hep-ph/***

FTUAM-24-xx IFT-UAM/CSIC-24-18

Superposition of gravitational fields.

Enrique Álvarez and Jesús Anero.

Departamento de Física Teórica and Instituto de Física Teórica, IFT-UAM/CSIC,

Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Linear superposition of gravitational fields is shown to be possible for a
large class of spacetimes, in some specific coordinates. Explicit examples are
presented.

###### Contents

  1. 1 Introduction.
  2. 2 When is GR linear?
  3. 3  Kerr-Schild coordinates.
    1. 3.1  Superposition of Schwarzschild metrics in different frames.
    2. 3.2 Superposition of plane waves
  4. 4 Conclusions
  5. 5 Acknowledgements.

##  1 Introduction.

It is well-known that gravitational fields in the framework of General
Relativity (GR), do not obey the superposition law; that is, the addition of
two free gravitational fields characterized locally by two metrics,
$g^{(1)}_{\mu\nu}(x)$ and $g^{(2)}_{\mu\nu}(x)$, say
$g^{(12)}_{\mu\nu}(x)=g^{(1)}_{\mu\nu}(x)+g^{(2)}_{\mu\nu}(x)$ is not itself a
free gravitational field. In the presence of an energy-momentum acting as a
source the same thing happens; the gravitational field induced by the addition
of two different energy-momentum tensors $T^{(1)}_{\mu\nu}(x)$ and
$T^{(2)}_{\mu\nu}(x)$, namely $T^{(12)}_{\mu\nu}(x)\equiv
T^{(1)}_{\mu\nu}(x)+T^{(2)}_{\mu\nu}(x)$ is not the sum of the two fields
(spacetime metrics) induced by each energy-momentum individually.

$g_{\mu\nu}\left[T^{(1)}_{\mu\nu}(x)+T^{(2)}_{\mu\nu}(x)\right]\neq
g_{\mu\nu}\left[T^{(1)}_{\mu\nu}(x)\right]+g_{\mu\nu}\left[T^{(2)}_{\mu\nu}(x)\right]$
(1)

All this of course stems from the fact that GR is a non-linear theory, and so
are Einstein’s equations of motion (EM), which reduce to Ricci flatness in the
free case. The addition of two Ricci flat metrics is not itself Ricci flat in
general.

One line of research [12][13] attempted to define the combined metric
$g_{\mu\nu}^{(12)}$ through some operation more complex that just a sum. It
would be nice to be able to define a star operation $g^{(1)}_{\mu\nu}(x)\star
g^{(2)}_{\mu\nu}(x)$ such that

$R_{\mu\nu}\left[g^{(1)}_{\alpha\beta}(x)\star
g^{(2)}_{\alpha\beta}(x)\right]=R_{\mu\nu}\left[g^{(1)}_{\alpha\beta}(x)\right]+R_{\mu\nu}\left[g^{(2)}_{\alpha\beta}(x)\right]$
(2)

This has however been met with only limited success.

There are exceptions however, and this in certain coordinates only. We shall
see in detail that in some important cases, superposition holds. Namely there
is a class of Ricci flat spacetimes introduced by Kerr-Schild (KS), for which
Gürses and Gürsey have shown that the equations of motion linearize. This
class is rather large; in it one can find some type D spacetimes, such as
Kerr-Newman (and of course some important particular cases, such as Kerr,
Schwarzschild and Reissner-Nordstrom) as well as some type N spacetimes, such
as gravitational plane waves. It follows that for these spacetimes and in the
right coordinates, superposition holds.

It may come at first sight as a surprise that this superposition property only
happens in certain reference frames. If one thinks about it, it is only
natural. After all, by the equivalence principle, an accelerated frame is
locally equivalent to a gravitational field, so that there is always a (free
falling) frame in which the gravitational field vanishes at a given point. Of
course this free falling frame depends on the particular gravitational field
under consideration and would be in general different for each field to be
superposed. The existence of linear frames  in our case follows the same
philosophy valid in this case for all KS spacetimes.

On the other hand, the idea that not all frames are equivalent from the
physical point of view is quite old, and can be traced back at least to Fock’s
work [8], who used to claim that physics laws hold simpler in harmonic
coordinates. That said, KS coordinates are not harmonic [3] ones in general.

It should be stressed that all our results are valid in GR; no extra symmetry
(e.g. supersymmetry) needs to be invoked.

##  2 When is GR linear?

There is an important class of metrics, namely the Kerr-Schild class
[11][1][5] characterized by the fact that

$g_{\mu\nu}=\eta_{\mu\nu}-2V(x)\lambda_{\mu}\lambda_{\nu}$ (3)

where $\eta_{\mu\nu}$ Minkowski’s metric and the vector $\lambda$ is null

$g_{\mu\nu}\lambda^{\mu}\lambda^{\nu}=\eta_{\mu\nu}\lambda^{\mu}\lambda^{\nu}=0$
(4)

This class contains many physically important type D metrics, like Kerr (and
of course Schwarzschild) as well as type N plane gravitational waves. The
electromagnetic generalizations like Kerr-Newman are also in this class.

In cartesian coordinates the metric is also unimodular

$|g|=1$ (5)

and in fact

$g^{\mu\nu}=\eta^{\mu\nu}+2V\lambda^{\mu}\lambda^{\nu}$ (6)

If the vector is also geodesic, that is

$\lambda^{\mu}\partial_{\mu}\lambda_{\alpha}=\lambda^{\mu}\nabla_{\mu}\lambda_{\alpha}=0$
(7)

then the spacetime is algebraically special.

There are two arguments to show that gravity linearizes in the KS frame, which
are to some stent independent.

  * •

A straightforward calculation shows that Ricci’s tensor in the KS frame
reduces to

$\displaystyle
R_{\mu\nu}=-\partial_{\mu}\partial_{\alpha}(l^{\alpha}l_{\nu})-\partial_{\nu}\partial_{\alpha}(l^{\alpha}l_{\mu})+\Box(l_{\mu}l_{\nu})+$
$\displaystyle+2\left[\partial_{\beta}l^{\alpha}\partial^{\beta}l_{\alpha}-L^{2}+A^{2}-\partial_{\alpha}(K\lambda^{\alpha})\right]l_{\mu}l_{\nu}$
(8)

where

$\displaystyle K\equiv-\partial_{\alpha}(V\lambda^{\alpha})$ $\displaystyle
l_{\mu}\equiv\sqrt{V}\lambda_{\mu}$ $\displaystyle
A\equiv-\lambda^{\alpha}\partial_{\alpha}\left(\sqrt{V}\right)$ $\displaystyle
L\equiv\partial_{\alpha}l^{\alpha}$ (9)

It easily follows that

$\displaystyle G^{\mu}_{\leavevmode\nobreak\
\nu}=\partial^{\mu}(K\lambda_{\nu})+\partial_{\nu}(K\lambda^{\mu})+\Box(l^{\mu}l_{\nu})-\delta^{\mu}_{\nu}\partial_{\alpha}(K\lambda^{\alpha})$
(10)

Bianchi identity implies

$\nabla_{\mu}G^{\mu}_{\leavevmode\nobreak\
\beta}=\partial_{\mu}G^{\mu}_{\leavevmode\nobreak\
\beta}+\Gamma^{\mu}_{\mu\lambda}G^{\lambda}_{\leavevmode\nobreak\
\beta}-\Gamma^{\lambda}_{\mu\beta}G^{\mu}_{\leavevmode\nobreak\ \lambda}=0$
(11)

but given that $\Gamma^{\mu}_{\mu\lambda}=0$ because $|g|=1$ and

$\Gamma^{\lambda}_{\mu\beta}G^{\mu}_{\leavevmode\nobreak\ \lambda}=0$ (12)

Einstein’s tensor is also conserved in the ordinary sense

$\nabla_{\mu}G^{\mu}_{\leavevmode\nobreak\
\beta}=\partial_{\mu}G^{\mu}_{\leavevmode\nobreak\ \beta}=0$ (13)

Now,

$2\eta^{\mu\rho}G^{\nu}_{\leavevmode\nobreak\
\rho}=\partial_{\alpha}\partial_{\beta}\left[\eta^{\mu\nu}g^{\alpha\beta}+\eta^{\alpha\beta}g^{\mu\nu}-\eta^{\mu\alpha}g^{\nu\beta}-\eta^{\nu\alpha}g^{\mu\beta}\right]=2\kappa^{2}T^{\mu\nu}$
(14)

Einstein’s equations in the KS frame are linear.

  * •

Let us now present another argument. Remember [6] the definition of Einstein’s
energy-momentum pseudotensor. Starting with Einstein’s lagrangian

$L=\sqrt{|g|}g^{\mu\nu}\left(\Gamma^{\sigma}_{\mu\nu}\Gamma^{\rho}_{\sigma\rho}-\Gamma^{\rho}_{\mu\sigma}\Gamma^{\sigma}_{\nu\rho}\right)$
(15)

which only involves the metric components and their first derivatives, we
define

$\sqrt{|g|}t_{\mu}^{\nu}\equiv-{\partial
L\over\partial\left(\partial^{\mu}g_{\alpha\beta}\right)}\partial^{\nu}g_{\alpha\beta}+\delta_{\mu}^{\nu}L$
(16)

The result reads

$2\kappa^{2}\sqrt{|g|}t_{\mu}^{\nu}=-\left(\Gamma^{\nu}_{\alpha\beta}-\delta_{\beta}^{\nu}\Gamma^{\sigma}_{\alpha\sigma}\right)\partial_{\mu}\left(\sqrt{|g|}g^{\alpha\beta}\right)+L\delta_{\mu}^{\nu}$
(17)

It can be proved [6] that

$\partial_{\nu}\left(\sqrt{|g|}t_{\mu}^{\nu}\right)=-{1\over
2\kappa^{2}}\sqrt{|g|}G^{\alpha\beta}\partial_{\mu}g_{\alpha\beta}=-{1\over
2}\sqrt{|g|}T^{\alpha\beta}\partial_{\mu}g_{\alpha\beta}$ (18)

that is

$\partial_{\nu}\left(\sqrt{|g|}\left(T_{\mu}\,^{\nu}+t_{\mu}\,^{\nu}\right)\right)=0$
(19)

where we have used the fact that

$0=\sqrt{|g|}\nabla_{\nu}T_{\mu}^{\nu}=\partial_{\nu}\left(\sqrt{|g|}T_{\mu}^{\nu}\right)-{1\over
2}\sqrt{|g|}T^{\alpha\beta}\partial_{\mu}g_{\alpha\beta}$ (20)

On the other hand [4][9]

$\sqrt{|g|}\left(T^{\mu}_{\leavevmode\nobreak\
\nu}+t^{\mu}_{\leavevmode\nobreak\
\nu}\right)=-\frac{1}{2\kappa^{2}}\partial_{\rho}\cal{U}^{\rho\mu}_{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \nu}$ (21)

where we have defined Freud’s superpotential [4]

$\cal{U}^{\mu\rho}_{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \nu}=-\cal{U}^{\rho\mu}_{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\nu}=\mathfrak{g}_{\nu\sigma}\partial_{\lambda}\left(\mathfrak{g}^{\sigma\mu}\mathfrak{g}^{\rho\lambda}-\mathfrak{g}^{\sigma\rho}\mathfrak{g}^{\mu\lambda}\right)$
(22)

with

$\displaystyle\mathfrak{g}^{\mu\nu}\equiv\sqrt{-g}g^{\mu\nu}$
$\displaystyle\mathfrak{g}_{\mu\nu}\equiv\frac{1}{\sqrt{-g}}g_{\mu\nu}$ (23)

In the particular case of the KS ansatz the superpotential reads

${\cal{U}}^{\mu\rho}_{\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\nu}=2\partial_{\beta}\left[\delta^{\mu}_{\nu}l^{\rho}l^{\beta}+\eta^{\beta\rho}l_{\nu}l^{\mu}-\eta^{\beta\mu}l_{\nu}l^{\rho}-\delta^{\rho}_{\nu}l^{\mu}l^{\beta}\right]$
(24)

Einstein equations tell us that

$G^{\mu}_{\leavevmode\nobreak\ \nu}=\kappa^{2}T^{\mu}_{\leavevmode\nobreak\
\nu}=-{1\over 2\sqrt{|g|}}\partial_{\rho}{\cal U}^{\rho\mu}\,_{\nu}$ (25)

It is an important fact [10] that for all KS spacetimes the pseudo tensor
$t^{\mu}_{\leavevmode\nobreak\ \nu}=0$.

This second argument is a little bit roundabout, but physically, all
nonlinearities are hidden in this pseudotensor, so that its vanishing is one
technical explanation as to why the equations of motion linearize.

We conclude that Einstein’s field equation take the linear form

$\partial_{\alpha}\partial_{\beta}\left[\eta^{\mu\nu}g^{\alpha\beta}+\eta^{\alpha\beta}g^{\mu\nu}-\eta^{\mu\alpha}g^{\nu\beta}-\eta^{\nu\alpha}g^{\mu\beta}\right]=2\kappa^{2}\eta^{\mu\rho}T^{\nu}_{\leavevmode\nobreak\
\rho}$ (26)

The Fierz-Pauli [7] equation (which are equivalent to the linearized
Einstein’s) reads

$\Box\left(h_{\mu\nu}-h\eta_{\mu\nu}\right)-\partial_{\mu}\partial^{\lambda}h_{\lambda\nu}-\partial_{\nu}\partial^{\lambda}h_{\lambda\mu}+\partial_{\mu}\partial_{\nu}h+\eta_{\mu\nu}\partial^{\lambda}\partial^{\delta}h_{\lambda\sigma}=0$
(27)

(where $h\equiv\eta^{\alpha\beta}h_{\alpha\beta}$. The KS equation of motion
is then equivalent to Fierz-Pauli’s with $h=0$ (which stems from
unimodularity).

In conclusion, all this means that in all KS spacetimes, there is
superposition of gravitational fields, provided we are using coordinates in
which the basic equation (3) holds true.

##  3  Kerr-Schild coordinates.

The general Kerr-Newman metric reads [1]

$g_{\mu\nu}=\eta_{\mu\nu}-2V\lambda_{\mu}\lambda_{\nu}$ (28)

with

$2V={2Gmr^{3}-Q^{2}r^{2}\over r^{4}+a^{2}z^{2}}$ (29)

and

$\lambda_{\mu}dx^{\mu}=dt+{z\over r}dz+{r\over r^{2}+a^{2}}(xdx+ydy)-{a\over
r^{2}+a^{2}}(xdt-ydx)$ (30)

where $r$ is defined through

${x^{2}+y^{2}\over r^{2}+a^{2}}+{z^{2}\over r^{2}}=1$ (31)

Here the parameter $m$ defines the mass of the spacetime, $a$ is related to
the angular momentum, and $Q$ stands for the electromagnetic charge.

###  3.1  Superposition of Schwarzschild metrics in different frames.

  * •

In order to stress the importance of the reference system, let us consider the
simplest example of Schwarzschild space time in the KS frame where

$V={r_{s}\over 2r}$ (32)

and

$\lambda_{\mu}dx^{\mu}=dt+{xdx+ydy+zdz\over r}=dt+dr$ (33)

We use $r$ only as a abbreviation, not as a coordinate. To be specific, the
form of the metric in KS coordinates reads

$ds^{2}=\left(1-{r_{s}\over r}\right)dt^{2}-\left(\delta_{ij}+{r_{s}\over
r^{3}}\sum_{ij}x^{i}x^{j}\right)dx^{i}dx^{j}-2r_{s}\sum_{i}{x^{i}\over
r^{2}}dx^{i}dt$ (34)

It can be easily verified that this metric is Ricci flat and unimodular.

Let us now consider a superposition of two such metrics

$\text{d}s^{2}_{(12)}=\text{d}s^{2}_{(1)}+\text{d}s^{2}_{(2)}=2\left[\text{d}t^{2}-\text{d}x^{2}-\text{d}y^{2}-\text{d}z^{2}\right]-\frac{r_{1}+r_{2}}{r}\left[\text{d}t+\frac{1}{r}\left(x\text{d}x+\text{d}y+z\text{d}z\right)\right]^{2}$
(35)

It is possible to check that this superposition is Ricci flat as well.

  * •

In the usual Schwarzschild coordinates

$\text{d}s^{2}_{(1)}=\left(1-\frac{r_{1}}{r}\right)\text{d}t-\frac{1}{\left(1-\frac{r_{1}}{r}\right)}\text{d}r-\text{d}\Omega^{2}$
(36)

and

$\text{d}s^{2}_{(2)}=\left(1-\frac{r_{2}}{r}\right)\text{d}t-\frac{1}{\left(1-\frac{r_{2}}{r}\right)}\text{d}r-\text{d}\Omega^{2}$
(37)

for simplicity we call $r_{1}$ and $r_{2}$ the Schwarzschild radii.

In spite of the fact that both metrics are Ricci flat, their sum

$\text{d}s^{2}_{(12)}=\left(2-\frac{r_{1}+r_{2}}{r}\right)\text{d}t-\left[\frac{1}{\left(1-\frac{r_{1}}{r}\right)}+\frac{1}{\left(1-\frac{r_{2}}{r}\right)}\right]\text{d}r-2\text{d}\Omega^{2}$
(38)

is not. To be specific,

$\displaystyle
R_{tt}=\frac{(r_{1}-r_{2})^{2}(r_{1}+r_{2})}{4r^{3}\left(r_{1}+r_{2}-2r\right)^{2}}$
$\displaystyle
R_{rr}=\frac{(r_{1}-r_{2})^{2}\left[3(r_{1}+r_{2})-8r\right]}{4r(r-r_{1})(r-r_{2})\left(r_{1}+r_{2}-2r\right)^{2}}$
(39)

This means that there is no superposition in this frame.

  * •

In fact one can pass from the usual reference system to the Kerr-Schild one as
follows. Start from

$\text{d}s^{2}=\left(1-\frac{r_{s}}{r}\right)\text{d}t^{2}-\left(1+\frac{r_{s}}{r}\right)\text{d}r^{2}-\frac{2r_{s}}{r}\text{d}r\text{d}t-r^{2}\text{d}\Omega^{2}$
(40)

Define a new temporal coordinate

$\text{d}\bar{t}=f\left[A\text{d}t-B\text{d}r\right]$ (41)

Then

$A\text{d}t^{2}-2B\text{d}t\text{d}r=\frac{\text{d}\bar{t}^{2}}{f^{2}A}-\frac{B^{2}\text{d}r^{2}}{A}$
(42)

where in our case

$\displaystyle A=1-\frac{r_{s}}{r}$ $\displaystyle B=\frac{r_{s}}{r}$ (43)

and the metric reduces to

$\text{d}s^{2}=\frac{\text{d}\bar{t}^{2}}{f^{2}\left(1-\frac{r_{s}}{r}\right)}-\frac{1}{\left(1-\frac{r_{s}}{r}\right)}\text{d}r^{2}-r^{2}\text{d}\Omega^{2}$
(44)

Taking $f^{2}\equiv\frac{1}{\left(1-\frac{r_{s}}{r}\right)^{2}}$ (this makes
the second member of (41) a total differential) yields the traditional metric
in Schwarzschild coordinates

$\text{d}s^{2}=\left(1-\frac{r_{s}}{r}\right)\text{d}\bar{t}^{2}-\frac{1}{\left(1-\frac{r_{s}}{r}\right)}\text{d}r^{2}-r^{2}\text{d}\Omega^{2}$
(45)

  * •

In fact, it is even simpler. In Eddington-Filkenstein outgoing coordinates,
where

$u=t-r^{*}$ (46)

and

$r^{*}\equiv r+r_{s}\log\,(r-r_{s})$ (47)

is the usual tortoise coordinate, Schwarzschild’s metric reads

$ds^{2}=du^{2}+2dudr-r^{2}d\Omega^{2}-{r_{s}\over r}du^{2}$ (48)

which is already in the KS frame with

$-2V\lambda_{\mu}\lambda_{\nu}dx^{\mu}dx^{\nu}=-{r_{s}\over r}du^{2}$ (49)

It is however plain that this form is not unimodular. This can be fixed by
defining new coordinates $\mu$ and $x$ such that

$\displaystyle x={r^{3}\over 3}$ $\displaystyle\mu=\cos\,\theta$ (50)

The unimodular metric reads

$ds^{2}=\left(1-{r_{s}\over(3x)^{1/3}}\right)du^{2}-2(3x)^{-2/3}dudx-(3x)^{2/3}\left({d\mu^{2}\over
1-\mu^{2}}+(1-\mu^{2})d\phi^{2}\right)$ (51)

It is remarkable that superposition works with both metrics (48) and (51).

###  3.2 Superposition of plane waves

Consider now gravitational plane waves

$\text{d}s^{2}_{(1)}=\text{d}u\text{d}v-H^{(1)}_{ab}[u]x^{a}x^{b}\text{d}u^{2}-\delta_{ab}\text{d}x^{a}\text{d}x^{b}$
(52)

and

$\text{d}s^{2}_{(2)}=\text{d}u\text{d}v-H^{(2)}_{ab}[u]x^{a}x^{b}\text{d}u^{2}-\delta_{ab}\text{d}x^{a}\text{d}x^{b}$
(53)

which are Ricci flat if $H_{ab}$ is traceless (cf. [2] for a recent reference)

$H_{pq}[u]x^{p}x^{q}\equiv a_{+}[u](x^{2}-y^{2})+2a_{\times}[u]xy$ (54)

It is plain that the sum of two such Ricci flat waves is also another Ricci
flat wave of the same type.

##  4 Conclusions

In spite of the fact that GR is a non-linear theory, and so are Einstein’s
equations of motion (EM), which reduce to Ricci flatness in the free case, in
this paper we have pointed out that in some cases one can superpose
gravitational fields, but this in certain coordinates only. This happens in a
class of spacetimes introduced by Kerr-Schild, for which Einstein’s equations
linearize. This class is rather large; in it one can find some type D
spacetimes, such as Kerr-Newman (and of course some important particular
cases, such as Kerr, Schwarzschild and Reissner-Nordstrom) as well as some
type N spacetimes, such as gravitational plane waves. It follows that for
these spacetimes and in the right coordinates, superposition holds.

We have also discussed why it is not physically surprising that this
superposition property only takes place in certain reference frames. In fact,
it is only natural. Not all frames are equivalent from the physical point of
view. After all, by the equivalence principle, an accelerated frame is locally
equivalent to a gravitational field, so that there is always a (free falling)
frame in which the gravitational field vanishes at a given point. Of course
this free falling frame depends on the particular gravitational field under
consideration and would be in general different for each field to be
superposed.

The existence of linear frames  in our case follows the same philosophy valid
in this case for all Kerr-Schild spacetimes; this is the main surprising
result. In some sense, the KS frame is the generalization of the free falling
frame valid for all spacetimes in this family. In this frame and for those
spacetimes, gravity is linear.

We learn that superposition in this framework is not a diffeomorphism
invariant property.

We are still a long way from being able to elucidate the physical implications
of our results. One of the most important questions is what is the meaning of
the KS frame in the general case and how it can it be related to observables.
Some quantum effects in these spacetimes have been studied in [2]. It would be
very interesting to see whether some generalization of wave packets is
possible for gravitational plane waves.

It has been also pointed out that all our results are valid in GR; no extra
symmetry (e.g. supersymmetry) needs to be invoked.

##  5 Acknowledgements.

One of us (EA) is grateful for stimulating discussions with Luis Álvarez-Gaumé
and Carmelo P. Martín. We acknowledge partial financial support by the Spanish
MINECO through the Centro de excelencia Severo Ochoa Program under Grant
CEX2020-001007-S funded by MCIN/AEI/10.13039/501100011033. We also acknowledge
partial financial support by the Spanish Research Agency (Agencia Estatal de
Investigación) through the grant PID2022-137127NB-I00 funded by
MCIN/AEI/10.13039/501100011033/ FEDER, UE. All authors acknowledge the
European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie grant agreement No 860881-HIDDeN and also byGrant
PID2019-108892RB-I00 funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A
way of making Europe”.

## References

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multiplication and the antipode of the colored dual immaculate functions
remains an open problem. As shown in the example below, the product of two
colored dual immaculate functions does not have exclusively positive structure
constants, and their combinatorial description is not yet evident.

$\mathfrak{S}^{*}_{(ab)}\mathfrak{S}^{*}_{(c)}=\mathfrak{S}^{*}_{(abc)}+\mathfrak{S}^{*}_{(c,ab)}+\mathfrak{S}^{*}_{(ac,b)}-\mathfrak{S}^{*}_{(a,bc)}$

##  7\. A partially commutative generalization of the row-strict dual
immaculate functions

Niese, Sundaram, Van Willigenburg, Vega, and Wang define a pair of dual bases
in $QSym$ and $NSym$ in [23] by applying an involution $\psi$ to the
immaculate and dual immaculate bases. The row-strict dual immaculate basis has
extensive representation theoretic applications, specifically to 0-Hecke
algebras [23]. The combinatorics of this basis involve a variation of
immaculate tableaux with different conditions on the rows and columns. Note
that the original paper uses French notation for diagrams (the bottom row is
row 1) so the definitions here have been adapted to English notation. We first
review the theory of these two bases, then define their colored
generalizations, and finally extend our earlier results using a lift of the
original $\psi$.

###  7.1. Row-strict immaculate and dual immaculate functions

We begin by recalling several definitions and results from [23].

######  Definition 7.1.

Given a composition $\alpha$, a _row-strict immaculate tableau_ $T$ is a
filling of the diagram of $\alpha$ such that the entries in the leftmost
column weakly increase from top to bottom and the entries in each row strictly
increase from left to right. A row-strict immaculate tableau with $n$ boxes is
_standard_ if each integer $1$ through $n$ appears exactly once. The _type_ of
a row-strict immaculate tableau is defined the same way as the type of an
immaculate tableau.

Monomials are associated with row-strict immaculate tableaux according to
their type in the same fashion as immaculate tableaux.

######  Definition 7.2.

For a composition $\alpha$, the _row-strict dual immaculate function_ is
defined as

$\mathfrak{RS}^{*}_{\alpha}=\sum_{T}x^{T},$

where the sum runs over all row-strict immaculate tableaux $T$ of shape
$\alpha$.

To standardize a row-strict immaculate tableau $T$, replace the $1$’s in $T$
with $1,2,\ldots$ moving left to right and top to bottom, then continue with
the $2$’s, etc. Note also that the set of standard row-strict immaculate
tableaux is the same as the set of standard immaculate tableaux.

######  Definition 7.3.

A positive integer $i$ is a _row-strict descent_ of a standard row-strict
immaculate tableau $U$ if $U$ contains the entry $i+1$ in a weakly higher row
than entry $i$. The row-strict descent set of a standard row-strict immaculate
tableau $U$ is

$Des^{rs}(U)=\\{i:i+1\text{ is weakly above }i\text{ in }U\\}.$

The _row-strict descent composition_ of a standard row-strict immaculate
tableaux $U$ is defined as

$co^{rs}(U)=comp(Des^{rs}(U)).$

Following these definitions, the row-strict dual immaculate function for a
composition $\alpha$ can also be defined as

$\mathfrak{RS}^{*}_{\alpha}=\sum_{S}F_{co^{rs}(S)},$

where the sum is taken over all standard row-strict immaculate tableaux of
shape $\alpha$.

######  Definition 7.4.

The dual involutions $\psi:QSym\rightarrow QSym$ and $\psi:NSym\rightarrow
NSym$ are defined

$\psi(F_{\alpha})=F_{\alpha^{c}}\text{\quad and
\quad}\psi(R_{\alpha})=R_{\alpha^{c}}.$

Note that there are two separate $\psi$ involutions, although they are often
referred to together as if they are one map.

######  Theorem 7.5.

[23] Let $\alpha$ be a composition. Then,
$\psi(\mathfrak{S}^{*}_{\alpha})=\mathfrak{RS}^{*}_{\alpha^{c}}.$

Since $\psi$ is an involution and $\mathfrak{S}^{*}_{\alpha}$ is a basis,
$\\{\mathfrak{RS}^{*}_{\alpha}\\}_{\alpha}$ is a basis for $QSym$.

######  Definition 7.6.

For a composition $\alpha$ and weak composition $\beta$, let
$K^{rs}_{\alpha,\beta}$ be the number of row-strict immaculate tableaux of
shape $\alpha$ and type $\beta$, and $L^{rs}_{\alpha,\beta}$ be the number of
standard row-strict immaculate tableaux of shape $\alpha$ with row-strict
descent composition $\beta$.

######  Theorem 7.7.

[23] For a composition $\alpha$, the row-strict dual immaculate function
expands as

$\mathfrak{RS}^{*}_{\alpha}=\sum_{\beta}K^{rs}_{\alpha,\beta}M_{\beta}=\sum_{\gamma}L^{rs}_{\alpha,\gamma}F_{\gamma}.$

where the sums run over compositions $\beta$ and $\gamma$ such that
$|\beta|=|\alpha|$ and $|\gamma|=|\alpha|$.

The row-strict dual immaculate functions have a dual basis that can be
constructed similarly to the immaculate basis.

######  Definition 7.8.

For $m\in\mathbb{Z}$, the _non-commutative row-strict Bernstein operator_
$\mathbb{B}^{rs}_{m}$ is defined by

$\mathbb{B}^{rs}_{m}=\sum_{i\geq 0}(-1)^{i}E_{m+i}F^{\perp}_{(i)},\text{\qquad
and
\qquad}\mathbb{B}^{rs}_{\alpha}=\mathbb{B}^{rs}_{\alpha_{1}}\ldots\mathbb{B}^{rs}_{\alpha_{k}}\text{\qquad
for $\alpha\in\mathbb{Z}^{k}$}.$

For a composition $\alpha$, the _row-strict immaculate function_
$\mathfrak{RS}_{\alpha}$ is defined as

$\mathfrak{RS}_{\alpha}=\mathbb{B}^{rs}_{\alpha}(1).$

These functions are dual to the row-strict dual immaculate basis,
$\langle\mathfrak{RS}_{\alpha},\mathfrak{RS}^{*}_{\beta}\rangle=\delta_{\alpha,\beta}$,
and they are the image of the immaculate basis under $\psi$.

Applying $\psi$ to various results from [3] yields similar results for the
row-strict immaculate and row-strict dual immaculate bases, which are
summarized in the following result.

######  Theorem 7.9.

[23] For compositions $\alpha,\beta\models n$, $s\in\mathbb{Z}_{\geq 0}$,
$m\in\mathbb{Z}$, and $f\in NSym$,

  * (1)

$\mathbb{B}_{m}(f)H_{s}=\mathbb{B}_{m+1}(f)H_{s-1}+\mathbb{B}_{m}(fH_{s})\xLongleftrightarrow{\psi}\mathbb{B}_{m+1}^{rs}(f)E_{s-1}+\mathbb{B}^{rs}_{m}(fE_{s}).$

  * (2)

Multiplicity-free right Pieri rule:

$\mathfrak{S}_{\alpha}H_{s}=\sum_{\alpha\subset_{s}\beta}\mathfrak{S}_{\beta}\xLongleftrightarrow{\psi}\mathfrak{RS}_{\alpha}E_{s}=\sum_{\alpha\subset_{s}\beta}\mathfrak{RG}_{\beta}.$

  * (3)

Multiplicity-free right Pieri rule:

$\mathfrak{S}_{\alpha}\mathfrak{S}_{(1^{s})}=\mathfrak{S}_{\alpha}E_{s}=\sum_{\beta}\mathfrak{S}_{\beta}\xLongleftrightarrow{\psi}\mathfrak{RS}_{\alpha}\mathfrak{RG}_{(1^{s})}=\mathfrak{RS}_{\alpha}H_{s}=\sum_{\beta}\mathfrak{RS}_{\beta},$

where the sum runs over compositions $\beta\models|\alpha|+s$ such that
$\alpha_{i}\leq\beta_{i}\leq\alpha_{i}+1$ and $\alpha_{i}=0$ for
$i>\ell(\alpha)$.

  * (4)

$\mathfrak{S}_{(1^{n})}=\sum_{\alpha\models
n}(-1)^{n-\ell(\alpha)}H_{\alpha}=E_{n}\xLongleftrightarrow{\psi}\mathfrak{RG}_{(1^{n})}=\sum_{\alpha\models
n}(-1)^{n-\ell(\alpha)}E_{\alpha}=H_{n}.$

  * (5)

Complete homogeneous and elementary expansions:

$H_{\beta}=\sum_{\alpha\geq_{\ell}\beta}K_{\alpha,\beta}\mathfrak{S}_{\alpha}\xLongleftrightarrow{\psi}E_{\beta}=\sum_{\alpha\geq_{\ell}\beta}K_{\alpha,\beta}\mathfrak{RS}_{\alpha},\text{\quad\quad}H_{\beta}=\sum_{\alpha\geq_{\ell}\beta}K^{rs}_{\alpha,\beta}\mathfrak{RS}_{\alpha}\xLongleftrightarrow{\psi}E_{\beta}=\sum_{\alpha\geq_{\ell}\beta}K^{rs}_{\alpha,\beta}\mathfrak{S}_{\alpha}.$

  * (6)

Ribbon basis expansions:

$R_{\beta}=\sum_{\alpha\geq_{\ell}\beta}L_{\alpha,\beta}\mathfrak{S}_{\alpha}\xLongleftrightarrow{\psi}R_{\beta^{c}}=\sum_{\alpha\geq_{\ell}\beta}L_{\alpha,\beta}\mathfrak{RS}_{\alpha}.$

The immaculate poset also represents a poset of the standard row-strict
immaculate tableaux as a result of the equivalence between standard immaculate
tableaux and standard row-strict immaculate tableaux, thus results for the
row-strict skew case follow closely to those of the dual immaculate functions.

######  Definition 7.10.

Let $\alpha$ and $\beta$ be compositions with $\beta\subseteq\alpha$. A skew
row-strict immaculate tableau is a skew shape $\alpha/\beta$ filled with
positive integers such that the entries in the first column are weakly
increasing from top to bottom and the entries in each row strictly increase
from left to right.

######  Definition 7.11.

For compositions $\alpha,\beta$ such that $\beta\subseteq\alpha$, the skew
row-strict dual immaculate functions are defined as

$\mathfrak{RS}^{*}_{\alpha/\beta}=\sum_{\gamma}\langle\mathfrak{RS}_{\beta}H_{\gamma},\mathfrak{RS}^{*}_{\alpha}\rangle
M_{\gamma},$

where the sum runs over all $\gamma\in\mathfrak{P}$ such that
$|\alpha|-|\beta|=|\gamma|$.

As with the skew dual immaculate functions, these functions connect to the
multiplication of the row-strict immaculate functions and the comultiplication
of the row-strict dual immaculate functions.

######  Theorem 7.12.

[23] Let $\alpha$ and $\beta$ be compositions with $\beta\subseteq\alpha$.
Then,

$\mathfrak{RS}^{*}_{\alpha/\beta}=\sum_{T}x^{T},$

where the sum runs over all skew row-strict immaculate tableaux $T$ of shape
$\alpha/\beta$. Moreover,

$\mathfrak{RS}^{*}_{\alpha/\beta}=\psi(\mathfrak{S}^{*}_{\alpha/\beta})=\sum_{\gamma}\langle\mathfrak{RS}_{\beta}R_{\gamma},\mathfrak{RS}^{*}_{\alpha}\rangle
F_{\gamma}=\sum_{\gamma}\langle\mathfrak{RS}_{\beta}\mathfrak{RS}_{\gamma},\mathfrak{RS}^{*}_{\alpha}\rangle\mathfrak{RS}^{*}_{\gamma},$

where the sums run over all compositions $\gamma\in\mathfrak{P}$ such that
$|\alpha|-|\beta|=|\gamma|$.

Comultiplication on the row-strict dual immaculate functions is also defined
in terms of skew shapes.

######  Definition 7.13.

For a composition $\alpha$,

$\Delta(\mathfrak{RS}^{*}_{\alpha})=\sum_{\beta}\mathfrak{RG}^{*}_{\beta}\otimes\mathfrak{RS}^{*}_{\alpha/\beta},$

where the sum runs over all compositions $\beta$ such that
$\beta\subseteq\alpha$.

###  7.2. Colored row-strict dual immaculate functions in $QSym_{A}$

To generalize these definitions and results to the colored case, we first
define a lift of the involution $\psi$ to $QSym_{A}$ and $NSym_{A}$. Note that
we technically define two separate dual involutions $\psi$, one on $QSym_{A}$
and one on $NSym_{A}$, but we treat them as a single map that works on both
spaces.

######  Definition 7.14.

For a sentence $J$, define the linear maps $\psi:QSym_{A}\rightarrow QSym_{A}$
and $\psi:NSym_{A}\rightarrow NSym_{A}$ by

$\psi(F_{J})=F_{J^{c}}\text{\quad and \quad}\psi(R_{J})=R_{J^{c}}.$

######  Proposition 7.15.

The maps $\psi$ are involutions, and the duality between $QSym_{A}$ and
$NSym_{A}$ is invariant under $\psi$, meaning that

$\langle G,F\rangle=\langle\psi(G),\psi(H)\rangle.$

Furthermore, the map $\psi:NSym_{A}\rightarrow NSym_{A}$ is an isomorphism.

###### Proof.

To see that $\psi$ is invariant under duality, it suffices to observe that
$\langle R_{I},F_{J}\rangle=\langle
R_{I^{c}},F_{J^{c}}\rangle=\langle\psi(R_{I}),\psi(F_{J})\rangle$. The map
$\psi$ is an involution because $\psi(\psi(F_{I}))=F_{(I^{c})^{c}}=F_{I}$ and
$\psi(\psi(R_{I}))=R_{(I^{c})^{c}}=R_{I}$ and the map extends linearly. Next,
we show that $\psi$ is an isomorphism on $NSym_{A}$. For sentences $I$ and
$J$, we have $R_{I}R_{J}=R_{I\cdot J}+R_{I\odot J}$ [11] and thus
$\psi(R_{I}R_{J})=\psi(R_{I\cdot J})+\psi(R_{I\odot J})$. Observe that
$(I\cdot J)^{c}=I^{c}\odot J^{c}$ and $(I\odot J)^{c}=I^{c}\cdot J^{c}$.
Therefore, $\psi(R_{I}R_{J})=R_{I^{c}\odot J^{c}}+R_{I^{c}\cdot
J^{c}}=R_{I^{c}}R_{J^{c}}=\psi(R_{I})\psi(R_{J}).$ ∎

Note that $\psi:QSym_{A}\rightarrow QSym_{A}$ is not an isomorphism because it
fails to preserve multiplication. Now, we prove that $\psi$ maps the complete
homogenous basis to the elementary basis in $NSym$ and vice versa, which will
allow us to apply $\psi$ to both these bases.

######  Proposition 7.16.

For a sentence $J$, $\psi(E_{J})=H_{J}$.

###### Proof.

First, for a sentence $J$, we expand $E_{J}$ in terms of the colored ribbon
basis as

$\displaystyle E_{J}$ $\displaystyle=\sum_{K\preceq
J}(-1)^{|J|-\ell(K)}H_{K}=\sum_{K\preceq
J}(-1)^{|J|-\ell(K)}\left[\sum_{I\succeq
K}R_{I}\right]=\sum_{I}\left[\sum_{K\preceq
J,I}(-1)^{|J|-\ell(K)}\right]R_{I}.$ Next, we split the sum into two pieces
according to $I$: one where $I\succ J^{c}$ and the other where $I\preceq
J^{c}$, $\displaystyle E_{J}$ $\displaystyle=\sum_{I\succ
J^{c}}\left[\sum_{K\preceq J,I}(-1)^{|J|-\ell(K)}\right]R_{I}+\sum_{I\preceq
J^{c}}\left[\sum_{K\preceq J,I}(-1)^{|J|-\ell(K)}\right]R_{I}.$ In the first
case, observe that $I\succ J^{c}$ implies that $J\succ I$. Thus, $K\preceq
J,I$ becomes $K\preceq J$. Also notice that because $J$ is constant we can
write $(-1)^{|J|}=(-1)^{|J|-\ell(J)}(-1)^{\ell(J)}$ and factor the first term
out of the sum. In the second case, $I\preceq J^{c}$ means that $K\preceq I,J$
becomes $K\preceq J,J^{c}$. The only way for $K$ to be a refinement of a
sentence and its complement is if $K$ is a sentence made up of only single
letters. That is, $|K|=\ell(K)$. Thus the inner sum has only one summand,
which is $(-1)^{|J|-\ell(K)}=(-1)^{|J|-|K|}=1$. As a result, the equation
simplifies as $\displaystyle E_{J}$ $\displaystyle=\sum_{I\succ
J^{c}}(-1)^{|J|-\ell(J)}\left[\sum_{J^{c}\preceq K\preceq
I}(-1)^{\ell(J)-\ell(K)}\right]R_{I}+\sum_{I\preceq J^{c}}R_{I}.$ By
properties of the Möbius function [11], the coefficient of first section is 0
for all $K$ and we are left with $\displaystyle E_{J}$
$\displaystyle=\sum_{I\preceq J^{c}}R_{I}.$ Therefore, applying $\psi$ to
$E_{J}$ and noticing that $I\preceq J^{c}$ if and only if $J\preceq I^{c}$,
yields $\displaystyle\psi(E_{J})$ $\displaystyle=\sum_{I\preceq
J^{c}}\psi(R_{I})=\sum_{I\preceq J^{c}}R_{I^{c}}=\sum_{J\preceq
I^{c}}R_{I^{c}}=H_{J}.\qed$

We continue by defining and studying colored row-strict immaculate tableaux.
Their combinatorics in relation to those of the colored immaculate tableaux
will allow us to define the colored row-strict dual immaculate basis and
verify its relationship to the colored dual immaculate basis via $\psi$.

######  Definition 7.17.

A colored row-strict immaculate tableau (CRSIT) of shape $I$ is a colored
composition diagram of shape $I$ in which the sequence of integer entries is
strictly increasing from left to right in each row, and weakly increasing top
to bottom in the leftmost column. The _type_ of a colored row-strict
immaculate tableau $T$ is the sentence $C=(u_{1},\ldots,u_{g})$ such that for
each $i\in[g]$ the word $u_{i}$ lists the colors of all boxes in $T$ filled
with the integer $i$ in the order they appear when entries in $T$ are read
from left to right and top to bottom. A standard colored row-strict immaculate
tableau is a colored row-strict immaculate tableau of size $n$ with the
integer entries $1,\ldots,n$ each appearing exactly once. To _standardize_ a
colored row-strict tableau, replace its integer entries with the numbers
$1,2,\ldots$ based on the order they appear in the type, first replacing all
entries equal to $1$, then $2$, etc. just as in the standardization of non-
colored row-strict immaculate tableaux.

We also use the same notion of row-strict descents and the row-strict descent
set $Des^{rs}$ from row-strict immaculate tableaux, but define an additional
concept of colored row-strict descent composition.

######  Definition 7.18.

The _colored row-strict descent composition_ of a standard colored row-strict
immaculate tableau $U$, denoted $co^{rs}_{A}(U)$, is the sentence obtained by
reading the colors in each box in order of their number and splitting into a
new word after each row-strict descent.

######  Example 7.19.

A few CRSIT of shape $(ab,bca)$ along with their types and standardization, as
well as the row-strict descent sets and colored row-strict descent
compositions of these standardizations, are:

$T_{1}=$ a, 1 & b, 2
b, 1 c, 3 a, 4 $T_{2}=$ a, 1 & b, 3
b, 1 c, 2 a, 4

$T_{3}=$ a, 1 & b, 4
b, 2 c, 3 a, 3

$\quad\ \
(ab,b,c,a)\quad\quad\quad\quad\quad(ab,c,b,a)\quad\quad\quad\quad\quad\
(a,b,bc,a)$

$U_{2}=$ a, 1 & b, 3
b, 2 c, 4 a, 5

$U_{3}=$ a, 1 & b, 4
b, 2 c, 3 a, 5 $U_{4}=$ a, 1 & b, 5
b, 2 c, 3 a, 4

$\quad\quad\quad\begin{subarray}{c}Des^{rs}(U_{2})=\\{2,4\\}\\\
co^{rs}_{A}(U_{2})=(ab,bc,a)\end{subarray}\quad\quad\quad\begin{subarray}{c}Des^{rs}(U_{3})=\\{2,3\\}\\\
co^{rs}_{A}(U_{3})=(ab,c,ba)\end{subarray}\quad\quad\quad\begin{subarray}{c}Des^{rs}(U_{4})=\\{2,3,4\\}\\\
co^{rs}_{A}(U_{4})=(ab,c,a,b)\end{subarray}$

######  Definition 7.20.

For a sentence J, the _colored row-strict dual immaculate function_ is defined
as

$\mathfrak{RS}^{*}_{J}=\sum_{T}x_{T},$

where the sum is taken over all colored row-strict immaculate tableaux $T$ of
shape $J$.

######  Example 7.21.

For $J=(ab,bca)$, the colored row-strict dual immaculate function is

$\mathfrak{RS}^{*}_{ab,bca}=x_{ab,1}x_{b,2}x_{c,3}x_{a,4}+x_{ab,1}x_{c,2}x_{b,3}x_{a,4}+x_{a,1}x_{b,2}x_{bc,3}x_{a,4}+\ldots+2x_{a,1}x_{b,2}x_{b,3}x_{c,4}x_{a,5}+\ldots.$

######  Proposition 7.22.

For a sentence $J$,

$\mathfrak{RS}^{*}_{J}=\sum_{S}F_{co^{rs}_{A}(S)},$

where the sum runs over all standard colored row-strict immaculate tableaux
$S$ of shape $J$.

###### Proof.

Let $T$ be a colored row-strict immaculate tableau of shape $J$ that
standardizes to the standard colored row-strict immaculate tableau $S$. The
flattening of the type of $T$ must be a refinement of the colored row-strict
descent composition of $S$, which can be shown by applying the same reasoning
used in the proof of Proposition 4.14. In fact, each sentence $B$ that
flattens to a refinement of $co^{rs}_{A}(S)$ corresponds to a unique colored
row-strict immaculate tableau of type $B$ that standardizes to $S$. Therefore,

$F_{co^{rs}_{A}(S)}=\sum_{T_{S}}x_{T_{S}},$

where the sum runs over all colored row-strict immaculate tableaux $T_{S}$ of
shape $J$ that standardize to $S$. It follows that

$\mathfrak{RS}^{*}_{J}=\sum_{T}x_{T}=\sum_{S}\sum_{T_{S}}x_{T_{S}}=\sum_{S}F_{co^{rs}_{A}(S)},$

where the sums run over all CRSIT $T$ of shape $J$, all standard CRSIT $S$ of
shape $J$, and all CRSIT $T_{S}$ of shape $J$ that standardize to $S$. ∎

######  Theorem 7.23.

Let J be a sentence. Then,

$\psi(\mathfrak{S}^{*}_{J})=\mathfrak{RS}^{*}_{J}.$

###### Proof.

For a sentence $J$,

$\psi(\mathfrak{S}^{*}_{J})=\psi(\sum_{U}F_{co_{A}(U)})=\sum_{U}F_{(co_{A}(U))^{c}}.$

The complement of the colored descent composition of a standard colored
immaculate tableau $U$ splits exactly where $U$ does not have a descent. These
are exactly the locations of the row-strict descents in $U$, thus
$(co_{A}(U))^{c}=co^{rs}_{A}(U)$, and

∎
$\psi(\mathfrak{S}^{*}_{J})=\sum_{U}F_{(co^{rs}_{A}(U))}=\mathfrak{RS}^{*}_{J}.$

Because $\\{\mathfrak{S}^{*}_{J}\\}_{J}$ is a basis and $\psi$ is an
involution, Theorem 7.23 also implies the following.

######  Corollary 7.24.

$\\{\mathfrak{RS}^{*}_{J}\\}_{J}$ is a basis for $QSym_{A}.$

Using $\psi$, we extend each of our results on the colored dual immaculate
functions to the colored row-strict dual immaculate functions.

######  Definition 7.25.

For sentences $J,C$ and weak sentence $B$, define $K^{rs}_{J,B}$ as the number
of colored row-strict immaculate tableaux of shape J and type B, and
$L^{rs}_{J,C}$ as the number of standard colored row-strict immaculate
tableaux of shape J with row-strict descent composition C.

######  Proposition 7.26.

For a sentence $J$,

$\mathfrak{RS}^{*}_{J}=\sum_{B}K^{rs}_{J,B}M_{B}\text{\quad\quad and
\quad\quad}\mathfrak{RS}^{*}_{J}=\sum_{C}L^{rs}_{J,C}F_{C},$

where the sums run over sentences $B$ and $C$ such that $|B|=|J|$ and
$|C|=|J|$.

The above proposition follows from Definition 7.20 in the manner of Theorem
4.20. The results of Section 4.3 also extend nicely to the row-strict case
under the involution $\psi$.

######  Definition 7.27.

The _colored row-strict immaculate descent graph_ , denoted
${}^{rs}\mathfrak{D}^{n}_{A}$ is the edge-weighted directed graph with the set
of sentences on $A$ of size $n$ as its vertex set and an edge from each
sentence $I$ to $J$ if there exists a standard colored row-strict immaculate
tableau of shape $I$ with colored row-strict descent composition $J$. The edge
from $I$ to $J$ is weighted with the coefficient $L^{rs}_{I,J}$.

Due to the differing definitions of descents and descent compositions in row-
strict tableaux, the neighbors of $I$ in ${}^{rs}\mathfrak{D}^{n}_{A}$ are
exactly the (sentence) complements of $I$’s neighbors in
$\mathfrak{D}^{n}_{A}$ and the complement of $I$ itself. Here, we say two
vertices are _neighbors_ if they are adjacent by an edge in either direction.

######  Example 7.28.

The standard colored row-strict immaculate tableaux of shape $(ab,cbb)$ have
colored row-strict descent compositions $(a,bc,b,b)$, $(ac,bb,b)$,
$(ac,b,bb)$, and $(ac,b,bb)$, so $(ab,cbb)$ has outgoing edges to these
sentences in ${}^{rs}\mathfrak{D}^{5}_{\\{a,b,c\\}}$. Notice that if we take
the complement of each of these sentences we get $(ab,cbb)$, $(a,cb,bb)$,
$(a,cbbb)$, and $(a,cbb,b)$ which are exactly $(ab,cbb)$ itself and the
sentences to which it has outgoing edges to in
$\mathfrak{D}^{5}_{\\{a,b,c\\}}$, as seen in Figure 1.

######  Proposition 7.29.

For a sentence $I$, the colored fundamental functions expand into the colored
row strict immaculate basis as

$F_{I}=\sum_{J}L^{rs(-1)}_{I,J}\mathfrak{RS}^{*}_{J}\text{ \qquad with
coefficients
\qquad}L^{rs(-1)}_{I,J}=\sum_{\mathcal{P}}(-1)^{\ell(\mathcal{P})}prod(\mathcal{P}),$

where the sums run over all sentences $J$ below $I$ in
${}^{rs}\mathfrak{D}^{n}_{A}$ and all directed paths $\mathcal{P}$ from $I$ to
$J$ in ${}^{rs}\mathfrak{D}^{n}_{A}$.

The proof follows that of Theorem 4.29 using Proposition 7.26 in place of
Theorem 4.25. Similarly, this proposition specializes to the non-colored case
in the same manner as Corollary 4.31.

###  7.3. Colored row-strict immaculate functions

We define the colored row-strict immaculate functions as the image of the
colored immaculate functions under $\psi$, and thus also as the basis dual to
the colored row-strict dual immaculate functions.

######  Definition 7.30.

For a sentence $J$, the _colored row-strict immaculate function_ is defined as

$\mathfrak{RS}_{J}=\psi(\mathfrak{S}_{J}).$

Equivalently, due to the invariance of $\psi$ under duality, we have
$\langle\mathfrak{RS}_{I},\mathfrak{RS}^{*}_{J}\rangle=\delta_{I,J}$.

Applying $\psi$ to the colored immaculate functions yields row-strict versions
of our earlier results and colored generalizations of the results in Theorem
7.9. Note that certain results from Theorem 7.9 are not generalized here
because we lack the corresponding result on the colored immaculate functions
or due to the fact that $\psi$ is not an isomorphism on $QSym_{A}$. The non-
colored analogues of $\psi$ are automorphisms on both $QSym$ and $NSym$.

######  Theorem 7.31.

For words $w$ and $v$, sentences $J$ and $C$, and $f\in NSym_{A}$

  * (1)

Right Pieri rule:

$\mathfrak{S}_{J}H_{w}=\sum_{J\subset_{w}K}\mathfrak{S}_{K}\xLongleftrightarrow{\psi}\mathfrak{RS}_{J}E_{w}=\sum_{J\subset_{w}K}\mathfrak{RS}_{K}.$

  * (2)

Colored complete homogeneous and colored elementary expansions:

$H_{C}=\sum_{J}K_{J,C}\mathfrak{S}_{J}\xLongleftrightarrow{\psi}E_{C}=\sum_{J}K_{J,C}\mathfrak{RS}_{J},\text{\quad\quad}H_{C}=\sum_{J}K^{rs}_{J,C}\mathfrak{RS}_{J}\xLongleftrightarrow{\psi}E_{C}=\sum_{J}K^{rs}_{J,C}\mathfrak{S}_{J}.$

  * (3)

Colored ribbon expansions:

$R_{C}=\sum_{J}L_{J,C}\mathfrak{S}_{J}\xLongleftrightarrow{\psi}R_{C^{c}}=\sum_{J}L_{J,C}\mathfrak{RS}_{J}.$

The application of Proposition 2.14 to Proposition 7.29 also yields the
following result. The analogous result is also true in $NSym$, as in Corollary
5.20.

######  Corollary 7.32.

For a sentence $J$, the colored row-strict immaculate functions expand into
the colored ribbon basis as

$\mathfrak{RS}_{J}=\sum_{I}L^{rs(-1)}_{I,J}R_{I}\text{\qquad with coefficients
\qquad}L^{rs(-1)}_{I,J}=\sum_{\mathcal{P}}(-1)^{\ell(\mathcal{P})}prod(\mathcal{P})$

where the sums run over all $I$ above $J$ in ${}^{rs}\mathfrak{D}^{n}_{A}$ and
all paths $\mathcal{P}$ from $I$ to $J$ in ${}^{rs}\mathfrak{D}^{n}_{A}$.

###  7.4. The colored immaculate poset and skew colored row-strict dual
immaculate functions

The set of standard colored immaculate tableaux is equal to the set of
standard colored row-strict immaculate tableaux, meaning that many of our
results on the colored immaculate poset immediately extend to the row-strict
setting.

######  Definition 7.33.

For sentences $I$ and $J$ where $J\subseteq_{L}I$, a _skew colored row-strict
immaculate tableau_ of shape $I/J$ is a colored skew shape $I/J$ filled with
positive integers such that the sequences of entries in the first column is
weakly increasing top to bottom and the sequence of integers in each row is
strictly increasing left to right.

######  Definition 7.34.

For sentences $I$ and $J$ where $J\subseteq_{L}I$, define the skew colored
row-strict dual immaculate function as

$\mathfrak{RS}^{*}_{I/J}=\sum_{K}\langle\mathfrak{RS}_{J}H_{K},\mathfrak{RS}^{*}_{I}\rangle
M_{K},$

where the sum runs over all sentences $K\in\mathfrak{P}_{A}$ such that
$|I|-|J|=|K|$.

Applying $\psi$ to the equations in Proposition 6.8 yields the following
results.

######  Theorem 7.35.

For sentences $I$ and $J$ with $J\subseteq_{L}I$,

$\mathfrak{RS}^{*}_{I/J}=\sum_{K}\langle\mathfrak{RS}_{J}R_{K},\mathfrak{RS}^{*}_{I}\rangle
F_{K}=\sum_{K}\langle\mathfrak{RS}_{J}\mathfrak{RS}_{K},\mathfrak{RS}^{*}_{I}\rangle\mathfrak{RS}^{*}_{K},$

where the sums run over all sentences $K\in\mathfrak{P}_{A}$ such that
$|I|-|J|=|K|$.

######  Proposition 7.36.

For sentences $I$ and $J$ such that $J\subseteq_{L}I$,

$\psi(\mathfrak{S}^{*}_{I/J})=\mathfrak{RS}^{*}_{I/J}.$

###### Proof.

Let $I$ and $J$ be sentences such that $J\subseteq_{L}I$. Then,

$\displaystyle\psi(\mathfrak{RS}^{*}_{I/J})$
$\displaystyle=\sum_{K}\langle\mathfrak{RS}_{J}R_{K},\mathfrak{RS}^{*}_{I}\rangle\psi(F_{K})=\sum_{K}\langle\mathfrak{RS}_{J}R_{K},\mathfrak{RS}^{*}_{I}\rangle
F_{K^{c}}$
$\displaystyle=\sum_{K}\langle\psi(\mathfrak{S}_{J}R_{K^{c}}),\psi(\mathfrak{S}^{*}_{I})\rangle
F_{K^{c}}\text{\quad by Theorem \ref{involutionbasis}}$
$\displaystyle=\sum_{K}\langle\mathfrak{S}_{J}R_{K^{c}},\mathfrak{S}^{*}_{I}\rangle
F_{K^{c}}=\mathfrak{S}^{*}_{I/J}.\text{\quad By Proposition
\ref{rowstrictdual}}\qed$

Comultiplication on the colored row-strict immaculate basis can be defined in
terms of skew colored row-strict immaculate functions. The proof follows that
of Proposition 6.10 using Theorem 7.35.

######  Proposition 7.37.

Let $I$ be a sentence. Then,

$\Delta(\mathfrak{RS}^{*}_{I})=\sum_{J}\mathfrak{RS}^{*}_{J}\otimes\mathfrak{RS}^{*}_{I/J},$

where the sum runs over all sentences $J$ such that $J\subseteq_{L}I$.

Multiplication and antipode of the colored row-strict dual immaculate
functions are closely related to the multiplication and antipode of colored
dual immaculate functions, and thus also remain open.

###  7.5. Future work

Our future work on this project will take three directions. First, we hope to
continue exploring properties of the colored immaculate and dual immaculate
bases by looking at multiplicative structures, potential Jacobi-Trudi
formulas, possible rim hook generalizations, and expansions to and from more
bases. Second, we will continue to generalize other Schur-like bases to
$QSym_{A}$ and $NSym_{A}$, specifically the quasisymmetric shin functions and
Young quasisymmetric Schur functions, as well as their duals. Finally, we are
interested in defining and studying the colored generalization of the
symmetric functions that would be a subset of $QSym_{A}$ and the image of
$NSym_{A}$ under a forgetful map.

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##  8\. Appendix

Let $A$ be a finite alphabet with a total order $\leq$. Let
$I=(w_{1},\ldots,w_{k})$ and $J=(v_{1},\ldots,v_{h})$ be sentences and
$K=(u_{1},\ldots,u_{g})$ a weak sentence. Let $w=a_{1}\ldots a_{n}$ and
$v=b_{1}\ldots b_{m}$. Let $U$ be a standard colored immaculate tableau and
thus also a standard colored row-strict immaculate tableau.

  *   * $|w|=n$ (size)

  * $w\cdot v=a_{1}\ldots a_{n}b_{1}\ldots b_{m}$ (concatenation of words)

  * $w\preceq_{\ell}v$ if $a_{i}<b_{i}$ for the first positive integer $i$ such that $a_{i}\not=b_{i}$ (lexicographic order)

  *   * $|I|=\sum_{i=1}^{k}|w_{i}|$ (size)

  * $\ell(I)=k$ (length)

  * $w(I)=w_{1}\cdot w_{2}\cdots w_{k}$ (maximal word)

  * $w\ell(I)=(|w_{1}|,|w_{2}|,\ldots,|w_{k}|)$ (word length)

  *   * $J\preceq I$ for $w(I)=w(J)$, if $J$ splits at each location that $I$ splits (refinement order)

  * $\mu(J,I)=(-1)^{\ell(J)-\ell(I)}\text{ for }J\preceq I$ (Möbius function on the poset of sentences ordered by $\preceq$)

  * $J\subset_{w}I$ if $w_{i}=v_{i}q_{i}$ for $1\leq i\leq k$ such that $q_{k}\cdots q_{1}=w$ and $k\leq h+1$ where $v_{h+1}=\emptyset$

  *   * $I\cdot J=(w_{1},\ldots,w_{n},v_{1},\ldots,v_{m})$ (concatenation of sentences)

  * $I\odot J=(w_{1},\ldots,w_{n-1},w_{n}\cdot v_{1},v_{2},\ldots,v_{m})$ (near-concatenation)

  *   * $I^{r}=(w_{k},\ldots,w_{2},w_{1})$ (reversal)

  * $I^{c}$ the unique sentence with $w(I^{c})=w(I)$ that splits exactly where $I$ does not (complement)

  * $\tilde{K}$ the sentence obtained by removing all empty words from K (flattening)

  *   * $K\subseteq_{L}I$ there exists a weak sentence $I/_{L}K$ as defined below (left-containment)

  * $I/_{L}K=(q_{1},\ldots,q_{t})$ such that $w_{i}=u_{i}q_{i}$ for all $i\in[k]$

  *   * $K\subseteq_{R}I$ there exists a weak sentence $I/_{R}K$ as defined below (right-containment)

  * $I/_{R}K=(q_{1},\ldots,q_{t})$ such that $w_{i}=q_{i}u_{i}$ for all $i\in[k]$

  *   * $\\{M_{I}\\}_{I}=\sum_{1\leq j_{1}<j_{2}<\ldots<j_{k}}x_{w_{1},j_{1}}x_{w_{2},j_{2}}\cdots x_{w_{k},j_{k}}$ (the monomial basis of $QSym_{A}$)

  * $\\{F_{I}\\}_{I}=\sum_{J\preceq I}M_{J}$ (the fundamental basis of $QSym_{A}$)

  *   * $\\{H_{I}\\}_{I}$ (the complete homogeneous basis of $NSym_{A}$)

  * $\\{R_{I}\\}_{I}=\sum_{I\preceq J}(-1)^{\ell(J)-\ell(I)}H_{J}$ (the ribbon basis of $NSym_{A}$)

  * $\\{E_{I}\\}_{I}=\sum_{J\preceq I}(-1)^{|I|-\ell(J)}H_{J}$ (the elementary basis of $NSym_{A}$)

  *   * $\langle\cdot,\cdot\rangle:NSym\otimes QSym\rightarrow\mathbb{Q}$ $\langle H_{I},M_{J}\rangle=\delta_{I,J}$ (inner product)

  *   * _Colored immaculate tableau (CIT)_ a colored composition diagram filled with integers such that rows are weakly increasing left to right and the first column is strictly decreasing top to bottom

  * $shape(T)=(w_{1},\ldots,w_{k})$ where the colors of row $i$ read left to right give $w_{i}$

  * $type(T)=(v_{1},\ldots,v_{h})$ $v_{i}=$ colors of boxes filled with $i$ read from left to right, bottom to top

  *   * $Des(U)=\\{i:\text{$i+1$ is on a strictly lower row of $U$ than $i$}\\}$ (descent set)

  * $Co_{A}(U)$ a sentence formed by reading the color in each box of $U$ in order of number, splitting into a new word when moving to a strictly lower row (colored descent composition)

  * $std(T)$ (the standardization of a tableau $T$)

  *   * $\\{\mathfrak{S}^{*}_{I}\\}_{I}$ (the colored dual immaculate basis of $QSym_{A}$)

  * $\\{\mathfrak{S}_{I}\\}_{I}$ (the colored immaculate basis of $NSym_{A}$)

  *   * $\mathfrak{P}_{A}$ (the colored immaculate poset)

  * $\mathfrak{D}_{A}^{n}$ (the colored immaculate descent graph)

  * ${}^{rs}\mathfrak{D}_{A}^{n}$ (the colored row-strict immaculate descent graph)

  *   * $Des^{rs}(U)=\\{i:\text{$i+1$ is on a weakly higher row of $U$ than $i$}\\}$ (row-strict descent set)

  * $Co^{rs}_{A}(U)$ a sentence formed by reading the color in each box of $U$ in order of number, splitting into a new word when moving to a weakly higher row (colored row-strict descent composition)

  *   * $\\{\mathfrak{RS}^{*}_{I}\\}_{I}$ (the row-strict dual immaculate basis of $QSym_{A}$)

  * $\\{\mathfrak{RS}_{I}\\}_{I}$ (the row-strict immaculate basis of $NSym_{A}$)