###  8.1. Examples distinguished by their 0th homology

We shall use the results of the previous section to construct, for each $k\geq
2$ and each $k$-tuple of integers $(m_{1},m_{2},\ldots,m_{k})$, two series of
pairwise non-isomorphic $k$-graphs with two vertices and different groups

$\mathbb{Z}C_{o}/\Big{(}\sum^{k}_{i=1}\operatorname{im}(I-M(C)^{t}_{i})\Big{)}.$

Our construction consists of two steps: first, we construct a family of cube
complexes with two vertices, covered by products of $k$ trees, and second, we
explain how to get a $k$-graph from each complex.

For background on cube complexes covered by products of $k$ trees see [26] and
references in the paper.

Step 1. Let $X_{1},...,X_{k}$ be distinct alphabets, such that
$\left|X_{i}\right|=m_{i}$ and

$X_{i}=\\{x_{1}^{i},x_{2}^{i},...,x_{m_{i}}^{i}\\}.$

Let $F_{i}$ be the free group generated by $X_{i}$. The direct product

$G=F_{1}\times F_{2}\times\ldots\times F_{k}$

has presentation

$G=\langle X_{1},X_{2},...,X_{k}|[x^{i}_{s},x^{j}_{l}]=1,i\neq
j=1,...,k;s=1,...,m_{i};l=1,...,m_{j}\rangle,$

where $[x,y]$ denotes the commutator $xyx^{-1}y^{-1}$.

The group $G$ acts simply and transitively on a Cartesian product $\Delta$ of
$k$ trees $T_{1},T_{2},..,T_{k}$ of valencies $2m_{1},2m_{2},...,2m_{k}$
respectively: each $T_{i}$ is identified with the Cayley tree of $F_{i}$, and
the action of $G$ is the coordinatewise action of the component groups
$F_{i}$.

The quotient of this action is a cube complex $P$ with one vertex $v$ such
that the universal cover of $P$ is $\Delta$. The edges of the cube complex $P$
are naturally labelled by elements of $X=X_{1}\cup X_{2}...\cup X_{k}$, and
are naturally oriented by the usual algebraic ordering on $F_{i}$. The
$1$-skeleton of $P$ is a wedge of $\sum_{i=1}^{k}m_{i}$ circles.

We construct a family of double covers of $P$ in the following way. Consider a
labelling $\ell:X\to\mathbb{Z}_{2}$ of the elements of $X$ (equivalently the
edges of the $1$-skeleton of $P$). We obtain a cover $P^{2}_{\ell}$ of $P$
whose vertex set is $\\{v\\}\times\mathbb{Z}_{2}$ and whose set of $1$-cubes
is $X\times\mathbb{Z}_{2}$, with range and domain maps given by
$r(x,i)=(r(x),i)$ and $s(x,i)=(s(x),i+\ell(x))$. Specifically, $P^{2}_{\ell}$
is the quotient of $\Delta$ by the action of the kernel of the homomorphism
$G\to\mathbb{Z}_{2}$ induced by $\ell$. Observe that in $P^{2}_{\ell}$, for a
given $x\in X$ either $(x,0)$ and $(x,1)$ are loops based at $(v,0)$ and
$(v,1)$ (if $\ell(x)=0$), or $(x,0)$ is an edge from $(v,1)$ to $(v,0)$ and
$(x,1)$ is an edge from $(v,0)$ to $(v,1)$.

So Figure 1 illustrates the $2$-cubes and part of the $1$-skeleton of $P$
corresponding to symbols $a\in X_{i}$ and $b\in X_{j}$ with $i\not=j$ in which
$\ell(a)=1$ and $\ell(b)=0$; Figure 2 illustrates the corresponding $2$-cubes
and part of the $1$-skeleton if $\ell(a)=\ell(b)=1$.

$(v,0)$$(v,1)$$(v,0)$$(v,1)$$(a,1)$$(b,0)$$(a,1)$$(b,1)$$(v,1)$$(v,0)$$(v,1)$$(v,0)$$(a,0)$$(b,0)$$(a,0)$$(b,1)$$(v,0)$$(v,1)$$(a,1)$$(a,0)$$(b,0)$$(b,1)$
Figure 1. $\ell(a)=1$ and $\ell(b)=0$
$(v,0)$$(v,1)$$(v,1)$$(v,0)$$(a,1)$$(b,1)$$(a,0)$$(b,0)$$(v,1)$$(v,0)$$(v,0)$$(v,1)$$(a,0)$$(b,0)$$(a,1)$$(b,1)$$(v,0)$$(v,1)$$(b,1)$$(b,0)$$(a,1)$$(a,0)$
Figure 2. $\ell(a)=\ell(b)=1$

We will consider two specific labellings $\ell_{u}$ and $\ell_{m}$ (the $u$
and $m$ stand for “uniform” and “mixed”). The uniform labelling $\ell_{u}$ is
given by $\ell_{u}(x)=1$ for all $x$, whereas the mixed labelling $\ell_{m}$
satisfies $\ell_{m}|_{X_{1}}\equiv 0$ and $\ell_{m}|_{X\setminus X_{1}}\equiv
1$. So under $\ell_{u}$ all $2$-squares are as in Figure 2; but under
$\ell_{m}$ squares in which $b$ belongs to $X_{1}$ are as in Figure 1, and the
remaining squares are as in Figure 2. Figure 3 illustrates a $3$-cube in the
cube complex for $\ell_{u}$ (left) and a $3$-cube in the cube complex for
$\ell_{m}$ in which $a$ belongs to $X_{1}$ and the edges $b,c$ belong to
$X\setminus X_{1}$ (right).

$(v,0)$$(v,1)$$(v,1)$$(v,0)$$(v,1)$$(v,0)$$(v,0)$$(v,1)$$(a,1)$$(b,1)$$(c,1)$$(b,0)$$(c,0)$$(a,0)$$(b,0)$$(a,0)$$(c,0)$$(c,1)$$(b,1)$$(a,1)$
$(v,0)$$(v,0)$$(v,1)$$(v,1)$$(v,1)$$(v,1)$$(v,0)$$(v,0)$$(a,0)$$(b,1)$$(c,1)$$(b,1)$$(c,1)$$(a,1)$$(b,0)$$(a,1)$$(c,0)$$(c,0)$$(b,0)$$(a,0)$
Figure 3. A $3$-cube for $\ell_{u}$ (left) and for $\ell_{m}$ (right)

Step 2. For each of $\star=u$ and $\star=m$, we explain how to construct a
$k$-graph $C_{\star}$ from $P^{2}_{\ell_{\star}}$, by specifying their
skeletons and factorisation rules as in [7]. For either value of $\star$, we
define $(C_{\star})_{o}=\\{(v,0),(v,1)\\}$, and for each $i\leq k$, we define
$(C_{\star})_{e_{i}}$ to be the set

$X_{i}\times\mathbb{Z}_{2}\sqcup\\{(\overline{x},j):x\in
X_{i},j\in\mathbb{Z}_{2}\\},$

a disjoint union of two copies $X\times\mathbb{Z}_{2}$. The range and source
maps on $(C_{\star})_{e_{i}}$ restrict to those in $P^{2}_{\ell_{\star}}$ on
$X_{i}\times\mathbb{Z}_{2}$, and we define $r(\overline{x},j)=s(x,j)$ and
$s(\overline{x},j)=r(x,j)$. The factorisation rules are as follows: for each
$2$-cube

$(v,0)$$(v,1)$$(v,0)$$(v,1)$$(a,i_{a})$$(b,i_{b})$$(a,j_{a})$$(b,j_{b})$

in $P^{2}_{\ell_{\star}}$, we have four factorisation rules:

$\displaystyle(b,i_{b})(a,i_{a})$ $\displaystyle=(a,j_{a})(b,j_{b}),$
$\displaystyle\qquad(a,i_{a})(\overline{b},j_{b})$
$\displaystyle=(\overline{b},i_{b})(a,j_{a}),$
$\displaystyle(\overline{a},j_{a})(b,i_{b})$
$\displaystyle=(b,j_{b})(\overline{a},i_{a}),$
$\displaystyle\qquad(\overline{b},j_{b})(\overline{a},j_{a})$
$\displaystyle=(\overline{a},i_{a})(\overline{b},i_{b})$

That these factorisation rules satisfy the associativity condition of [7]
follows from a routine calculation using that the $P^{2}_{\ell_{\star}}$ is a
quotient of of a direct product of trees.

To proceed, we define matrices

$D_{i}:=\begin{pmatrix}2m_{i}&0\\\ 0&2m_{i}\end{pmatrix}\quad\text{ and }\quad
T_{i}:=\begin{pmatrix}0&2m_{i}\\\ 2m_{i}&0\end{pmatrix}.$

Observe that in $C_{u}$ the adjacency matrices $M(C)_{i}$ are equal to
$T_{i}$, while in $C_{m}$, we have $M(C)_{1}=D_{1}$ and $M(C)_{i}=T_{i}$ for
$2\leq i\leq k$. Routine calculations using this show that

$\displaystyle\mathbb{Z}(C_{u})_{o}/\Big{(}\sum^{k}_{i=1}\operatorname{im}(I-M(C_{u})^{t}_{i})\Big{)}$
$\displaystyle\cong\mathbb{Z}/\gcd(4m_{1}^{2}-1,\dots,4m_{k}^{2}-1)\mathbb{Z},\quad\text{
and}$
$\displaystyle\mathbb{Z}(C_{m})_{o}/\Big{(}\sum^{k}_{i=1}\operatorname{im}(I-M(C_{m})^{t}_{i})\Big{)}$
$\displaystyle\cong\mathbb{Z}/\gcd(2m_{1}-1,4m_{2}^{2}-1,\dots,4m_{k}^{2}-1)\mathbb{Z}.$

######  Corollary 8.1.

There are at least two non-isomorphic groups $\mathscr{G}(C)$ for each $k\geq
2$ and infinitely many choices of alphabets $Y_{1},\ldots,Y_{k}$ of sizes
$2m,\ldots,2m$.

###### Proof.

Take $m_{1}=\cdots=m_{k}$ in the examples above. We obtain
$H_{0}(D^{C_{u}}_{*},\partial_{*})\cong\mathbb{Z}/(4m^{2}-1)\mathbb{Z}$, and
since $2m-1$ divides $4m^{2}-1$ we have
$H_{0}(D^{C_{m}}_{*},\partial_{*})\cong\mathbb{Z}/(2m-1)\mathbb{Z}$. The
result then follows from Corollary 7.3. ∎

Note that the known groups $nV$ can be presented in our language using an
$n$-graph $C$ with $1\times 1$ adjacency matrices with single entry equal $2$.
It is relatively easy to check that each $\ker(\partial_{j})$ in Evans’
complex is a subgroup of a direct sum of kernels of the maps $I-M(C)^{t}_{i}$,
which in this instance are all equal to the $1\times 1$ matrix $(-1)$. So
$H_{j}(C)=0$ for $1\leq j\leq n$. Also,
$H_{0}(C)=\mathbb{Z}/\big{(}\sum^{n}_{i=1}I-M(C)^{t}_{i}\mathbb{Z})=\mathbb{Z}/(-1)\mathbb{Z}=0$.
Hence the homology groups of all of these $n$-graphs are trivial. In
particular none of the groups $\mathscr{G}(C)$ discussed above are isomorphic
to the groups $nV$.

###  8.2. Examples with nontrivial $k$th homology

Fix integers $k,R\geq 1$. We will construct a $k$-graph $C_{k,R}$ by
specifying its skeleton and factorisation rules as in [7].

The vertex set $V$ of the skeleton $E$ has $R+1$ elements (for example, we
could take $V=\\{1,\dots,R+1\\}$, but to lighten notation, we will avoid
choosing a particular enumeration).

For each $i\leq k$, the set of edges of $E$ of colour $i$ is

$\\{e^{i}_{v,m,w}:v\not=w\in V\text{ and }1\leq m\leq
2\\}\cup\\{e^{i}_{v,m,v}:v\in V\text{ and }1\leq m\leq 3\\}$

and the range and domain maps are given by $\mathbf{r}(e^{i}_{v,m,w})=v$ and
$\mathbf{d}(e^{i}_{v,m,w})=w$.

So for any two distinct vertices $v,w$, there are $2$ edges of each colour
pointing from $v$ to $w$ (and also from $w$ to $v$), and there are $3$ loops
of each colour at each vertex.

For example, in the skeleton of $C_{k,2}$, each of the singly-coloured
subgraphs is as in Figure 4.

$v$$w$$e^{i}_{w,1,v}$$e^{i}_{w,2,v}$$e^{i}_{v,1,w}$$e^{i}_{v,2,w}$$e^{i}_{v,1,v}$$e^{i}_{v,2,v}$$e^{i}_{v,3,v}$$e^{i}_{v,1,v}$$e^{i}_{v,2,v}$$e^{i}_{v,3,v}$
Figure 4. One of the singly-coloured subgraphs in the skeleton of $C_{k,2}$

We must now specify factorisation rules. For $j\not=l\leq k$, the
$jl$-coloured paths are those of the form $e^{j}_{u,m,v}e^{l}_{v,n,w}$, and we
must specify factorisation rules that determine range and source preserving
bijections between $jl$-coloured paths and $lj$-coloured paths and that
satisfy the associativity condition [7, Equation (3.2)].

We define them in two cases:

  * (F1)

if $u,v\in V$ are distinct, then we define
$e^{i}_{u,m,u}e^{j}_{u,n,v}=e^{j}_{u,n,v}e^{i}_{v,m,v}$.

  * (F2)

if either $u=v=w$ or $u\not=v$ and $v\not=w$, we define
$e^{i}_{u,m,v}e^{j}_{v,n,w}=e^{j}_{u,n,v}e^{i}_{v,m,w}$.

It is routine to check that these factorisation rules determine a complete
collection of squares as in [7, p.578]. Routine but tedious calculations also
verify the associativity condition [7, Equation (3.2)].

By [7, Theorem 4.4] there is a $k$-graph $C_{k,R}$ whose skeleton is the
coloured graph $E$, and whose factorisation rules are given by (F1) and (F2).
Since for all $v,w\in V$ we have $v(C_{k,R})_{e_{1}}w\not=\emptyset$, it is
immediate that $C_{k,R}$ is cofinal. To see that it is aperiodic, let
$B_{R+1}$ be the $1$-graph whose skeleton is a bouquet of $R+1$ loops at a
single vertex. Fix a vertex $v\in V$ and observe that the sub-$k$-graph
generated by the edges $\\{e^{i}_{(v,m,v)}:i\leq k\text{ and }m\leq R+1\\}$ is
isomorphic to the cartesian product $\prod^{k}_{i-1}B_{R+1}$ of $k$ copies of
$B_{R+1}$. Since the $1$-graph $B_{R+1}$ is aperiodic, so is the $k$-graph
$\prod^{k}_{i-1}B_{R+1}$, and so it contains an aperiodic infinite path. This
is then an aperiodic infinite path in $C_{k,R}$.

We now observe that for each $i\leq k$, the matrix $(1-M(C_{k,R})^{t}_{i})$ is
the $(R+1)\times(R+1)$ integer matrix

$A_{R}:=\left(\begin{matrix}-2&-2&\cdots&-2\\\ -2&-2&\cdots&-2\\\
\vdots&\vdots&&\vdots\\\ -2&-2&\cdots&-2\end{matrix}\right).$

Consequently $H_{k}(C_{k,R})=\ker(A_{R})\cong\mathbb{Z}^{R}$.

We have now proved the following.

######  Corollary 8.2.

For each positive integer $k$, there exists a family $\\{C_{k,R}:R\geq 2\\}$
of $k$-graphs, such that
$\mathscr{G}(C_{k,R})\cong\mathscr{G}(C_{k^{\prime},R^{\prime}})$ only if
$k=k^{\prime}$ and $R=R^{\prime}$. Moreover, for any $k,R$, the group
$\mathscr{G}(C_{k,R})$ is not isomorphic to $\mathscr{G(C^{\prime})}$ for any
$l$-graph $C^{\prime}$ with $l<k$.

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#  Microscopic derivation of transition-state theory for complex quantum
systems

K. Hagino  Department of Physics, Kyoto University, Kyoto 606-8502, Japan
G.F. Bertsch  Department of Physics and Institute for Nuclear Theory, Box
351560, University of Washington, Seattle, Washington 98915, USA

###### Abstract

The decay of quantum complex systems through a potential barrier is often
described with transition-state theory, which is also known as RRKM theory in
chemistry. Here we derive the basic formula for transition-state theory based
on a generic configuration-interaction Hamiltonian. To this end, we consider
two random Hamiltonians, which are coupled to intermediate configurations at a
barrier. Under a condition that the total decay probability of the post-
barrier configurations to open channels is large, we show that the
transmission coefficient from the first random Hamiltonian to the second is
given as a factorized form of the formation and the decay probabilities of
transition states. In that limit the transmission coefficient is found to be
independent of the decay widths of the configurations in the random
Hamiltonians. We also show that the transmission coefficient is reduced to a
Breit-Wigner form, which is well known for electronic conductance through
resonant tunneling states.

##  I Introduction

Transition-state theory is ubiquitous in physics and chemistry to calculate
reaction and decay rates for many-particle systems in the presence of a
barrier hanggi1990 ; tr96 ; weiss2012 ; BW39 ; MR51 . The assumptions in the
theory are clear in classical dynamics but less so in the quantum regime. For
fermionic systems of equal-mass particles, the Hamiltonian is often formulated
in a configuration-interaction (CI) representation. This motivates considering
models that exhibit the barrier dynamics in the CI framework to understand
conditions to support transition-state approximations.

##  II Model Hamiltonian

Following previous recent work, we consider here a Hamiltonian composed of
three parts: the barrier Hamiltonian $H_{2}$ and two statistical reservoirs
$H_{1},H_{3}$ for the states on either side, and the matrix $V_{12}$ and
$V_{23}$ coupling them together,

$H=\left(\begin{matrix}H_{1}&V_{12}&0\cr V_{12}^{T}&H_{2}&V_{32}^{T}\cr
0&V_{32}&H_{3}\end{matrix}\right).$ (1)

Here, $H_{k}~{}(k=$ 1,2, and 3) is a $N_{k}\times N_{k}$ Hermitian matrix,
while $V_{kl}$ is a $N_{k}\times N_{l}$ real matrix. The statistical
reservoirs are described by Hamiltonians of the Gaussian Orthogonal Ensemble
(GOE). The barrier Hamiltonian $H_{2}$ is composed of an arbitrary set of
configurations, each connected to states in the two reservoirs by the
submatrices $V_{12}$ and $V_{23}$. In this respect the models discussed here
are generalizations of the model in Ref. weidenmuller2022 , which has a single
state at the barrier top.

To complete a model for reactions, one also needs the coupling matrix elements
between the CI Hamiltonian and the external reaction channels. With those
ingredients the $S$-matrix for transitions from one channel to another can be
computed by standard linear algebra manipulations. If one is only interested
in reaction probabilities, the linear algebra can be collapsed to Datta’s
formula da01 ; da95 ; al21  for inelastic reactions, given by

$|S_{ab}(E)|^{2}={\rm
Tr}\left(\tilde{\Gamma}_{a}G(E)\tilde{\Gamma}_{b}G^{\dagger}(E)\right)$ (2)

Here $S_{ab}$ is the scattering $S$-matrix element connecting channels $a$ and
$b$ and $G$ is the Green’s function of the Hamiltonian in presence of decay
channels $i$ at reaction energy111It is implicitly assumed in Eq. (3) that the
$\tilde{\Gamma}_{i}$ are independent of $E$. $E$,

$G(E)=\left(H-i\sum_{c}\tilde{\Gamma}_{c}/2-E\right)^{-1}.$ (3)

The $\tilde{\Gamma}_{c}$ are matrices (of the same dimension as $H$) of decay
widths to external channels $i$. They have diagonal block structure given by

$\left(\begin{matrix}\tilde{\Gamma}_{c,\rm in}&0&0\cr 0&0&0\cr
0&0&0\end{matrix}\right)\,\,{\rm
or}\,\,\left(\begin{matrix}\tilde{\Gamma}_{c,\rm 1}&0&0\cr 0&0&0\cr
0&0&0\end{matrix}\right)\,\,{\rm or}\,\,\left(\begin{matrix}0&0&0\cr 0&0&0\cr
0&0&\tilde{\Gamma}_{c,\rm 3}\end{matrix}\right)$ (4)

depending on which subblock the channel $c$ connects to.

In general, one is interested in the reaction probability $T_{a3}$ from an
entrance channel $a={\rm in}$ to all possible decay channels in the second
reservoir,

$T_{a3}=\sum_{b\in 3}|S_{ab}|^{2}.$ (5)

Due to the block structures of $H$ and $\tilde{\Gamma}$ we only need the
$G_{13}$ block of the Green’s function

$G=\left(\begin{matrix}G_{11}&G_{12}&G_{13}\cr G_{21}&G_{22}&G_{23}\cr
G_{31}&G_{32}&G_{33}\end{matrix}\right)$ (6)

in Eq. (2). As derived in Appendix A, the submatrix $G_{13}$ reduces to

$G_{13}=G_{1}V_{12}G_{2}V_{32}^{T}G_{3},$ (7)

where $G_{1}$, $G_{3}$, and $G_{2}$ are given by

$\displaystyle G_{1}$ $\displaystyle=$
$\displaystyle\left(H_{1}-i\tilde{\Gamma}_{c,\rm
in}/2-i\tilde{\Gamma}_{c,1}/2-E\right)^{-1},$ (8) $\displaystyle G_{3}$
$\displaystyle=$
$\displaystyle\left(H_{3}-i\tilde{\Gamma}_{c,3}/2-E\right)^{-1},$ (9)
$\displaystyle G_{2}$ $\displaystyle=$
$\displaystyle\left(H_{2}-V_{12}^{T}G_{1}V_{12}-V_{32}^{T}G_{3}V_{32}-E\right)^{-1}.$
(10)

Substituting Eq. (7) into Eq. (2), the transmission coefficient is obtained as

$\displaystyle T_{\rm in,3}(E)$ $\displaystyle=$ $\displaystyle{\rm
Tr}[\tilde{\Gamma}_{c,\rm
in}(G_{1}V_{12}G_{2}V_{32}^{T}G_{3})\tilde{\Gamma}_{c,3}(G_{3}^{\dagger}V_{32}G_{2}^{\dagger}V_{12}^{T}G_{1}^{\dagger})],$
$\displaystyle=$ $\displaystyle{\rm
Tr}[(V_{12}^{T}G_{1}^{\dagger}\tilde{\Gamma}_{c,\rm
in}G_{1}V_{12})G_{2}(V_{23}G_{3}\tilde{\Gamma}_{c,3}G_{3}^{\dagger}V_{32})G_{2}^{\dagger}].$

We write the elements of the two GOE Hamiltonians as

$(H_{i})_{jk}=(H_{i})_{kj}=v_{i}\sqrt{1+\delta_{kj}}\,r_{ijk},$ (13)

where $r_{ijk}$ is a random number from a Gaussian distribution of unit
dispersion, $\langle r_{ijk}^{2}\rangle=1$. Then the average level density
$\rho_{i}$ at $E=0$ at the centers of the GOE Hamiltonians is given by

$\rho_{i}=\frac{N_{i}^{1/2}}{\pi v_{i}}.$ (14)

We set $E=0$ for the rest of this paper. Each state in $H_{2}$ is assumed to
couple to specific states in the GOE reservoirs. We parameterize the couplings
as 222We implicitly assume $N_{1}>N_{2}$ and $N_{3}>N_{2}$.

$(V_{12})_{jk}=v_{12}N_{1}^{1/2}\delta_{jk},~{}~{}~{}(V_{32})_{jk}=v_{32}N_{3}^{1/2}\delta_{jk}.$
(15)

This parameterization is not as restrictive as it may seem. Due to the GOE
invariance, the couplings can be to arbitrary orthogonal vectors in the GOE
spaces. The specific form of the coupling is such that the average matrix
element is the same as in $H_{i}$ if $v_{i2}=v_{i}$.

The matrices for the decay widths are assumed to be diagonal with elements

$(\tilde{\Gamma}_{1})_{jk}=\gamma_{1}\,\delta_{j,k},~{}~{}~{}(\tilde{\Gamma}_{3})_{kj}=\gamma_{3}\,\delta_{j,k},$
(16)

except for the entrance channel $a={\rm in}$, which couples to a single state
in the first reservoir,

$(\tilde{\Gamma}_{\rm in})_{jk}=\gamma_{\rm in}\,\delta_{j,{\rm
in}}\delta_{k,{\rm in}}.$ (17)

##  III Transition-state theory

We now examine how the average reaction probability depends on the parameters
of the model. Since transition-state theory deals with fluxes into or from a
statistical reservoir, it is convenient to define a transmission coefficient
$\mathcal{T}_{ci}$ of a channel $c$ into the reservoir $i$

$\mathcal{T}_{ci}=2\pi\rho_{i}\gamma_{c}$ (18)

and its sum over channels,

$\mathcal{T}_{i}=\sum_{c\in i}\mathcal{T}_{ci}.$ (19)

For small values of $\mathcal{T}_{i}$ it has the physical significance of the
transmission factor from an external channel into the statistical reservoir.
As shown in Ref. weidenmuller2022 , it is straightforward to carry out the
statistical averaging for Eq. (2) in the limit that $\mathcal{T}_{i}\gg 1$ for
both reservoirs.

We first examine the Green’s function $G_{2}$ and the coupling terms
$V_{i2}^{T}G_{i}V_{i2}$ in it. The average and standard deviation of the GOE
Green’s functions333 We note that there are mild restrictions on the ranges of
the parameters in Eq. (20). In practice, the widths associated with the
individual channels should be large compared to the level spacing in the GOE
but small with respect to the boundaries of its eigenspectrum. including the
decay-width matrices are given by weidenmuller2022 ; Lo2000 ; Fe2020 ; hagino-
bertsch2021

$\displaystyle\langle G_{1}\rangle_{jk}$ $\displaystyle=$
$\displaystyle\frac{\pi\rho_{1}}{N_{i}}\left[i\delta_{jk}\pm\left(\frac{2(1+\delta_{j,k})N_{i}}{\mathcal{T}_{1}}\right)^{1/2}\right.$
$\displaystyle\pm
i\left.\left(\frac{2(1+\delta_{j,k})N_{i}}{\mathcal{T}_{1}}\right)^{1/2}\right]$
(20)

and similarly for $G_{3}$. The fluctuations go to zero in the limit
$\mathcal{T}_{i}\gg 1$ and these terms in $G_{2}$ can be be replaced by $i\pi
v_{i2}^{2}\rho_{i}$ times the unit matrix. Thus the correlations between
$G_{2}$ and the other terms in Eq. (LABEL:eq:datta2) vanish, allowing it to be
evaluated as

$\bar{G}_{2}=(H_{2}-V_{12}^{T}\langle G_{1}\rangle V_{12}-V_{32}^{T}\langle
G_{3}\rangle V_{32}-E)^{-1}.$ (21)

The two terms in parentheses in Eq. (LABEL:eq:datta2) are independent of each
other so can also be replaced by their ensemble averages. As is shown in
Appendix B, they are given by

$\langle(V_{12}^{T}G_{1}\tilde{\Gamma}_{\rm
in}G_{1}^{\dagger}V_{12})_{jk}\rangle=\frac{\gamma_{\rm
in}}{N_{1}\gamma_{1}}2\pi v_{i2}^{2}\rho_{i}\delta_{jk}$ (22)
$\langle(V_{32}^{T}G_{3}\tilde{\Gamma}_{3}G_{3}^{\dagger}V_{32})_{jk}\rangle=2\pi
v_{32}^{2}\rho_{i}\delta_{jk}.$ (23)

in the limit $\mathcal{T}_{i}\gg 1$. A computer program comparing these
formulas with the full numerical evaluation of Eq. (2) is provided in the
Supplementary Material.

We can cast the formulas in a more transparent notation by defining decay
widths of the transition states to the right-hand and left-hand reservoirs as

$\Gamma_{R}=2\pi v_{32}^{2}\rho_{3}$ (24)

and

$\Gamma_{L}=2\pi v_{12}^{2}\rho_{1}.$ (25)

Using Eq. (48) in Appendix B, one obtains

$\displaystyle\langle T_{\rm in,3}\rangle$ $\displaystyle=$
$\displaystyle\frac{\mathcal{T}_{\rm
in}}{\mathcal{T}_{1}}\,\Gamma_{L}\Gamma_{R}\sum_{i,j}\langle|(G_{2})_{ij}|^{2}\rangle\,\sum_{b\in
3}\left(\frac{T_{b}}{\sum_{b^{\prime}\in 3}T_{b^{\prime}}}\right),$ (27)
$\displaystyle=$ $\displaystyle\frac{\mathcal{T}_{\rm
in}}{\mathcal{T}_{1}}\,\Gamma_{L}\Gamma_{R}\sum_{i,j}\langle|(G_{2})_{ij}|^{2}\rangle$

where $\mathcal{T}_{\rm in}=2\pi\Gamma_{\rm in}\rho_{1}/N_{1}$. It is
remarkable that the ensemble average of the transmission coefficient is
independent of $\Gamma_{3}$, and thus the insensitive property bertsch-
hagino2021 ; hagino-bertsch2021 ; bertsch-hagino2023  is realized.

Notice that $G_{2}$ in Eq. (10) can be written

$G_{2}=(H_{2}-i(\Gamma_{L}/2+\Gamma_{R}/2)\mathbb{1})^{-1}.$ (28)

Then if $H_{2}$ is diagonal with matrix elements
$(H_{2})_{ij}=E_{i}\delta_{i,j}$, the transmission coefficient becomes

$\langle T_{\rm in,3}\rangle=\frac{\mathcal{T}_{\rm
in}}{\mathcal{T}_{1}}\sum_{i}\frac{\Gamma_{L}\Gamma_{R}}{E_{i}^{2}+(\Gamma_{L}+\Gamma_{R})^{2}/4}.$
(29)

This is a well-known formula for electron transport through intermediate
resonances alhassid2000 ; bertsch2014 . It agrees with an underlying
assumption in transition-state models, that the contributions of the
individual transition states are additive to the total transmission
probability, provided flux conservation is maintained through the calculated
decay rates444This is not the case for the probability fluxes through the
individual transition states. $\Gamma_{i}$.

The formula also shows that the contribution to the transmission coefficient
is suppressed when the energy of a bridge state is outside the range of
$\pm(\Gamma_{L}+\Gamma_{R})/2$ around the incident energy. This is marked
contrast to models in which the transition state is an internal channel that
remains open at all energies above the threshold. To maintain the
correspondence to the CI formulation, one would have to include highly excited
configurations that carry momentum along a collective coordinate. At some
point the model would break down because the coupling matrix element would
become small compared to $v_{i}$, the coupling strength of the configurations
within the GOE’s.

##  IV summary

While transition-state theory for decay of quantum complex systems is usually
derived with a statistical approach, we have successfully derived it starting
from a matrix Hamiltonian as is commonly used in configuration-interaction
formulations. To this end, we considered two reservoirs described by random
matrices. One of the configurations in the first reservoir undergoes a
transitions to configurations in the second reservoir through bridge
configurations between them. A potential barrier may exist for the bridge
configurations. This generalizes a model with a single barrier configuration
that was discussed by Weidenmüller weidenmuller2022 .

As in Ref. weidenmuller2022 , we have shown that the average transmission
coefficient from the entrance configuration to configurations in the second
reservoir can be factorized into a product form of the formation and the decay
probabilities of transition channels, in the limit of $\mathcal{T}_{3}\gg 1$.
This is also a consequence of the usual starting point of transition state
theory, that once the system passes the barrier, it never comes back.

If the condition $\mathcal{T}_{1}\gg 1$ is also satisfied, the transmission
coefficient is further simplified to a product of the population probability
of the first reservoir, the transmission coefficient over the barrier, and the
decay probability of the configurations in the second reservoir. In that case
the transmission coefficient can be expressed in terms of Breit-Wigner
resonance decays, as has been long known in the field of electron transport.

Transition-state theory is a landmark framework for decays of quantum complex
systems, but conditions for transition-state theory to work have not yet been
clarified well. The microscopic derivation based on the random matrix approach
shown in this paper provides a necessary condition for transition-state theory
to work. Such consideration would be important in the decay of complex systems
at energies close to the barrier top.

###### Acknowledgements.

We thank Hans Weidenmüller for useful discussions. This work was supported in
part by JSPS KAKENHI Grant Numbers JP19K03861 and JP23K03414.

##  Appendix A the Green’s function for a block-triagonal Hamiltonian

We invert the matrix

$H-i\tilde{\Gamma}_{\rm
in}/2-i\tilde{\Gamma}_{1}/2-i\tilde{\Gamma}_{3}/2-E=\left(\begin{matrix}\tilde{H}_{1}&V_{12}&0\cr
V_{12}^{T}&\tilde{H}_{2}&V_{23}\cr
0&V_{23}^{T}&\tilde{H}_{3}\end{matrix}\right),$ (30)

where $\tilde{H}_{i}$ are defined as

$\displaystyle\tilde{H}_{1}$ $\displaystyle\equiv$ $\displaystyle
H_{1}-i\Gamma_{\rm in}/2-i\Gamma_{1}/2-E,$ (31) $\displaystyle\tilde{H}_{2}$
$\displaystyle\equiv$ $\displaystyle H_{2}-E,$ (32)
$\displaystyle\tilde{H}_{3}$ $\displaystyle\equiv$ $\displaystyle
H_{1}-i\Gamma_{3}/2-E.$ (33)

The Green’s function (6) satisfies the relation,

$\left(\begin{matrix}\tilde{H}_{1}&V_{12}&0\cr
V_{12}^{T}&\tilde{H}_{2}&V_{32}^{T}\cr
0&V_{32}&\tilde{H}_{3}\end{matrix}\right)\left(\begin{matrix}G_{11}&G_{12}&G_{13}\cr
G_{21}&G_{22}&G_{23}\cr
G_{31}&G_{32}&G_{33}\end{matrix}\right)=\left(\begin{matrix}1&0&0\cr 0&1&0\cr
0&0&1\end{matrix}\right),$ (34)

from which one finds

$\displaystyle\tilde{H}_{1}G_{13}+V_{12}G_{32}^{T}=0,$ (35) $\displaystyle
V_{12}^{T}G_{13}+\tilde{H}_{2}G_{23}+V_{23}G_{33}=0,$ (36) $\displaystyle
V_{32}G_{23}+\tilde{H}_{3}G_{33}=1.$ (37)

From Eqs. (35) and (37), $G_{13}$ and $G_{33}$ read

$G_{13}=-\tilde{H}_{1}^{-1}V_{12}G_{23},$ (38)

and

$G_{33}=\tilde{H}_{3}^{-1}-\tilde{H}_{3}^{-1}V_{32}G_{23},$ (39)

respectively. Substituting these into Eq. (36), one obtains

$G_{23}=-(\tilde{H}_{2}-V_{12}^{T}\tilde{H}_{1}^{-1}V_{12}-V_{32}^{T}\tilde{H}_{3}^{-1}V_{32})^{-1}V_{32}^{T}\tilde{H}_{3}^{-1}.$
(40)

Combining Eqs. (38) and (40), one finally obtains Eq. (7).

Following a similar procedure, one can also derive

$G_{11}=G_{1}+G_{1}V_{12}G_{2}V_{12}^{T}G_{1}.$ (41)

##  Appendix B Ensemble average of $VG\tilde{\Gamma}G^{\dagger}V^{T}$

In this Appendix, we evaluate the ensemble average of a matrix
$VG\tilde{\Gamma}G^{\dagger}V^{T}$, where the elements of $V$ are Gaussian-
distributed random numbers with $\langle v_{jk}^{2}\rangle=v^{2}$ and the
Green’s function $G$ is given by $G=(H-i\tilde{\Gamma}/2)^{-1}$. Here $H$ is
an element of the $N\times N$ GOE, and $\tilde{\Gamma}$ is a constant times
the unit matrix $\tilde{\Gamma}_{jk}=\gamma\delta_{k,j}$. We closely follow
Refs. bertsch-hagino2021 ; hagino-bertsch2021  to carry out the ensemble
averaging. We first write the elements of the Green’s function as

$G_{jk}=\sum_{\lambda}\frac{\phi_{\lambda,j}\phi_{\lambda,k}}{E_{\lambda}-i\gamma/2},$
(42)

where $E_{\lambda}$ are the eigenvalues of the Hamiltonian $H$ and
$\phi_{\lambda}$ are the corresponding eigenfunctions. The ensemble average of
$VG\tilde{\Gamma}G^{\dagger}V^{T}$ then reads,

$\displaystyle\langle(VG\tilde{\Gamma}G^{\dagger}V^{T})_{jk}\rangle$
$\displaystyle=\gamma\sum_{m}\sum_{\lambda,\lambda^{\prime}}\left\langle\frac{(\mbox{\boldmath$V$}_{j}\cdot\mbox{\boldmath$\phi$}_{\lambda})(\mbox{\boldmath$V$}_{k}\cdot\mbox{\boldmath$\phi$}_{\lambda^{\prime}})\phi_{\lambda}(m)\phi_{\lambda^{\prime}}(m)}{(E_{\lambda}-i\gamma/2)(E_{\lambda^{\prime}}+i\gamma/2)}\right\rangle,$
(43)

where $\mbox{\boldmath$V$}_{i}\cdot\mbox{\boldmath$\phi$}_{\lambda}$ is
defined as
$\mbox{\boldmath$V$}_{i}\cdot\mbox{\boldmath$\phi$}_{\lambda}=\sum_{k}V_{ik}\phi_{\lambda,k}$.
We now assert that in the limit $N\gg 1$ the ensemble average can be factored
into a product of three averages:

$\langle\sum_{m}\phi_{\lambda,m}\phi_{\lambda^{\prime},m}\rangle=\delta_{\lambda,\lambda^{\prime}},$
(44)
$\langle(\mbox{\boldmath$V$}_{i}\cdot\mbox{\boldmath$\phi$}_{\lambda})(\mbox{\boldmath$V$}_{j}\cdot\mbox{\boldmath$\phi$}_{\lambda})\rangle=v^{2}\delta_{i,j},$
(45)

and

$\left\langle\sum_{\lambda}\frac{1}{E_{\lambda}^{2}+\gamma^{2}/4}\right\rangle=\frac{2\pi}{\gamma}.$
(46)

The last two equations are derived in more detail in Ref. bertsch-hagino2021 ;
hagino-bertsch2021 . The final result is

$\left\langle(VG\tilde{\Gamma}G^{\dagger}V^{T})_{jk}\right\rangle=2\pi
v^{2}\rho_{0}\,\delta_{j,k}.$ (47)

We also need the ensemble average of $VG\tilde{\Gamma}_{\rm
in}G^{\dagger}V^{T}$ with $G=(H-i\tilde{\Gamma}/2-i\tilde{\Gamma}_{\rm
in}/2)^{-1}$, where $\tilde{\Gamma}_{\rm in}$ is given by Eq. (17). In this
case, the sum over $m$ is restricted to $m={\rm in}$ in Eq. (43). Due to the
invariance of the averages under unitary transformations, Eq. (44) becomes
$\langle\phi_{\lambda,{\rm
in}}\phi_{\lambda^{\prime},\rm{in}}\rangle=\delta_{\lambda,\lambda^{\prime}}/N$.
The final result is

$\left\langle(VG\tilde{\Gamma}_{\rm
in}G^{\dagger}V^{T})_{jk}\right\rangle=\frac{\Gamma_{\rm in}}{N\gamma}2\pi
v^{2}\rho_{0}\,\delta_{j,k}.$ (48)

.

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Lower Bounds for Numbers of Real Self-Dual Spaces in Problems of Schubert
Calculus

Lower Bounds for Numbers of Real Self-Dual Spaces
in Problems of Schubert Calculus

Kang LU

K. Lu

Department of Mathematical Sciences, Indiana University – Purdue University
Indianapolis,
402 North Blackford St, Indianapolis, IN 46202-3216, USA<EMAIL_ADDRESS>
Received November 27, 2017, in final form May 07, 2018; Published online May
14, 2018

The self-dual spaces of polynomials are related to Bethe vectors in the Gaudin
model associated to the Lie algebras of types B and C. In this paper, we give
lower bounds for the numbers of real self-dual spaces in intersections of
Schubert varieties related to osculating flags in the Grassmannian. The higher
Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate
indefinite Hermitian form. Our bound comes from the computation of the
signature of this form.

real Schubert calculus; self-dual spaces; Bethe ansatz; Gaudin model

14N99; 17B80; 82B23

##  1 Introduction

It is well known that the problem of finding the number of real solutions to
algebraic systems is very difficult, and not many results are known. In
particular, the counting of real points in problems of Schubert calculus in
the Grassmannian has received a lot of attention, see [2, 5, 6, 7, 13, 19, 20]
for example. In this paper, we give lower bounds for the numbers of real self-
dual spaces in intersections of Schubert varieties related to osculating flags
in the Grassmannian.

We define the Grassmannian $\operatorname{Gr}(N,d)$ to be the set of all
$N$-dimensional subspaces of the $d$-dimensional space $\mathbb{C}_{d}[x]$ of
polynomials in $x$ of degree less than $d$. In other words, we always assume
for $X\in\operatorname{Gr}(N,d)$, we have $X\subset\mathbb{C}_{d}[x]$. Set
$\mathbb{P}^{1}=\mathbb{C}\cup\\{\infty\\}$. Then, for any
$z\in{\mathbb{P}}^{1}$, we have the osculating flag $\mathcal{F}(z)$, see
(4.1), (4.2). Denote the Schubert cells corresponding to $\mathcal{F}(z)$ by
$\Omega_{\xi}(\mathcal{F}(z))$, where
$\xi=(d-N\geqslant\xi_{1}\geqslant\xi_{2}\geqslant\cdots\geqslant\xi_{N}\geqslant
0)$ are partitions. Then the set $\Omega_{\bm{\xi},\bm{z}}$ consists of spaces
$X\in\operatorname{Gr}(N,d)$ such that $X$ belongs to the intersection of
Schubert cells $\Omega_{\xi^{(i)}}(\mathcal{F}(z_{i}))$ for
$\bm{z}=(z_{1},\dots,z_{n})$ and
$\bm{\xi}=\big{(}\xi^{(1)},\dots,\xi^{(n)}\big{)}$, where all
$z_{i}\in{\mathbb{P}}^{1}$ are distinct and $\xi^{(i)}$ are partitions, see
(4.3). A point $X\in\operatorname{Gr}(N,d)$ is called _real_ if it has a basis
consisting of polynomials with all coefficients real. A lower bound for the
number of real points in $\Omega_{\bm{\xi},\bm{z}}$ is given in [13].

Let $X\in\operatorname{Gr}(N,d)$ be an $N$-dimensional subspace of polynomials
in $x$. Let $X^{\vee}$ be the $N$-dimensional space of polynomials which are
Wronskian determinants of $N-1$ elements of $X$

$\displaystyle
X^{\vee}=\big{\\{}\det\big{(}d^{i-1}\varphi_{j}/dx^{i-1}\big{)}_{i,j=1}^{N-1},\,\varphi_{j}(x)\in
X\big{\\}}.$

The space $X$ is called _self-dual_ if $X^{\vee}=\psi\cdot X$ for some
polynomial $\psi(x)$, see [16]. We define $\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$
the subset of $\Omega_{\bm{\xi},\bm{z}}$ consisting of all self-dual spaces.
Our main result of this paper is a lower bound for the number of real self-
dual spaces in $\Omega_{\bm{\xi},\bm{z}}$, see Corollary 7.4, i.e., a lower
bound for the number of real points in $\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$,
by following the idea of [13].

Let $\mathfrak{g}_{N}$ be the Lie algebra $\mathfrak{so}_{2r+1}$ if $N=2r$ or
the Lie algebra $\mathfrak{sp}_{2r}$ if $N=2r+1$. We also set
$\mathfrak{g}_{3}=\mathfrak{sl}_{2}$. It is known from [10], see also [16,
Section 6.1], that if $\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$ is nonempty, then
$\xi_{i}^{(s)}-\xi_{i+1}^{(s)}=\xi_{N-i}^{(s)}-\xi_{N-i+1}^{(s)}$ for
$i=1,\dots,N-1$. Hence the $\mathfrak{sl}_{N}$-weight corresponding to the
partition $\xi^{(s)}$ has certain symmetry and thus induces a
$\mathfrak{g}_{N}$-weight $\lambda^{(s)}$, cf. (4.4). Therefore, the sequence
of partitions $\bm{\xi}$ with nonempty $\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$
can be expressed in terms of a sequence of dominant integral
$\mathfrak{g}_{N}$-weights
$\bm{\lambda}=\big{(}\lambda^{1},\dots,\lambda^{(n)}\big{)}$ and a sequence of
nonnegative integers $\bm{k}=(k_{1},\dots,k_{n})$, see Lemma 4.1. In
particular, $k_{i}=\xi_{N}^{(i)}$. We call $\bm{\xi},\bm{z}$ or
$\bm{\lambda},\bm{k},\bm{z}$ the _ramification data_.

As a subset of $\Omega_{\bm{\xi},\bm{z}}$,
$\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$ can be empty even if
$\Omega_{\bm{\xi},\bm{z}}$ is infinite. However, if
$\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$ is nonempty, then
$\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$ is finite if and only if
$\Omega_{\bm{\xi},\bm{z}}$ is finite. More precisely, if

$\displaystyle|\bm{\xi}|:=\sum_{i=1}^{n}\big{|}\xi^{(i)}\big{|}=N(d-N),$

then the number of points in $\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$ counted with
multiplicities equals the multiplicity of the trivial
$\mathfrak{g}_{N}$-module in the tensor product
$V_{\lambda^{(1)}}\otimes\dots\otimes V_{\lambda^{(n)}}$ of irreducible
$\mathfrak{g}_{N}$-modules of highest weights
$\lambda^{(1)},\dots,\lambda^{(n)}$. Since we are interested in the counting
problem, from now on, we always assume that $|\bm{\xi}|=N(d-N)$.

For brevity, we consider $\infty$ to be real. If all $z_{1},\dots,z_{n}$ are
real, it follows from [14, Theorem 1.1] that all points in
$\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$ are real. Hence the number of real points
is maximal possible in this case. Moreover, it follows from [15, Corollary
6.3] that all points in $\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$ are multiplicity-
free.

Then we want to know how many real points we can guarantee in other cases. In
general, a necessary condition for the existence of real points is that the
set $\\{z_{1},\dots,z_{n}\\}$ should be invariant under the complex
conjugation and the partitions at the complex conjugate points are the same.
In other words,
$\big{(}\lambda^{(i)},k_{i}\big{)}=\big{(}\lambda^{(j)},k_{j}\big{)}$ provided
$z_{i}=\bar{z}_{j}$. In this case we say that $\bm{z}$, $\bm{\lambda}$,
$\bm{k}$ are _invariant under conjugation_. Moreover, the greatest common
divisor of $X\in\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$ in this case is a real
polynomial. Hence we reduce the problem to the case that $k_{i}=0$, for all
$i=1,\dots,n$.

The derivation of the lower bounds is based on the identification of the self-
dual spaces of polynomials with points of spectrum of higher Gaudin
Hamiltonians of types B and C ($\mathfrak{g}_{N}$, $N\geqslant 4$) built in
[10] and [16], see Theorem 5.2. We show that higher Gaudin Hamiltonians of
types B and C have certain symmetry with respect to the Shapovalov form which
is positive definite Hermitian, see Proposition 6.1. In particular, these
operators are self-adjoint with respect to the Shapovalov form for real
$z_{1},\dots,z_{n}$ and hence have real eigenvalues. Therefore, it follows
from Theorem 5.2 that self-dual spaces with real $z_{1},\dots,z_{n}$ are real.

If some of $z_{1},\dots,z_{n}$ are not real, but the data $\bm{z}$,
$\bm{\lambda}$, $\bm{k}$ are invariant under the complex conjugation, the
higher Gaudin Hamiltonians are self-adjoint with respect to a nondegenerate
(indefinite) Hermitian form. One of the key observations for computing the
lower bound for the number of real points in
$\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$ is the fact that the number of real
eigenvalues of such operators is at least the absolute value of the signature
of the Hermitian form, see Lemma 6.4.

The computation of the signature of the form is reduced to the computation of
the character values of products of symmetric groups on products of commuting
transpositions. The formula for such character, similar to the Frobenius
formula in [4] and [13, Proposition 2.1], is given in Proposition 3.1.
Consequently, we obtain our main result, a lower bound for the number of real
points in $\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$ for $N\geqslant 4$, see
Corollary 7.4. The case $N=2$ is the same as that of [13] since every
2-dimensional space of polynomials is self-dual. By the proof of [10, Theorem
4.19], the case $N=3$ is reduced to the case of [13], see Section 7.2.

Based on the identification of the self-self-dual spaces of polynomials with
points of spectrum of higher Gaudin Hamiltonians of type G2 built in [1] and
[8], we expect that lower bounds for the numbers of real self-self-dual spaces
in $\Omega_{\bm{\xi},\bm{z}}$ with $N=7$ can also be given in a similar way as
conducted in this paper.

It is also interesting to find an algorithm to compute all (real) self-dual
spaces with prescribed ramification data. The solutions to the Bethe ansatz
equations described in [9] can be used to find nontrivial examples of self-
dual spaces.

The paper is organized as follows. We start with the standard notation of Lie
theory in Section 2 and computations of characters of a product of symmetric
groups in Section 3. Then we recall notation and definitions for osculating
Schubert calculus and self-dual spaces in Section 4. In Section 5 we recall
the connections between Gaudin model of types B, C and self-dual spaces of
polynomials. The symmetry of higher Gaudin Hamiltonians with respect to
Shapovalov form and the key lemma from linear algebra are discussed in Section
6. In Section 7 we prove our main results, see Theorem 7.2 and Corollary 7.4.
Finally, we display some simple data computed from Corollary 7.4 in Section 8.

##  2 Simple Lie algebras

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$ with Cartan
matrix $A=(a_{i,j})_{i,j=1}^{r}$, where $r$ is the rank of $\mathfrak{g}$. Let
$D=\operatorname{diag}(d_{1},\dots,d_{r})$ be the diagonal matrix with
positive relatively prime integers $d_{i}$ such that $DA$ is symmetric.

Let $\mathfrak{h}\subset\mathfrak{g}$ be a Cartan subalgebra with the Cartan
decomposition
$\mathfrak{g}=\mathfrak{n}_{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{+}$. Fix
simple roots $\alpha_{1},\dots,\alpha_{r}$ in $\mathfrak{h}^{*}$. Let
$\check{\alpha}_{1},\dots,\check{\alpha}_{r}\in\mathfrak{h}$ be the
corresponding coroots. Fix a nondegenerate invariant bilinear form $(\ ,\ )$
on $\mathfrak{g}$ such that
$(\check{\alpha}_{i},\check{\alpha}_{j})=a_{i,j}/d_{j}$. The corresponding
bilinear form on $\mathfrak{h}^{*}$ is given by
$(\alpha_{i},\alpha_{j})=d_{i}a_{i,j}$. We have
$\langle\lambda,\check{\alpha}_{i}\rangle=2(\lambda,\alpha_{i})/(\alpha_{i},\alpha_{i})$
for $\lambda\in\mathfrak{h}^{*}$. In particular,
$\langle\alpha_{j},\check{\alpha}_{i}\rangle=a_{i,j}$. Let
$\omega_{1},\dots,\omega_{r}\in\mathfrak{h}^{*}$ be the fundamental weights,
$\langle\omega_{j},\check{\alpha}_{i}\rangle=\delta_{i,j}$.

Let
$\mathcal{P}=\\{\lambda\in\mathfrak{h}^{*}\,|\,\langle\lambda,\check{\alpha}_{i}\rangle\in\mathbb{Z},\,i=1,\dots,r\\}$
and
$\mathcal{P}^{+}=\\{\lambda\in\mathfrak{h}^{*}\,|\,\langle\lambda,\check{\alpha}_{i}\rangle\in\mathbb{Z}_{\geqslant
0},\,i=1,\dots,r\\}$ be the weight lattice and the cone of dominant integral
weights.

Let $e_{1},\dots,e_{r}\in\mathfrak{n}_{+}$,
$\check{\alpha}_{1},\dots,\check{\alpha}_{r}\in\mathfrak{h}$,
$f_{1},\dots,f_{r}\in\mathfrak{n}_{-}$ be the Chevalley generators of
$\mathfrak{g}$.

Given a $\mathfrak{g}$-module $M$, denote by $(M)^{\mathfrak{g}}$ the subspace
of $\mathfrak{g}$-invariants in $M$. The subspace $(M)^{\mathfrak{g}}$ is the
multiplicity space of the trivial $\mathfrak{g}$-module in $M$.

A sequence of nonnegative integers $\xi=(\xi_{1},\dots,\xi_{k})$ such that
$\xi_{1}\geqslant\xi_{2}\geqslant\dots\geqslant\xi_{k}\geqslant 0$ is called a
partition with at most $k$ parts. Set $|\xi|=\sum\limits_{i=1}^{k}\xi_{i}$.

For $\lambda\in\mathfrak{h}^{*}$, let $V_{\lambda}$ be the irreducible
$\mathfrak{g}$-module with highest weight $\lambda$. For any
$\mathfrak{g}$-weights $\lambda$ and $\mu$, it is well known that
$\dim(V_{\lambda}\otimes V_{\mu})^{\mathfrak{g}}=\delta_{\lambda,\mu}$ for
$\mathfrak{g}=\mathfrak{so}_{2r+1},\mathfrak{sp}_{2r}$.

For any Lie algebra $\mathfrak{g}$, denote by $\mathcal{U}(\mathfrak{g})$ the
universal enveloping algebra of $\mathfrak{g}$.

##  3 Characters of the symmetric groups

Let $\mathfrak{g}_{N}$ be the Lie algebra $\mathfrak{so}_{2r+1}$ if $N=2r$ or
the Lie algebra $\mathfrak{sp}_{2r}$ if $N=2r+1$, $r\geqslant 2$. We also set
$\mathfrak{g}_{3}=\mathfrak{sl}_{2}$. Let $\mathrm{G}_{N}$ be the respective
classical group with Lie algebra $\mathfrak{g}_{N}$.

Let $\mathfrak{S}_{k}$ be the symmetric group permuting a set of $k$ elements.
In this section we deduce a formula for characters of a product of the
symmetric groups acting on a tensor product of finite-dimensional irreducible
$\mathfrak{g}_{N}$-modules.

For each dominant integral $\mathfrak{g}_{N}$-weight $\lambda$, denote by
$\bar{\lambda}=(\bar{\lambda}_{1},\dots,\bar{\lambda}_{r})$ the partition with
at most $r$ parts such that

$\displaystyle
2\langle\lambda,\check{\alpha}_{i}\rangle=\bar{\lambda}_{i}-\bar{\lambda}_{i+1},\qquad
i=1,\dots,r-1,\qquad\text{and}\qquad\bar{\lambda}_{r}=\begin{cases}\langle\lambda,\check{\alpha}_{r}\rangle,&\text{if
}N=2r,\\\ 2\langle\lambda,\check{\alpha}_{r}\rangle,&\text{if
}N=2r+1.\end{cases}$

Define an anti-symmetric Laurent polynomial $\Delta_{N}$ in
$x_{1},\dots,x_{r}$ as follows

$\displaystyle\Delta_{N}=\det\big{(}x_{i}^{N+1-2j}-x_{i}^{-(N+1-2j)}\big{)}_{i,j=1}^{r}.$
(3.1)

We call $\Delta_{N}$ the _Vandermonde determinant of_ $\mathfrak{g}_{N}$.

Let $\lambda$ be a dominant integral $\mathfrak{g}_{N}$-weight. It is well
known that the character of the module $V_{\lambda}$ is given by

$\displaystyle\mathcal{S}_{\lambda}^{N}(x_{1},\dots,x_{r})=\operatorname{tr}^{\vphantom{1}}_{V_{\lambda}}X_{N}=\dfrac{\det\big{(}x_{i}^{\bar{\lambda}_{j}+N+1-2j}-x_{i}^{-(\bar{\lambda}_{j}+N+1-2j)}\big{)}_{i,j=1}^{r}}{\Delta_{N}},$
(3.2)

where $X_{N}\in\mathrm{G}_{N}$ is given by

$\displaystyle
X_{N}=\begin{cases}\operatorname{diag}\big{(}x_{1}^{2},\dots,x_{r}^{2},1,x_{r}^{-2},\dots,x_{1}^{-2}\big{)},&\text{if
}N=2r,\\\
\operatorname{diag}\big{(}x_{1}^{2},\dots,x_{r}^{2},x_{r}^{-2},\dots,x_{1}^{-2}\big{)},&\text{if
}N=2r+1.\end{cases}$

We call $\mathcal{S}_{\lambda}^{N}$ the _Schur function of $\mathfrak{g}_{N}$
associated with the weight $\lambda$_.

Note that $\mathcal{S}_{\lambda}^{N}$ are symmetric Laurent polynomials in
$x_{1},\dots,x_{r}$,
$\mathcal{S}_{\lambda}^{N}\in\big{(}\mathbb{C}\big{[}x_{1}^{\pm
1},\dots,x_{r}^{\pm 1}\big{]}\big{)}^{\mathfrak{S}_{r}}$.

Let $\lambda^{(1)},\dots,\lambda^{(s)}$ be a sequence of dominant integral
$\mathfrak{g}_{N}$-weights and $k_{1},\dots,k_{s}$ a sequence of positive
integers. Consider the tensor product of $\mathfrak{g}_{N}$-modules

$\displaystyle V_{\bm{\lambda}}=V_{\lambda^{(1)}}^{\otimes k_{1}}\otimes
V_{\lambda^{(2)}}^{\otimes k_{2}}\otimes\dots\otimes
V_{\lambda^{(s)}}^{\otimes k_{s}}$

and its decomposition into irreducible $\mathfrak{g}_{N}$-submodules

$\displaystyle V_{\bm{\lambda}}=\bigoplus_{\mu}V_{\mu}\otimes
M_{\bm{\lambda},\mu}.$

By permuting the corresponding tensor factors of $V_{\bm{\lambda}}$, the
product of symmetric groups
$\mathfrak{S}_{\bm{k}}=\mathfrak{S}_{k_{1}}\times\mathfrak{S}_{k_{2}}\times\cdots\times\mathfrak{S}_{k_{s}}$
acts naturally on $V_{\bm{\lambda}}$. Note that the
$\mathfrak{S}_{\bm{k}}$-action commutes with the $\mathfrak{g}_{N}$-action,
therefore the group $\mathfrak{S}_{\bm{k}}$ acts on the multiplicity space
$M_{\bm{\lambda},\mu}$ for all $\mu$.

For
$\sigma=\sigma_{1}\times\sigma_{2}\times\dots\times\sigma_{s}\in\mathfrak{S}_{\bm{k}}$,
$\sigma_{i}\in\mathfrak{S}_{k_{i}}$. Suppose all $\sigma_{i}$ are written as a
product of disjoint cycles. Denote by $c_{i}$ the number of cycles in the
product representing $\sigma_{i}$ and $l_{ij}$, $j=1,\dots,c_{i}$, the lengths
of cycles. Note that $l_{i,1}+\dots+l_{i,c_{i}}=k_{i}$.

We then consider the value of the character of $\mathfrak{S}_{\bm{k}}$
corresponding to the representation $M_{\bm{\lambda},\mu}$ on $\sigma$. Let
$\chi^{\vphantom{1}}_{\bm{\lambda},\mu}=\operatorname{tr}^{\vphantom{1}}_{M_{\bm{\lambda},\mu}}$.

######  Proposition 3.1.

The character value $\chi^{\vphantom{1}}_{\bm{\lambda},\mu}(\sigma)$ equals
the coefficient of the monomial

$\displaystyle x_{1}^{\bar{\mu}_{1}+N-1}x_{2}^{\bar{\mu}_{2}+N-3}\cdots
x_{r}^{\bar{\mu}_{r}+N+1-2r}$

in the Laurent polynomial

$\displaystyle\Delta_{N}\cdot\prod_{i=1}^{s}\prod_{j=1}^{c_{i}}\mathcal{S}_{\lambda^{(i)}}^{N}\big{(}x_{1}^{l_{ij}},\dots,x_{r}^{l_{ij}}\big{)}.$

###### Proof.

The proof of the statement is similar to that of [13, Proposition 2.1]. ∎

##  4 Osculating Schubert calculus and self-dual spaces

Let $N$, $d\in\mathbb{Z}_{>0}$ be such that $N\leqslant d$. Consider
$\mathbb{P}^{1}:=\mathbb{C}\cup\\{\infty\\}$. Set

$\displaystyle{\mathring{\mathbb{P}}}_{n}:=\big{\\{}\bm{z}=(z_{1},\dots,z_{n})\in(\mathbb{P}^{1})^{n}\,|\,z_{i}\neq
z_{j}\ \text{for}\ 1\leqslant i<j\leqslant n\big{\\}},$
$\displaystyle\mathbb{R}{\mathring{\mathbb{P}}}_{n}:=\big{\\{}\bm{z}=(z_{1},\dots,z_{n})\in{\mathring{\mathbb{P}}}_{n}\,|\,z_{i}\in\mathbb{R}\
\text{or}\ z_{i}=\infty,\ \text{for}\ 1\leqslant i\leqslant n\big{\\}}.$

###  4.1 Osculating Schubert calculus

Let $\mathbb{C}_{d}[x]$ be the space of polynomials in $x$ with complex
coefficients of degree less than $d$. We have $\dim\mathbb{C}_{d}[x]=d$. Let
$\operatorname{Gr}(N,d)$ be the Grassmannian of all $N$-dimensional subspaces
in $\mathbb{C}_{d}[x]$. The Grassmannian $\operatorname{Gr}(N,d)$ is a smooth
projective complex variety of dimension $N(d-N)$.

Let $\mathbb{R}_{d}[x]\subset\mathbb{C}_{d}[x]$ be the set of polynomials in
$x$ with real coefficients of degree less than $d$. Let
$\operatorname{Gr}^{\mathbb{R}}(N,d)\subset\operatorname{Gr}(N,d)$ be the set
of subspaces which have a basis consisting of polynomials with all
coefficients real. For $X\in\operatorname{Gr}(N,d)$ we have
$X\in\operatorname{Gr}^{\mathbb{R}}(N,d)$ if and only if
$\dim_{\mathbb{R}}(X\cap\mathbb{R}_{d}[x])=N$. We call such points $X$ real.

For a complete flag
$\mathcal{F}=\\{0\subset\mathscr{F}_{1}\subset\mathscr{F}_{2}\subset\dots\subset\mathscr{F}_{d}=\mathbb{C}_{d}[x]\\}$
and a partition $\xi=(\xi_{1},\dots,\xi_{N})$ such that $\xi_{1}\leqslant
d-N$, the Schubert cell
$\Omega_{\xi}(\mathcal{F})\subset\operatorname{Gr}(N,d)$ is given by

$\displaystyle\Omega_{\xi}(\mathcal{F})=\big{\\{}X\in\operatorname{Gr}(N,d)\,|\,\dim(X\cap\mathscr{F}_{d-j-\xi_{N-j}})=N-j,$
$\displaystyle\hphantom{\Omega_{\xi}(\mathcal{F})=\big{\\{}}{}\dim(X\cap\mathscr{F}_{d-j-\xi_{N-j}-1})=N-j-1\big{\\}}.$

Note that $\operatorname{codim}\Omega_{\xi}(\mathcal{F})=|\xi|$.

Let $\mathcal{F}(\infty)$ be the complete flag given by

$\displaystyle\mathcal{F}(\infty)=\big{\\{}0\subset\mathbb{C}_{1}[x]\subset\mathbb{C}_{2}[x]\subset\dots\subset\mathbb{C}_{d}[x]\big{\\}}.$
(4.1)

The subspace $X$ is a point of $\Omega_{\xi}(\mathcal{F}(\infty))$ if and only
if for every $i=1,\dots,N$, it contains a polynomial of degree
$d-i-\xi_{N+1-i}$.

For $z\in\mathbb{C}$, consider the complete flag

$\displaystyle\mathcal{F}(z)=\big{\\{}0\subset(x-z)^{d-1}\mathbb{C}_{1}[x]\subset(x-z)^{d-2}\mathbb{C}_{2}[x]\subset\dots\subset\mathbb{C}_{d}[x]\big{\\}}.$
(4.2)

The subspace $X$ is a point of $\Omega_{\xi}(\mathcal{F}(z))$ if and only if
for every $i=1,\dots,N$, it contains a polynomial with a root at $z$ of order
exactly $\xi_{i}+N-i$.

A point $z\in\mathbb{C}$ is called a base point for a subspace
$X\subset\mathbb{C}_{d}[x]$ if $\varphi(z)=0$ for every $\varphi\in X$.

Let $\bm{\xi}{=}\big{(}\xi^{(1)},{\dots},\xi^{(n)}\big{)}$ be a sequence of
partitions with at most $N$ parts and
$\bm{z}{=}(z_{1},{\dots},z_{n}){\in}{\mathring{\mathbb{P}}}_{n}$. Set
$|\bm{\xi}|=\sum\limits_{s=1}^{n}\big{|}\xi^{(s)}\big{|}$.

Assuming $|\bm{\xi}|=N(d-N)$, denote by $\Omega_{\bm{\xi},\bm{z}}$ the
intersection of the Schubert cells

$\displaystyle\Omega_{\bm{\xi},\bm{z}}=\bigcap_{s=1}^{n}\Omega_{\xi^{(s)}}(\mathcal{F}(z_{s})).$
(4.3)

Note that due to our assumption, $\Omega_{\bm{\xi},\bm{z}}$ is a finite subset
of $\operatorname{Gr}(N,d)$.

Define a sequence of polynomials $\bm{T}=(T_{1},\dots,T_{N})$ by the formulas

$\displaystyle
T_{i}(x)=\prod_{s=1}^{n}(x-z_{s})^{\xi_{i}^{(s)}-\xi_{i+1}^{(s)}},\qquad
i=1,\dots,N,$

where $\xi_{N+1}^{(s)}=0$. Here and in what follows we use the convention that
$x-z_{s}$ is considered as the constant function $1$ if $z_{s}=\infty$. We say
that $\bm{T}$ is _associated with $\bm{\xi}$, $\bm{z}$_.

###  4.2 Self-dual spaces

Let $X\in\operatorname{Gr}(N,d)$ be an $N$-dimensional subspace of polynomials
in $x$. Given a polynomial $\psi$ in $x$, denote by $\psi\cdot X$ the space of
polynomials of the form $\psi\cdot\varphi$ for all $\varphi\in X$.

Let $X^{\vee}$ be the $N$-dimensional space of polynomials which are Wronskian
determinants of $N-1$ elements of $X$

$\displaystyle
X^{\vee}=\big{\\{}\det\left(d^{i-1}\varphi_{j}/dx^{i-1}\right)_{i,j=1}^{N-1},\,\varphi_{j}(x)\in
X\big{\\}}.$

The space $X$ is called _self-dual_ if $X^{\vee}=\psi\cdot X$ for some
polynomial $\psi(x)$, see [16].

Let $\operatorname{sGr}(N,d)$ be the set of all self-dual spaces in
$\operatorname{Gr}(N,d)$. We call $\operatorname{sGr}(N,d)$ the _self-dual
Grassmannian_. The self-dual Grassmannian $\operatorname{sGr}(N,d)$ is an
algebraic subset of $\operatorname{Gr}(N,d)$.

Denote by $\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$ the set of all self-dual spaces
in $\Omega_{\bm{\xi},\bm{z}}$

$\displaystyle\mathrm{s}\Omega_{\bm{\xi},\bm{z}}=\Omega_{\bm{\xi},\bm{z}}\bigcap\operatorname{sGr}(N,d).$

Let $\mu$ be a dominant integral $\mathfrak{g}_{N}$-weight and
$k\in\mathbb{Z}_{\geqslant 0}$. Define a partition $\mu_{A,k}$ with at most
$N$ parts by the rule: $(\mu_{A,k})_{N}=k$ and

$\displaystyle(\mu_{A,k})_{i}-(\mu_{A,k})_{i+1}=\begin{cases}\langle\mu,\check{\alpha}_{i}\rangle,&\text{if
}1\leqslant i\leqslant\big{[}\frac{N}{2}\big{]},\\\
\langle\mu,\check{\alpha}_{N-i}\rangle,&\text{if
}\big{[}\frac{N}{2}\big{]}<i\leqslant N-1.\end{cases}$ (4.4)

We call $\mu_{A,k}$ the partition _associated with weight $\mu$ and integer
$k$_.

Let $\bm{\lambda}=\big{(}\lambda^{(1)},{\dots},\lambda^{(n)}\big{)}$ be a
sequence of dominant integral $\mathfrak{g}_{N}$-weights and let
$\bm{k}=(k_{1},{\dots},k_{n})$ be an $n$-tuple of nonnegative integers. Then
denote
$\bm{\lambda}_{A,\bm{k}}=\big{(}\lambda_{A,k_{1}}^{(1)},\dots,\lambda_{A,k_{n}}^{(n)}\big{)}$
the sequence of partitions associated with $\lambda^{(s)}$ and $k_{s}$,
$s=1,\dots,n$.

We use the notation $\mu_{A}=\mu_{A,0}$ and
$\bm{\lambda}_{A}=\bm{\lambda}_{A,(0,\dots,0)}$.

######  Lemma 4.1 ([10]).

If $\bm{\xi}$ is a sequence of partitions with at most $N$ parts such that
$|\bm{\xi}|=N(d-N)$ and $\mathrm{s}\Omega_{\bm{\xi},\bm{z}}$ is nonempty, then
$\bm{\xi}$ has the form $\bm{\xi}=\bm{\lambda}_{A,\bm{k}}$ for a sequence of
dominant integral $\mathfrak{g}_{N}$-weights
$\bm{\lambda}=\big{(}\lambda^{(1)},\dots,\lambda^{(n)}\big{)}$ and a sequence
of nonnegative integers $\bm{k}=(k_{1},\dots,k_{n})$. The pair
$(\bm{\lambda},\bm{k})$ is uniquely determined by $\bm{\xi}$.

In what follows we write $\Omega_{\bm{\lambda},\bm{z}}$,
$\Omega_{\bm{\lambda},\bm{k},\bm{z}}$,
$\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}$,
$\mathrm{s}\Omega_{\bm{\lambda},\bm{k},\bm{z}}$ for
$\Omega_{\bm{\lambda}_{A},\bm{z}}$, $\Omega_{\bm{\lambda}_{A,\bm{k}},\bm{z}}$,
$\mathrm{s}\Omega_{\bm{\lambda}_{A},\bm{z}}$,
$\mathrm{s}\Omega_{\bm{\lambda}_{A,\bm{k}},\bm{z}}$, respectively.

Note that $|\bm{\lambda}_{A,\bm{k}}|=|\bm{\lambda}_{A}|+N|\bm{k}|$, where
$|\bm{k}|=k_{1}+\dots+k_{n}$. Suppose $|\bm{\lambda}_{A}|=N(d-N)$, there
exists a bijection between $\Omega_{\bm{\lambda},\bm{z}}$ in
$\operatorname{Gr}(N,d)$ and $\Omega_{\bm{\lambda},\bm{k},\bm{z}}$ in
$\operatorname{Gr}(N,d+|\bm{k}|)$ given by

$\displaystyle\Omega_{\bm{\lambda},\bm{z}}\rightarrow\Omega_{\bm{\lambda},\bm{k},\bm{z}},\qquad
X\mapsto\prod_{s=1}^{n}(x-z_{s})^{k_{s}}\cdot X.$ (4.5)

Moreover, (4.5) restricts to a bijection between
$\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}$ in $\operatorname{sGr}(N,d)$ and
$\mathrm{s}\Omega_{\bm{\lambda},\bm{k},\bm{z}}$ in
$\operatorname{sGr}(N,d+|\bm{k}|)$.

##  5 Gaudin model

Let $\mathfrak{g}[t]=\mathfrak{g}\otimes\mathbb{C}[t]$ be the Lie algebra of
$\mathfrak{g}$-valued polynomials with the pointwise commutator. We call it
the _current algebra_ of $\mathfrak{g}$. We identify the Lie algebra
$\mathfrak{g}$ with the subalgebra $\mathfrak{g}\otimes 1$ of constant
polynomials in $\mathfrak{g}[t]$.

It is convenient to collect elements of $\mathfrak{g}[t]$ in generating series
of a formal variable $x$. For $g\in\mathfrak{g}$, set

$\displaystyle g(x)=\sum_{k=0}^{\infty}\big{(}g\otimes t^{k}\big{)}x^{-k-1}.$
(5.1)

For each $a\in\mathbb{C}$, we have the evaluation homomorphism
$\mathrm{ev}_{a}\colon\mathfrak{g}[t]\to\mathfrak{g}$ where $\mathrm{ev}_{a}$
sends $g\otimes t^{s}$ to $a^{s}g$ for all $g\in\mathfrak{g}$ and
$s\in\mathbb{Z}_{\geqslant 0}$. Its restriction to the subalgebra
$\mathfrak{g}\subset\mathfrak{g}[t]$ is the identity map. For any
$\mathfrak{g}$-module $M$, we denote by $M(a)$ the $\mathfrak{g}[t]$-module,
obtained by pulling $M$ back through the evaluation homomorphism
$\mathrm{ev}_{a}$. The $\mathfrak{g}[t]$-module $M(a)$ is called an
_evaluation module_. The generating series $g(x)$ acts on the evaluation
module $M(a)$ by $g/(x-a)$.

The Bethe algebra $\mathcal{B}$ (the algebra of higher Gaudin Hamiltonians)
for a simple Lie algebra $\mathfrak{g}$ was described in [3]. The Bethe
algebra $\mathcal{B}$ is a commutative subalgebra of
$\mathcal{U}(\mathfrak{g}[t])$ which commutes with the subalgebra
$\mathcal{U}(\mathfrak{g})\subset\mathcal{U}(\mathfrak{g}[t])$. An explicit
set of generators of the Bethe algebra in Lie algebras of types B, C, and D
was given in [11].

######  Proposition 5.1 ([3, 11]).

Let $N>3$. There exist elements $F_{ij}\in\mathfrak{g}_{N}$, $i,j=1,\dots,N$,
and polynomials $B_{s}(x)$ in $d^{k}F_{ij}(x)/dx^{k}$, $s=1,\dots,N$,
$k=0,\dots,N$, such that the Bethe algebra $\mathcal{B}$ of $\mathfrak{g}_{N}$
is generated by the coefficients of $B_{s}(x)$ considered as formal power
series in $x^{-1}$.

We denote $M(\infty)$ the $\mathfrak{g}_{N}$-module $M$ with the trivial
action of the Bethe algebra $\mathcal{B}$, see [10] for more detail.

For a collection of $\mathfrak{g}_{N}$-weights
$\bm{\lambda}=\big{(}\lambda^{(1)},\dots,\lambda^{(n)}\big{)}$ and
$\bm{z}=(z_{1},\dots,z_{n})\in\mathring{\mathbb{P}}_{n}$, we set

$\displaystyle
V_{\bm{\lambda},\bm{z}}=\bigotimes_{s=1}^{n}V_{\lambda^{(s)}}(z_{s}),$

considered as a $\mathcal{B}$-module. We also denote $V_{\bm{\lambda}}$ the
module $V_{\bm{\lambda},\bm{z}}$ considered as a $\mathfrak{g}_{N}$-module.

Let $\partial_{x}$ be the differentiation with respect to $x$. Define a formal
differential operator

$\displaystyle\mathcal{D}^{\mathcal{B}}=\partial_{x}^{N}+\sum_{i=1}^{N}B_{i}(x)\partial_{x}^{N-i},$

where

$\displaystyle B_{i}(x)=\sum_{j=i}^{\infty}B_{ij}x^{-j}$ (5.2)

and $B_{ij}\in\mathcal{U}(\mathfrak{g}_{N}[t])$, $j\in\mathbb{Z}_{\geqslant
i}$, $i=1,\dots,N$. The operator $\mathcal{D}^{\mathcal{B}}$ is called _the
universal operator_.

Let $\bm{z}=(z_{1},\dots,z_{n})\in\mathring{\mathbb{P}}_{n}$ and let
$\bm{\lambda}=\big{(}\lambda^{(1)},\dots,\lambda^{(n)}\big{)}$ be a sequence
of dominant integral $\mathfrak{g}_{N}$-weights. For every
$g\in\mathfrak{g}_{N}$, the series $g(x)$ acts on $V_{\bm{\lambda},\bm{z}}$ as
a rational function of $x$.

Since the Bethe algebra $\mathcal{B}$ commutes with $\mathfrak{g}_{N}$,
$\mathcal{B}$ acts on the invariant space
$(V_{\bm{\lambda},\bm{z}})^{\mathfrak{g}_{N}}$. For $b\in\mathcal{B}$, denote
by
$b(\bm{\lambda},\bm{z})\in\operatorname{End}((V_{\bm{\lambda},\bm{z}})^{\mathfrak{g}_{N}})$
the corresponding linear operator.

Given a common eigenvector $v\in(V_{\bm{\lambda},\bm{z}})^{\mathfrak{g}_{N}}$
of the operators $b(\bm{\lambda},\bm{z})$, denote by
$b(\bm{\lambda},\bm{z};v)$ the corresponding eigenvalues, and define the
scalar differential operator

$\displaystyle\mathcal{D}_{v}=\partial_{x}^{N}+\sum_{i=1}^{N}\sum_{j=i}^{\infty}B_{ij}(\bm{\lambda},\bm{z};v)x^{-j}\partial_{x}^{N-i}.$

The following theorem connects self-dual spaces in the Grassmannian
$\operatorname{Gr}(N,d)$ with the Gaudin model associated to
$\mathfrak{g}_{N}$.

######  Theorem 5.2 ([10]).

Let $N>3$. There exists a choice of generators $B_{i}(x)$ of the Bethe algebra
$\mathcal{B}$, such that for any sequence of dominant integral
$\mathfrak{g}_{N}$-weights
$\bm{\lambda}=\big{(}\lambda^{(1)},\dots,\lambda^{(n)}\big{)}$, any
$\bm{z}\in\mathring{\mathbb{P}}_{n}$, and any $\mathcal{B}$-eigenvector
$v\in(V_{\bm{\lambda},\bm{z}})^{\mathfrak{g}_{N}}$, we have

$\displaystyle\operatorname{Ker}\big{(}(T_{1}\cdots
T_{N})^{1/2}\cdot\mathcal{D}_{v}\cdot(T_{1}\cdots
T_{N})^{-1/2}\big{)}\in\mathrm{s}\Omega_{\bm{\lambda},\bm{z}},$

where $\bm{T}=(T_{1},\dots,T_{N})$ is associated with $\bm{\lambda}_{A}$,
$\bm{z}$.

Moreover, if $|\bm{\lambda}_{A}|=N(d-N)$, then this defines a bijection
between the joint eigenvalues of $\mathcal{B}$ on
$(V_{\bm{\lambda},\bm{z}})^{\mathfrak{g}_{N}}$ and
$\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}\subset\operatorname{Gr}(N,d)$.

##  6 Shapovalov form and the key lemma

###  6.1 Shapovalov form

Define the anti-involution $\varpi\colon\mathfrak{g}_{N}\to\mathfrak{g}_{N}$
sending $e_{1},\dots,e_{r}$, $\check{\alpha}_{1},\dots,\check{\alpha}_{r}$,
$f_{1},\dots,f_{r}$ to $f_{1},\dots,f_{r}$,
$\check{\alpha}_{1},\dots,\check{\alpha}_{r}$, $e_{1},\dots,e_{r}$,
respectively.

For any dominant integral $\mathfrak{g}_{N}$-weight $\lambda$, the irreducible
$\mathfrak{g}_{N}$-module $V_{\lambda}$ admits a positive definite Hermitian
form $(\cdot,\cdot)_{\lambda}$ such that
$(gv,w)_{\lambda}=(v,\overline{\varpi(g)}w)_{\lambda}$ for any $v,w\in
V_{\lambda}$ and $g\in\mathfrak{g}_{N}$. Such a form is unique up to
multiplication by a positive real number. We call this form the _Shapovalov
form_.

Let $\bm{\lambda}=\big{(}\lambda^{(1)},\dots,\lambda^{(n)}\big{)}$ be a
sequence of dominant integral $\mathfrak{g}_{N}$-weights. We define the
positive definite Hermitian form $(\cdot,\cdot)_{\bm{\lambda}}$ on the tensor
product $V_{\bm{\lambda}}$ as the product of Shapovalov forms on the tensor
factors. The form $(\cdot,\cdot)_{\bm{\lambda}}$ induces a positive definite
Hermitian form $(\cdot|\cdot)_{\bm{\lambda}}$ on
$(V_{\bm{\lambda},\bm{z}})^{\mathfrak{g}_{N}}$.

######  Proposition 6.1.

For any $i=1,\dots,N$, $j\in\mathbb{Z}_{\geqslant i}$, and any
$v,w\in(V_{\bm{\lambda},\bm{z}})^{\mathfrak{g}_{N}}$, we have

$\displaystyle\big{(}B_{ij}(\bm{\lambda},\bm{z})v|w\big{)}_{\bm{\lambda}}=\big{(}v|B_{ij}(\bm{\lambda},\bar{\bm{z}})w\big{)}_{\bm{\lambda}},$

where $B_{ij}$ are given by (5.2),
$\bar{\bm{z}}=(\bar{z}_{1},\dots,\bar{z}_{n})$ and the bar stands for the
complex conjugation.

###### Proof.

We prove the proposition in Section 6.3. ∎

If $\bm{z}\in\mathbb{R}{\mathring{\mathbb{P}}}_{n}$, then
$B_{ij}(\bm{\lambda},\bm{z})$ are self-adjoint with respect to the Shapovalov
form. Therefore all $B_{ij}(\bm{\lambda},\bm{z})$ are simultaneously
diagonalizable and all eigenvalues of $B_{ij}(\bm{\lambda},\bm{z})$ are real.

The following statement is also known.

######  Theorem 6.2 ([18]).

For generic $\bm{z}\in{\mathring{\mathbb{P}}}_{n}$, the action of the Bethe
algebra $\mathcal{B}$ on $(V_{\bm{\lambda},\bm{z}})^{\mathfrak{g}_{N}}$ is
diagonalizable and has simple spectrum. In particular, this statement holds
for any sequence $\bm{z}\in\mathbb{R}{\mathring{\mathbb{P}}}_{n}$.

If some of the partitions $\lambda^{(1)},\dots,\lambda^{(n)}$ coincide, the
operators $b(\bm{\lambda},\bm{z})$ admit additional symmetry. Assume that
$\lambda^{(i)}=\lambda^{(i+1)}$ for some $i$. Let
$P_{i}\in\operatorname{End}(V_{\bm{\lambda}})$ be the flip of the $i$-th and
$(i+1)$-st tensor factors and
$\tilde{{\bm{z}}}^{(i)}=(z_{1},\dots,z_{i-1},z_{i+1},z_{i},z_{i+2},\dots,z_{n})$.

######  Lemma 6.3.

For any $b\in\mathcal{B}$, we have
$P_{i}b(\bm{\lambda},\bm{z})P_{i}=b\big{(}\bm{\lambda},\tilde{{\bm{z}}}^{(i)}\big{)}$.

###  6.2 Self-adjoint operators with respect to indefinite Hermitian form

In this section we recall the key lemma from linear algebra, see [17].

Given a finite-dimensional vector space $M$, a linear operator
$\mathfrak{T}\in\operatorname{End}(M)$, and a number $\alpha\in\mathbb{C}$,
let $M_{\mathfrak{T}}(\alpha)=\operatorname{ker}(\mathfrak{T}-\alpha)^{\dim
M}$. When $M_{\mathfrak{T}}(\alpha)$ is not trivial, it is the subspace of
generalized eigenvectors of $\mathfrak{T}$ with eigenvalue $\alpha$.

######  Lemma 6.4 ([17]).

Let $M$ be a complex finite-dimensional vector space with a nondegenerate
Hermitian form of signature $\kappa$, and let
${\mathcal{A}}\subset\operatorname{End}(M)$ be a commutative subalgebra over
$\mathbb{R}$, whose elements are self-adjoint operators. Let
$R=\bigcap_{\mathfrak{T}\in\mathcal{A}}\bigoplus_{\alpha\in\mathbb{R}}M_{\mathfrak{T}}(\alpha)$.
Then the restriction of the Hermitian form on $R$ is nondegenerate and has
signature $\kappa$. In particular, $\dim R\geqslant|\kappa|$.

###  6.3 Proof of Proposition 6.1

In this section, we give the proof of Proposition 6.1. We follow the
convention of [12]. We only introduce the necessary notation and refer the
reader to [11, Section 5] and [12, Section 3] for more detail.

###### Proof of Proposition 6.1.

We prove it for the case $N=2r$ first.

Let $E_{ij}$ with $i,j=1,\dots,2r+1$ be the standard basis of
$\mathfrak{gl}_{2r+1}$. The Lie subalgebra of $\mathfrak{gl}_{2r+1}$ generated
by the elements $F_{ij}=E_{ij}-E_{2r+2-j,2r+2-i}$ is isomorphic to the Lie
algebra $\mathfrak{so}_{2r+1}=\mathfrak{g}_{N}$. With this isomorphism, the
anti-involution $\varpi\colon\mathfrak{g}_{N}\to\mathfrak{g}_{N}$ is realized
by taking transposition, $F_{ij}\mapsto F_{ji}$. To be consistent with the
notation in [12], we write $\mathfrak{g}$ for $\mathfrak{g}_{N}$. The number
$N$ in [12] is $2r+1$ in our notation.

We write $F_{ij}[s]$ for $F_{ij}\otimes t^{s}$ in the loop algebra
$\mathfrak{g}\big{[}t,t^{-1}\big{]}$. Consider the affine Lie algebra
$\widehat{\mathfrak{g}}=\mathfrak{g}\big{[}t,t^{-1}\big{]}\oplus\mathbb{C}K$,
which is the central extension of the loop algebra
$\mathfrak{g}\big{[}t,t^{-1}\big{]}$, where the element $K$ is central in
$\widehat{\mathfrak{g}}$ and

$\displaystyle[g_{1}[k],g_{2}[l]]=[g_{1},g_{2}][k+l]+k\delta_{k,-l}(g_{1},g_{2})K,\qquad
g_{1},g_{2}\in\mathfrak{g},\qquad k,l\in\mathbb{Z}.$

Consider the extended affine Lie algebra
$\widehat{\mathfrak{g}}\oplus\mathbb{C}\tau=\mathfrak{g}\big{[}t,t^{-1}\big{]}\oplus\mathbb{C}K\oplus\mathbb{C}\tau$,
where $\tau$ satisfies

$\displaystyle[\tau,F_{ij}[s]]=-sF_{ij}[s-1],\qquad[\tau,K]=0,\qquad
s\in\mathbb{Z}.$

Set $\mathcal{U}=\mathcal{U}(\widehat{\mathfrak{g}}\oplus\mathbb{C}\tau)$ and
fix $m\in\\{1,\dots,N\\}$. Introduce the element $F[s]_{a}$ of the algebra
$\big{(}\operatorname{End}\big{(}\mathbb{C}^{2r+1}\big{)}\big{)}^{\otimes
m}\otimes\mathcal{U}$, see [12, equation (3.5)], by

$\displaystyle F[s]_{a}=\sum_{i,j=1}^{2r+1}1^{\otimes(a-1)}\otimes
e_{ij}\otimes 1^{\otimes(m-a)}\otimes F_{ij}[s],$

where $e_{ij}\in\operatorname{End}\big{(}\mathbb{C}^{2r+1}\big{)}$ denote the
standard matrix units. The map $\varpi$ induces an anti-involution

$\displaystyle\varpi\colon\
\mathcal{U}\big{(}t^{-1}\mathfrak{g}\big{[}t^{-1}\big{]}\big{)}\to\mathcal{U}\big{(}t^{-1}\mathfrak{g}\big{[}t^{-1}\big{]}\big{)},\qquad
F_{ij}[s]\mapsto F_{ji}[s],\qquad s\in\mathbb{Z}_{\leqslant-1}.$

For $1\leqslant a<b\leqslant m$, consider the operators $P_{ab}$ and $Q_{ab}$
in $\big{(}\operatorname{End}\big{(}\mathbb{C}^{2r+1}\big{)}\big{)}^{\otimes
m}$ defined as follows

$\displaystyle P_{ab}=\sum_{i,j=1}^{2r+1}1^{\otimes(a-1)}\otimes e_{ij}\otimes
1^{\otimes(b-a-1)}\otimes e_{ji}\otimes 1^{\otimes(m-b)},$ $\displaystyle
Q_{ab}=\sum_{i,j=1}^{2r+1}1^{\otimes(a-1)}\otimes e_{ij}\otimes
1^{\otimes(b-a-1)}\otimes e_{2r+2-i,2r+2-j}\otimes 1^{\otimes(m-b)}.$

Set

$\displaystyle S^{(m)}=\frac{1}{m!}\prod_{1\leqslant a<b\leqslant
m}\left(1+\frac{P_{ab}}{b-a}-\frac{2Q_{ab}}{2r+2b-2a-1}\right),$

where the product is taken in the lexicographic order on the pairs $(a,b)$.
The element $S^{(m)}$ is the symmetrizer of the Brauer algebra acting on
$\big{(}\mathbb{C}^{2r+1}\big{)}^{\otimes m}$. In particular, for any
$1\leqslant a<b\leqslant m$, the operator $S^{(m)}$ satisfies

$\displaystyle S^{(m)}Q_{ab}=Q_{ab}S^{(m)}=0,\qquad
S^{(m)}P_{ab}=P_{ab}S^{(m)}=S^{(m)}.$

Replacing $\tau$ with $\partial_{x}$ and $F_{ij}[-\ell-1]$ with
$-\partial_{x}^{\ell}F_{ij}(x)/\ell!$, where $F_{ij}(x)$ is defined in (5.1),
for $\ell\in\mathbb{Z}_{\geqslant 0}$, in the element

$\displaystyle\frac{2r+m-1}{2r+2m-1}\operatorname{tr}S^{(m)}(\tau+F[-1]_{1})\cdots(\tau+F[-1]_{m}),$

see [12, formula (3.26)], where the trace is taken on all $m$ copies of
$\operatorname{End}\big{(}\mathbb{C}^{2r+1}\big{)}$, we get a differential
operator

$\displaystyle\vartheta_{m0}(x)\partial_{x}^{m}+\vartheta_{m1}(x)\partial_{x}^{m-1}+\dots+\vartheta_{mm}(x),$

where $\vartheta_{mi}(x)$ is a formal power series in $x^{-1}$ with
coefficients in $\mathcal{U}(\mathfrak{g}[t])$. The Bethe subalgebra
$\mathcal{B}$ of $\mathcal{U}(\mathfrak{g}[t])$ is generated by the
coefficients of $\vartheta_{mi}(x)$, $m=1,\dots,N$, $i=0,\dots,m$, see [11,
Section 5].

Therefore, to prove the proposition, it suffices to show that the element

$\displaystyle\frac{2r+m-1}{2r+2m-1}\operatorname{tr}S^{(m)}(\tau+F[-1]_{1})\cdots(\tau+F[-1]_{m}),$
(6.1)

is stable under the anti-involution $\varpi$. Here $\varpi$ maps $\tau$ to
$\tau$.

Applying transposition on $a$-th and $b$-th components to the commutator
relation

$\displaystyle
F[k]_{a}F[l]_{b}-F[l]_{b}F[k]_{a}=(P_{ab}-Q_{ab})F[k+l]_{b}-F[k+l]_{b}(P_{ab}-Q_{ab}),$

see the proof of [12, Lemma 3.6], we get

$\displaystyle
F^{\top}[k]_{a}F^{\top}[l]_{b}-F^{\top}[l]_{b}F^{\top}[k]_{a}=F^{\top}[k+l]_{b}(P_{ab}-Q_{ab})-(P_{ab}-Q_{ab})F^{\top}[k+l]_{b},$

for all $1\leqslant a<b\leqslant m$. Here $\top$ stands for transpose,
explicitly,

$\displaystyle F^{\top}[s]_{a}=\sum_{i,j=1}^{2r+1}1^{\otimes(a-1)}\otimes
e_{ij}\otimes 1^{\otimes(m-a)}\otimes F_{ji}[s].$

Thus one can use the same argument as in the proof of [12, Lemma 3.2] to show
that the image of (6.1) under the anti-involution $\varpi$ equals

$\displaystyle\frac{2r+m-1}{2r+2m-1}\operatorname{tr}S^{(m)}\big{(}\tau+F^{\top}[-1]_{1}\big{)}\cdots\big{(}\tau+F^{\top}[-1]_{m}\big{)}.$
(6.2)

By applying the simultaneous transposition $e_{ij}\to e_{ji}$ to all $m$
copies of $\operatorname{End}\big{(}\mathbb{C}^{2r+1}\big{)}$ we conclude that
(6.2) coincides with (6.1) because the transformation takes each factor
$\tau+F^{\top}[-1]_{a}$ to $\tau+F[-1]_{a}$ whereas the symmetrizer $S^{(m)}$
stays invariant. Hence we complete the proof of Proposition 6.1 for the case
$N=2r$.

The case $N=2r+1$ is proved similarly, see for example [12, Lemma 3.9]. ∎

##  7 The lower bound

In this section we prove our main results – the lower bound for the number of
real self-dual spaces in $\Omega_{\bm{\lambda},\bm{z}}$, see Theorem 7.2 and
Corollary 7.4.

Recall the notation from Section 4. For positive integers $N$, $d$ such that
$d\geqslant N$ we consider the Grassmannian $\operatorname{Gr}(N,d)$ of
$N$-dimensional planes in the space $\mathbb{C}_{d}[x]$ of polynomials of
degree less than $d$. A point $X\in\operatorname{Gr}(N,d)$ is called real if
it has a basis consisting of polynomials with all coefficients real.

###  7.1 The general case $\bm{N\geqslant 4}$

Let us first consider the case $N\geqslant 4$.

Let $\bm{\lambda}=\big{(}\lambda^{(1)},\dots,\lambda^{(n)}\big{)}$ be a
sequence of dominant integral $\mathfrak{g}_{N}$-weights,
$\bm{k}=(k_{1},\dots,k_{n})$ an $n$-tuple of nonnegative integers, and
$\bm{z}=(z_{1},\dots,z_{n})\in\mathring{\mathbb{P}}_{n}$. Suppose that
$|\bm{\lambda}_{A,\bm{k}}|=N(d-N)$. Denote by $d(\bm{\lambda},\bm{k},\bm{z})$
the number of real points counted with multiplicities in
$\mathrm{s}\Omega_{\bm{\lambda},\bm{k},\bm{z}}\subset\operatorname{Gr}(N,d)$.

Clearly, $d(\bm{\lambda},\bm{k},\bm{z})=0$ unless the set
$\\{z_{1},\dots,z_{n}\\}$ is invariant under the complex conjugation and
$\big{(}\lambda^{(i)},k_{i}\big{)}=\big{(}\lambda^{(j)},k_{j}\big{)}$ whenever
$z_{i}=\bar{z}_{j}$. In particular, the polynomial
$\prod\limits_{s=1}^{n}(x-z_{s})^{k_{s}}$ has only real coefficients. It
follows from (4.5) that the number of real points in
$\mathrm{s}\Omega_{\bm{\lambda},\bm{k},\bm{z}}\subset\operatorname{Gr}(N,d)$
is equal to that of
$\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}\subset\operatorname{Gr}(N,d-|\bm{k}|)$.
From now on, we shall only consider the case that $\bm{k}=(0,\dots,0)$. We
simply write $d(\bm{\lambda},\bm{z})$ for $d(\bm{\lambda},\bm{k},\bm{z})$ if
$\bm{k}=(0,\dots,0)$.

Let $\bm{T}=(T_{1},\dots,T_{N})$ be associated with $\bm{\lambda}_{A,\bm{k}}$,
$\bm{z}$. Note that if $\bm{z}$, $\bm{\lambda}$, $\bm{k}$ is invariant under
conjugation, then the polynomial $T_{1}\cdots T_{N}$ also has only real
coefficients.

In what follows we denote by $c$ the number of complex conjugate pairs in the
set $\\{z_{1},\dots,z_{n}\\}$ and without loss of generality assume that
$z_{1}=\bar{z}_{2},\dots,z_{2c-1}=\bar{z}_{2c}$ while $z_{2c+1},\dots,z_{n}$
are real (one of them can be infinity). We will also always assume that
$\lambda^{(1)}=\lambda^{(2)},\dots,\lambda^{(2c-1)}=\lambda^{(2c)}$.

Recall that for any $\bm{\lambda}$ and generic
$\bm{z}\in\mathring{\mathbb{P}}_{n}$, all points of
$\Omega_{\bm{\lambda},\bm{z}}$ are multiplicity-free. The same also holds true
with $\bm{\lambda}$ imposed above for any $c$.

Consider the decomposition of $V_{\bm{\lambda}}$ into irreducible
$\mathfrak{g}_{N}$-submodules

$\displaystyle V_{\bm{\lambda}}=\bigoplus_{\mu}V_{\mu}\otimes
M_{\bm{\lambda},\mu}.$

Then $M_{\bm{\lambda},0}=(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}$. Since
$\lambda^{(2i-1)}=\lambda^{(2i)}$ for $i=1,\dots,c$, the flip $P_{2i-1}$ of
the $(2i-1)$-st and $2i$-th tensor factors of $V_{\bm{\lambda}}$ commutes with
the $\mathfrak{g}_{N}$-action and thus acts on
$(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}$. Denote by
$P_{\bm{\lambda},c}\in\operatorname{End}((V_{\bm{\lambda}})^{\mathfrak{g}_{N}})$
the action of the product $P_{1}P_{3}\cdots P_{2c-1}$ on
$(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}$.

The operator $P_{\bm{\lambda},c}$ is self-adjoint relative to the Hermitian
form $(\cdot|\cdot)_{\bm{\lambda}}$ on $(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}$
given in Section 6. Define a new Hermitian form
$(\cdot,\cdot)_{\bm{\lambda},c}$ on $(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}$ by
the rule: for any $v,w\in(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}$

$\displaystyle(v,w)_{\bm{\lambda},c}=(P_{\bm{\lambda},c}v|w)_{\bm{\lambda}}.$

Denote by $q(\bm{\lambda},c)$ the signature of the form
$(\cdot,\cdot)_{\bm{\lambda},c}$.

######  Proposition 7.1.

The signature $q(\bm{\lambda},c)$ equals the coefficient of the monomial

$\displaystyle x_{1}^{N-1}x_{2}^{N-3}\cdots x_{r}^{N+1-2r},$

in the Laurent polynomial

$\displaystyle\Delta_{N}\cdot\prod_{i=1}^{c}\mathcal{S}_{\lambda^{(2i)}}^{N}\big{(}x_{1}^{2},\dots,x_{r}^{2}\big{)}\prod_{j=2c+1}^{n}\mathcal{S}_{\lambda^{(j)}}^{N}(x_{1},\dots,x_{r}).$

Here $\Delta_{N}$ and $\mathcal{S}_{\lambda^{(s)}}^{N}$ are given by (3.1) and
(3.2), respectively.

###### Proof.

Since $P_{\bm{\lambda},c}^{2}=1$ and
$M_{\bm{\lambda},0}=(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}$, we have
$q(\bm{\lambda},c)=\operatorname{tr}^{\vphantom{1}}_{M_{\bm{\lambda},0}}P_{\bm{\lambda},c}$,
and the claim follows from Proposition 3.1. ∎

######  Theorem 7.2.

The number $d(\bm{\lambda},\bm{z})$ of real self-dual spaces in
$\Omega_{\bm{\lambda},\bm{z}}$ is no less than $|q(\bm{\lambda},c)|$.

###### Proof.

Our proof is parallel to that of [13, Theorem 7.2].

By Proposition 6.1 and Lemma 6.3, the operators
$B_{ij}(\bm{\lambda},\bm{z})\in\operatorname{End}((V_{\bm{\lambda}})^{\mathfrak{g}_{N}})$
are self-adjoint relative to the form $(\cdot,\cdot)_{\bm{\lambda},c}$. By
Lemma 6.4,

$\displaystyle\dim\bigg{(}\bigcap_{i,j}\bigoplus_{\alpha\in\mathbb{R}}\big{(}(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}\big{)}_{B_{ij}(\bm{\lambda},\bm{z})}(\alpha)\bigg{)}\geqslant|q(\bm{\lambda},c)|.$

By Theorem 6.2, for any $\bm{\lambda}$ and generic
$\bm{z}\in\mathring{\mathbb{P}}_{n}$ the operators
$B_{ij}(\bm{\lambda},\bm{z})$ are diagonalizable and the action of the Bethe
algebra $\mathcal{B}$ on $(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}$ has simple
spectrum. The same also holds true with $\bm{\lambda}$ imposed above for any
$c$. Thus for generic $\bm{z}$, the operators $B_{ij}(\bm{\lambda},\bm{z})$
have at least $|q(\bm{\lambda},c)|$ common eigenvectors with distinct real
eigenvalues, which provides $|q(\bm{\lambda},c)|$ distinct real points in
$\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}$ by Theorem 5.2. Hence,
$d(\bm{\lambda},\bm{z})\geqslant|q(\bm{\lambda},c)|$ for generic $\bm{z}$, and
therefore, for any $\bm{z}$, due to counting with multiplicities. ∎

######  Remark 7.3.

If $\dim(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}$ is odd, it follows from Theorem
7.2 by counting parity that

$\displaystyle d(\bm{\lambda},\bm{z})\geqslant|q(\bm{\lambda},c)|\geqslant 1.$

Therefore, there exists at least one real point in
$\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}$. In particular, if
$\dim(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}=1$, then the only point in
$\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}$ is always real.

The following corollary of Proposition 7.1 and Theorem 7.2 is our main result.

######  Corollary 7.4.

The number $d(\bm{\lambda},\bm{z})$ of real self-dual spaces in
$\Omega_{\bm{\lambda},\bm{z}}$ $($real points in
$\mathrm{s}\Omega_{\bm{\lambda},\bm{z}})$ is no less than
$|a(\bm{\lambda},c)|$, where $a(\bm{\lambda},c)$ is the coefficient of the
monomial $x_{1}^{N-1}x_{2}^{N-3}\cdots x_{r}^{N+1-2r}$ in the Laurent
polynomial

$\displaystyle\Delta_{N}\cdot\prod_{i=1}^{c}\mathcal{S}_{\lambda^{(2i)}}^{N}\big{(}x_{1}^{2},\dots,x_{r}^{2}\big{)}\prod_{j=2c+1}^{n}\mathcal{S}_{\lambda^{(j)}}^{N}(x_{1},\dots,x_{r}).$

Here $\Delta_{N}$ is the Vandermonde determinant of $\mathfrak{g}_{N}$ and
$\mathcal{S}_{\lambda^{(s)}}^{N}$ is the Schur function of $\mathfrak{g}_{N}$
associated with $\lambda^{(s)}$, $s=1,\dots,n$, see (3.1) and (3.2).

######  Remark 7.5.

Recall that the total number of points (counted with multiplicities) in
$\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}$ equals
$\dim(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}=q(\bm{\lambda},0)$. Hence if
$\bm{z}\in\mathbb{R}\mathring{\mathbb{P}}_{n}$, Theorem 7.2 claims that all
points in $\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}$ are real. It is proved in
[15, Corollary 6.3] that for $\bm{z}\in\mathbb{R}\mathring{\mathbb{P}}_{n}$
all points in $\Omega_{\bm{\lambda},\bm{z}}$ are real and multiplicity-free,
so are the points in $\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}$.

###  7.2 The case $\bm{N=2,3}$

Now let us consider the case $N=2,3$. Note that
$\operatorname{sGr}(2,d)=\operatorname{Gr}(2,d)$, this case is the usual
Grassmannian, which has already been discussed in [13].

Let $N=3$ and $\mathfrak{g}_{3}=\mathfrak{sl}_{2}$. It suffices for us to
consider the case that points in $\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}$ have
no base points, see the beginning of Section 7.1 for more detail. We shall
consider $\operatorname{sGr}(3,2d-1)$ instead of $\operatorname{sGr}(3,d)$,
see [10, Section 4.6]. We identify the dominant integral
$\mathfrak{sl}_{2}$-weights with nonnegative integers. Let
$\bm{\lambda}=\big{(}\lambda^{(1)},\dots,\lambda^{(n)}\big{)}$ be a sequence
of nonnegative integers and
$\bm{z}=(z_{1},\dots,z_{n})\in{\mathring{\mathbb{P}}}_{n}$. Then
$\bm{\lambda}_{A}$ has coordinates

$\displaystyle\lambda_{A}^{(s)}=\big{(}2\lambda^{(s)},\lambda^{(s)},0\big{)},\qquad
s=1,\dots,n.$

We also assume $|\bm{\lambda}_{A}|=6(d-2)$.

Recall from [10, Theorem 4.19], if
$X\in\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}$, then there exist monic
polynomials $\varphi$ and $\psi$ such that $\varphi^{2}$, $\varphi\psi$,
$\psi^{2}$ form a basis of $X$. Denote by $\sqrt{X}$ the space of polynomials
spanned by $\varphi$ and $\psi$. Let $\xi^{(i)}$ be the partitions with at
most two parts defined by $\big{(}\lambda^{(i)},0\big{)}$, $i=1,\dots,n$. Set
$\bm{\xi}=\big{(}\xi^{(1)},\dots,\xi^{(n)}\big{)}$, then $|\bm{\xi}|=2(d-2)$.
It follows from the proof of [10, Theorem 4.19] that
$\sqrt{X}\in\Omega_{\bm{\xi},\bm{z}}\subset\operatorname{Gr}(2,d)$. The map
$\Omega_{\bm{\xi},\bm{z}}\to\mathrm{s}\Omega_{\bm{\lambda},\bm{z}}$ given by
$\sqrt{X}\mapsto X$ is bijective.

######  Lemma 7.6.

The self-dual space $X$ is real if and only if $\sqrt{X}$ is real.

###### Proof.

It is obvious that $X$ is real if $\sqrt{X}$ is real.

Conversely, if $X$ is real, then there exist complex numbers $a_{i}$, $b_{i}$,
$c_{i}$, $i=1,2,3$, such that

$\displaystyle a_{i}\varphi^{2}+b_{i}\varphi\psi+c_{i}\psi^{2},\qquad
i=1,2,3,$

are real polynomials and form a basis of $X$. Without loss of generality, we
assume $\deg\varphi<\deg\psi$. Since $\deg\varphi<\deg\psi$, we have
$c_{i}\in\mathbb{R}$, $i=1,2,3$. At least one of $c_{i}$ is nonzero. We assume
$c_{3}\neq 0$. By subtracting a proper real multiple of
$a_{3}\varphi^{2}+b_{3}\varphi\psi+c_{3}\psi^{2}$, we assume further
$c_{1}=c_{2}=0$. Continuing with the previous step, we assume that $b_{1}=0$,
$b_{2}\neq 0$, $a_{1}\neq 0$ and hence obtain that
$a_{1},b_{2},c_{3}\in\mathbb{R}$. Then $a_{1}\varphi^{2}$ is a real
polynomial, so is $\varphi$. Therefore, $a_{2}\varphi+b_{2}\psi$ is also a
real polynomial, which implies that the space of polynomials $\sqrt{X}$ is
also real. ∎

Because of Lemma 7.6, the case $N=3$ is reduced to the lower bound for the
number of real solutions to osculating Schubert problems of
$\operatorname{Gr}(2,d)$, see [13]. Moreover, Corollary 7.4 also applies for
this case by putting $N=3$, $r=1$, and $\mathfrak{g}_{N}=\mathfrak{sl}_{2}$.

##  8 Some data for small $\bm{N}$

In this section, we give some data obtained from Corollary 7.4 when $N$ is
small. Since the cases $N=2,3$ reduce to the cases of [13], we start with
$N=4$.

We always assume that $\bm{\lambda}$, $\bm{k}$, $\bm{z}$ are invariant under
conjugation. By Remark 7.3, we shall only consider the cases that
$\dim(V_{\bm{\lambda}})^{\mathfrak{g}_{N}}\geqslant 2$. We also exclude the
cases that $\bm{z}\in\mathbb{R}\mathring{\mathbb{P}}_{n}$. In particular, the
cases that all pairs $\big{(}\lambda^{(s)},k_{s}\big{)}$, $s=1,\dots,n$, are
different.

We write the highest weights in terms of fundamental weights, for example
$(1,0,0,1)=\omega_{1}+\omega_{4}$. We also write
$\big{(}\lambda^{(1)}\big{)}_{k_{1}},\dots,\big{(}\lambda^{(n)}\big{)}_{k_{n}}$
for $(\bm{\lambda},\bm{k})$ and simply write $\lambda^{(s)}$ for
$(\lambda^{(s)})_{0}$. We use
$\big{(}\lambda_{1}^{(s)},\lambda_{2}^{(s)}\big{)}_{k_{s}}^{\otimes m}$ to
indicate that the pair
$\big{(}\big{(}\lambda_{1}^{(s)},\lambda_{2}^{(s)}\big{)},k_{s}\big{)}$
appears in $(\bm{\lambda},\bm{k})$ exactly $m$ times. For instance,
$(0,1)_{1},(0,1)^{\otimes 3}$ represents the pair $(\bm{\lambda},\bm{k})$
where $\bm{\lambda}=((0,1),(0,1),(0,1),(0,1))$ and $\bm{k}=(1,0,0,0)$.

###  8.1 The case $\bm{N=4,5}$

For each $\mathfrak{g}_{4}$-weight $\lambda=(\lambda_{1},\lambda_{2})$, denote
by $\lambda_{C}$ the $\mathfrak{g}_{5}$-weight $(\lambda_{2},\lambda_{1})$.
Note that $\mathfrak{g}_{4}=\mathfrak{so}_{5}$ is isomorphic to
$\mathfrak{g}_{5}=\mathfrak{sp}_{4}$, the lower bound obtained from the
ramification data
$\bm{\lambda}=\big{(}\lambda^{(1)},\dots,\lambda^{(n)}\big{)}$ and
$\bm{k}=(k_{1},\dots,k_{n})$ of $\mathfrak{g}_{4}$ is the same as that
obtained from the ramification data
$\bm{\lambda_{C}}=\big{(}\lambda_{C}^{(1)},\dots,\lambda_{C}^{(n)}\big{)}$ and
$\bm{k}=(k_{1},\dots,k_{n})$ of $\mathfrak{g}_{5}$.

ramification data | dimension | $c=1$ | $c=2$ | $c=3$
---|---|---|---|---
$(0,1)^{\otimes 6}$ | 14 | 2 | 2 | 6
$(1,0)^{\otimes 3},(0,1)^{\otimes 2}$ | 4 | 0,2 | 2 |
$(1,0)^{\otimes 3},(1,0)_{1}$ | 3 | 1 |  |
$(1,0)^{\otimes 4},(0,0)_{1}$ | 3 | 1 | 3 |
$(0,2),(0,1)^{\otimes 4}$ | 6 | 0 | 2 |
$(0,0)_{1},(0,1)^{\otimes 4}$ | 3 | 1 | 3 |
$(1,0),(0,1)^{\otimes 4}$ | 5 | 1 | 1 |
$(1,1),(0,1)^{\otimes 3}$ | 2 | 0 |  |
$(0,1)_{1},(0,1)^{\otimes 3}$ | 3 | 1 |  |
$(0,2)^{\otimes 2},(0,1)^{\otimes 2}$ | 3 | 1 | 3 |
$(1,0)^{\otimes 2},(0,1)^{\otimes 2}$ | 2 | 0 | 2 |
$(0,2),(1,0),(0,1)^{\otimes 2}$ | 2 | 0 |  |
$(1,0)_{1},(1,0),(0,1)^{\otimes 2}$ | 2 | 0 |  |
$(1,0)^{\otimes 2},(0,1)_{1},(0,1)$ | 2 | 0 |  |
$(1,1),(1,0)^{\otimes 2},(0,1)$ | 2 | 0 |  |
Table 1: The case $N=4,5$.

In Table 1, we give lower bounds for the cases from $\operatorname{Gr}(4,7)$
and $\operatorname{Gr}(5,10)$. By the observation above, we transform the case
from $\operatorname{Gr}(5,10)$ to its counter part in $\operatorname{Gr}(4,d)$
for some $d$ depending on the ramification data. The number in the column of
dimension is equal to $\dim(V_{\bm{\lambda}})^{\mathfrak{g}_{4}}$ for the
corresponding ramification data $\bm{\lambda}$ in each row. The numbers in the
column of $c=i$ equal the lower bounds computed from Corollary 7.4 with the
corresponding $c$.

For a given $c$, there may exist several choices of complex conjugate pair
corresponding to different pairs of $\mathfrak{g}_{N}$-weights. If the
corresponding lower bounds are the same, we just write one number. For
example, in the case of $(0,2)^{\otimes 2}$, $(0,1)^{\otimes 2}$ and $c=1$ of
Table 1, the complex conjugate pair may correspond to the weights
$(0,2)^{\otimes 2}$ or $(0,1)^{\otimes 2}$. However, they give the same lower
bound $1$. Hence we just write $1$ for $c=1$. If the bounds are different, we
write the lower bound with the conjugate pairs corresponding to the leftmost
$2c$ weights first while the one with the conjugate pairs corresponding to the
rightmost $2c$ weights last, in terms of the order of the ramification data
displayed on each row. Since we have at most 3 cases, the possible remaining
case is clear. For instance, in the case $(0,1,0)^{\otimes 4}$,
$(0,0,1)^{\otimes 4}$ and $c=2$ of Table 2, the two complex conjugate pairs
corresponding to $(0,1,0)^{\otimes 4}$ give the lower bound $12$ while the two
complex conjugate pairs corresponding to $(0,0,1)^{\otimes 4}$ give the lower
bound $24$. The remaining case, where the two conjugate pairs corresponding to
$(0,1,0)^{\otimes 2}$ and $(0,0,1)^{\otimes 2}$, gives the lower bound $2$.

###  8.2 The case $\bm{N=6}$

In what follows, we give lower bounds for ramification data consisting of
fundamental weights when $N=6$. We follow the same convention as in Section
8.1.

ramification data | dimension | $c=1$ | $c=2$ | $c=3$ | $c=4$
---|---|---|---|---|---
$(0,0,1)^{\otimes 4}$ | 4 | 0 | 4 |  |
$(0,1,0)^{\otimes 4}$ | 6 | 2 | 6 |  |
$(1,0,0)^{\otimes 4}$ | 3 | 1 | 3 |  |
$(0,0,1)^{\otimes 2},(0,1,0)^{\otimes 2}$ | 3 | 1 | 3 |  |
$(0,0,1)^{\otimes 2},(1,0,0)^{\otimes 2}$ | 2 | 0 | 2 |  |
$(0,1,0)^{\otimes 2},(1,0,0)^{\otimes 2}$ | 3 | 1 | 3 |  |
$(0,0,1)^{\otimes 6}$ | 30 | 2 | 2 | 10 |
$(0,1,0)^{\otimes 6}$ | 130 | 8 | 14 | 36 |
$(1,0,0)^{\otimes 6}$ | 15 | 3 | 3 | 7 |
$(0,1,0)^{\otimes 2},(0,0,1)^{\otimes 4}$ | 34 | 4,2 | 0,6 | 16 |
$(0,1,0)^{\otimes 4},(0,0,1)^{\otimes 2}$ | 55 | 3,1 | 3,7 | 19 |
$(1,0,0)^{\otimes 2},(0,0,1)^{\otimes 4}$ | 16 | 2 | 0,4 | 10 |
$(1,0,0)^{\otimes 4},(0,0,1)^{\otimes 2}$ | 10 | 0,2 | 2,0 | 6 |
$(1,0,0)^{\otimes 2},(0,1,0)^{\otimes 4}$ | 46 | 2 | 6 | 18 |
$(1,0,0)^{\otimes 4},(0,1,0)^{\otimes 2}$ | 21 | 1,3 | 5,3 | 11 |
$(1,0,0)^{\otimes 2},(0,1,0)^{\otimes 2},(0,0,1)^{\otimes 2}$ | 20 | 2 | 0,4,0 | 10 |
$(0,0,1)^{\otimes 8}$ | 330 | 20 | 6 | 0 | 50
$(0,1,0)^{\otimes 8}$ | 6111 | 69 | 59 | 113 | 311
$(1,0,0)^{\otimes 8}$ | 105 | 15 | 9 | 7 | 25
$(0,1,0)^{\otimes 4},(0,0,1)^{\otimes 4}$ | 984 | 22,28 | 12,2,24 | 0,38 | 108
$(1,0,0)^{\otimes 4},(0,0,1)^{\otimes 4}$ | 116 | 6,12 | 8,2,12 | 0,10 | 32
$(1,0,0)^{\otimes 4},(0,1,0)^{\otimes 4}$ | 510 | 6,12 | 22,4,18 | 28,18 | 74
Table 2: The case $N=6$.

### Acknowledgements

The author thanks E. Mukhin and V. Tarasov for useful discussions. The author
also thanks the referees for their comments and suggestions that substantially
improved the first version of this paper. This work was partially supported by
Zhejiang Province Science Foundation, grant No. LY14A010018.

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