)-1},1^{2(k+3r)-1},2^{2(k+r)-4},m^{3},m+1\\}$
$\displaystyle\cup\bigcup_{i=2}^{m-2}\\{(-m+i)^{4i-2},(m-i+1)^{4i-2}\\},$

if $k>3$.

######  Remark 67.

Utilizing Theorem 66, we can construct examples of Frobenius, type-A seaweeds
which do not have the log-concave spectrum property. For example, let
$\mathfrak{g}=\mathfrak{p}^{A}\frac{5|2}{7}$ and
$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{5|4|4|2}{7|4|4}.$ Recall from
Theorem 38 that the spectrum of $\mathfrak{g}$ is
$\\{-3,-2^{3},-1^{6},0^{9},1^{9},2^{6},3^{3},4\\}$. On the other hand, by
Theorem 64, we find that the spectrum of $\mathfrak{g}^{\prime}$ is
$\\{-3,-2^{3},-1^{10},0^{21},1^{21},2^{10},3^{3},4\\}.$ Clearly,
$\mathfrak{g}^{\prime}$ does not have the log-concave spectrum property.

######  Remark 68.

Note that for fixed $k$ and varying values of $r$, the collection of distinct
eigenvalues in the spectra of the Frobenius, type-A seaweeds considered in
Theorem 66 is fixed.

######  Remark 69.

Combining Theorems 64 and 66 and Corollary 65 with Corollaries 14, 17, and 18,
we obtain similar results for related families of Frobenius, type-A seaweeds.

Considering the results found above, as well as some experimental evidence, we
are naturally led to the following conjectures.

######  Conjecture 70.

Let
$\mathfrak{g}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|k}{b_{1}|\ldots|b_{t}}$,
for $k\geq 1$, be a Frobenius, type-A seaweed with spectrum $S$. If

$\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{2k|\cdots|2k}^{r}}{b_{1}|\ldots|b_{t}|\underbrace{2k|\cdots|2k}_{r-1}|k}\quad\text{or}\quad\mathfrak{g}^{\prime}=\mathfrak{p}^{A}\frac{a_{1}|\ldots|a_{m}|\overbrace{2k|\cdots|2k}^{r}|k}{b_{1}|\ldots|b_{t}|\underbrace{2k|\cdots|2k}_{r}},$

for $r\geq 1$, then $\mathfrak{g}^{\prime}$ is a Frobenius, type-A seaweed
with spectrum $S\cup S^{\prime}$, where $S^{\prime}$ is a multiset consisting
of values contained in $S$. Moreover, if $\mathfrak{g}$ has the unimodal
spectrum property, then so does $\mathfrak{g}^{\prime}$.

######  Conjecture 71.

If

$\mathfrak{g}=\mathfrak{p}^{A}\frac{\overbrace{2k|\cdots|2k}^{r}|1}{2kr+1},$

for $k,r\geq 1$, then $\mathfrak{g}$ is Frobenius, and the set of distinct
eigenvalues contained in the spectrum of $\mathfrak{g}$ is equal to the
collection of integers contained in the interval

$\begin{cases}[-2k+1,2k],&\text{if}~{}r~{}\text{is odd};\\\
[-k,k+1],&\text{if}~{}r~{}\text{is even}.\end{cases}$

Moreover, $\mathfrak{g}$ has the unimodal spectrum property.

######  Remark 72.

The family of type-A seaweeds considered in Conjecture 71 provide us with yet
another example of a Frobenius, type-A seaweed which does not have the log-
concave spectrum property. One can compute that the spectrum of
$\mathfrak{g}=\mathfrak{p}^{A}\frac{8|8|8|1}{25}$ is equal to

$\\{-7^{2},-6^{5},-5^{8},-4^{13},-3^{23},-2^{37},-1^{52},0^{64},1^{64},2^{52},3^{37},4^{23},5^{13},6^{8},7^{5},8^{2}\\}.$

Notice that $8^{2}=64<65=5*13,$ so $\mathfrak{g}$ does not have the log-
concave spectrum property.

######  Conjecture 73.

If

$\mathfrak{g}_{k,r}=\mathfrak{p}^{A}\frac{\overbrace{2k|\cdots|2k}^{r}|1}{1|\underbrace{2k|\cdots|2k}_{r}},$

for $k,r\geq 1$, then the set of distinct eigenvalues contained in the
spectrum of $\mathfrak{g}_{k,r}$ is equal to the collection of integers
contained in the interval $[-k+1,k]$, and $\mathfrak{g}_{k,r}$ has the log-
concave spectrum property. Moreover, if $i\in[-k+1,0]$, then the multiplicity
of $i$ in the spectrum of $\mathfrak{g}_{k,r}$ is equal to the multiplicity of
$i-1$ in the spectrum of $\mathfrak{g}_{k+1,r}$; similarly, if $i\in(0,k]$,
then the multiplicity of $i$ in the spectrum of $\mathfrak{g}_{k,r}$ is equal
to the multiplicity of $i+1$ in the spectrum of $\mathfrak{g}_{k+1,r}$.

##  5 Epilogue

The inductive proof methods outlined in this article correspond naturally to a
set of “winding moves” on the meander of a Frobenius, type-A seaweed. Such
winding moves were first introduced by Panyushev ([16], 2001) – and later
recast graph-theoretically by Coll et al. ([2], 2012) – for the purpose of
simplifying computations involving seaweeds. The initial, overall goal of our
study was to prove Conjecture 8 by tracking the effects of the winding moves
on the spectra of Frobenius, type-A seaweeds via (extended) spectrum matrices.
It quickly became clear, however, that the spectrum of a generic Frobenius,
type-A seaweed grew increasingly wild upon iterative applications of winding
moves; e.g., compare Theorems 38 and 45. Thus, we still have the following
general question.

######  Question 74.

How do the winding moves on the meander of a Frobenius, type-A seaweed
$\mathfrak{g}$ affect the spectrum of $\mathfrak{g}?$

Note that upon successfully answering this question, one should obtain a
combinatorial proof of Conjecture 8.

While the results of Section 3 provide significant insight into answering
Question 74, such results have led to further questions. In particular, to the
authors’ knowledge, this article marks the first acknowledgement of the log-
concave spectrum property for seaweeds. While not a property of Frobenius,
type-A seaweeds in general, per Section 4, the results of Section 3 suggest
that all Frobenius, maximal parabolic, type-A seaweeds do possess the log-
concave spectrum property. Thus, we have the following conjecture.

######  Conjecture 75.

If $\mathfrak{g}$ is a Frobenius, maximal parabolic, type-A seaweed, then
$\mathfrak{g}$ has the log-concave spectrum property.

The conjecture above is certainly not comprehensive. For example, the seaweed
$\mathfrak{p}^{A}\frac{1|2|2|4|4|2}{2|2|2|1|4|4}$ is not parabolic, yet it
possesses the log-concave spectrum property. We pose the following question.

######  Question 76.

Which Frobenius, type-A seaweeds admit the log-concave spectrum property?

In contrast to Section 3, in which the focus was primarily on applications of
winding moves that increase the values of the parts in the defining
compositions of the seaweeds while leaving the number of parts fixed, the
motivation for Section 4 was to increase the number of parts without altering
the size of each. In addition to providing a construction procedure for
Frobenius, type-A seaweeds without the log-concave spectrum property,
applications of such winding moves preserve the set of distinct eigenvalues in
the spectra of the associated algebras. To the author’s knowledge, this is the
first instance of such behavior being noted, and it leads to the following
question.

######  Question 77.

Are the methods discussed in Section 4 the only means by which to construct a
family of Frobenius, type-A seaweeds with a fixed set of distinct eigenvalues.

Recall that, in Section 3, our proof methods required the determination of
formulas for the extended spectrum of each Frobenius, type-A seaweed. While it
follows from its definition that the extended spectrum of a Frobenius, type-A
seaweed consists entirely of integers centered at 0, it is unclear whether
these integers are generally unbroken and whether their multiplicities
generally form a log-concave (resp., unimodal) sequence. Thus, we ask

######  Question 78.

Is the set of values in the extended spectrum of a Frobenius type-A seaweed
unbroken?

and

######  Question 79.

If the values in the extended spectrum of a Frobenius, type-A seaweed
$\mathfrak{g}$ are written in increasing order, is the corresponding sequence
of multiplicities log-concave (resp., unimodal)?

While an answer to Question 78 would be an interesting result in its own
right, consideration of Question 79 may also provide insight into an answer to
Question 74. Part of the challenge with using meanders when attempting to
answer Question 74 is that it can be difficult to discern which paths in a
given meander contribute their weights to the spectrum of the associated
algebra. However, this particular difficulty is resolved when computing the
extended spectrum of a Frobenius, type-A seaweed $\mathfrak{g}$, since the
weights of all paths in $\overrightarrow{M}(\mathfrak{g})$ are counted. Thus,
it is plausible that a purely graph-theoretic approach may yield answers to
Question 79, which, in turn, may provide an avenue toward a proof (or
counterexample) of Conjecture 8.

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