# On Equivariant flag $f$-vectors for balanced relative simplicial complexes

Jacob A. White  School of Mathematical and Statistical Sciences, University of
Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539 USA

###### Abstract.

We study the equivariant flag $f$-vector and equivariant flag $h$-vector of a
balanced relative simplicial complex with respect to a group action. When the
complex satisfies Serre’s condition $(S_{\ell}),$ we show that the equivariant
flag $h$-vector, the equivariant $h$-vector, and the equivariant $f$-vector
satisfy several inequalities.

We apply these results to the study of $P$-partitions of double posets, and
weak colorings of mixed graphs.

###### Key words and phrases:

Balanced simplicial complexes, flag f-vectors, group actions

###### 2020 Mathematics Subject Classification:

05E05, 05E18

##  1\. Introduction

In Stanley’s seminal paper [Sta82], he studies group actions on graded posets.
Given a group $\mathfrak{G}$ acting on a graded poset $P$, it also acts on the
order complex $\Delta(P)$ of chains of $P$. Stanley proves several results
about this action.

Stanley also proved several results about balanced Cohen-Macaulay complexes
[Sta79]. A _balanced simplicial complex_ is a pure simplicial complex $\Delta$
on vertex set $V$ of dimension $d-1$, with a function $f:V\rightarrow[d]$ such
that, for all $\sigma\in\Delta$ and $i\in[d]$, we have $|\sigma\cap
f^{-1}(i)|\leq 1.$ Given $S\subseteq[d]$, we let
$\Delta|_{S}=\\{\sigma\in\Delta:f(v)\in S\mbox{ for every }v\in\sigma\\}$.
Stanley showed that if $\Delta$ is Cohen-Macaulay, then so is $\Delta|_{S}$
for all $S\subseteq[d]$. This result was recently generalized, by replacing
the Cohen-Macaulay property with Serre’s condition.

######  Theorem 1.

Let $\Delta$ be a balanced simplicial complex of dimension $d-1$. Given
$S\subseteq[d]$, if $\Delta$ satisfies Serre’s condition $(S_{\ell})$, then so
does $\Delta|_{S}$.

Our goal is to study balanced relative simplicial complexes, equipped with a
group action by a finite group $\mathfrak{G}$. Let $\Phi$ be a balanced
relative simplicial complex of dimension $d-1$. That is, $\Phi$ is a
collection of subsets of $V$ such that, for
$\rho\subseteq\sigma\subseteq\tau\subseteq V$, if $\rho,\tau\in\Phi$, then
$\sigma\in\Phi$. We also require that the maximal faces all have the same size
$d$, and assume that there is a function $\kappa:V\rightarrow[d]$ such that,
for every maximal face $\sigma\in\Phi$, $\kappa$ restricted to $\sigma$ is a
bijection. Given $S\subseteq[d]$, we let
$\Phi|_{S}=\\{\sigma\in\Phi:\\{\kappa(v):v\in\sigma\\}\subseteq S\\}$.

######  Theorem 2.

Let $\Phi$ be a balanced relative simplicial complex on vertex set $V$ of
dimension $d-1$ with coloring $\kappa:V\rightarrow[d].$ Suppose that $\Phi$
satisfies Serre’s condition $(S_{\ell})$. Given $S\subseteq[d]$, we have
$\Phi|_{S}$ also satisfies Serre’s condition $(S_{\ell})$.

If $\Phi$ is relatively Cohen-Macaulay, then so is $\Phi|_{S}$.

Our primary goal is to apply this result to obtain information about flag
enumeration for $\Phi$ with respect to a group action $\mathfrak{G}$. Let
$\mathfrak{G}$ be a group which acts on $V$. We say $\mathfrak{G}$ acts on
$\Phi$ if for every $\mathfrak{g}\in\mathfrak{G}$, the following two
conditions are satisfied:

  1. (1)

For every $\sigma\in\Phi$, we have $\\{\mathfrak{g}v:v\in\sigma\\}\in\Phi$.

  2. (2)

For every $v\in V$, we have $\kappa(\mathfrak{g}v)=\kappa(v)$.

Let $x_{1},x_{2},\ldots,$ be a sequence of commuting indeterminates. A power
series $f$ in $x_{1},\ldots$ is a _quasisymmetric function_ if
$[x_{i_{1}}^{a_{1}}\cdots x_{i_{k}}^{a_{k}}]f=[x_{1}^{a_{1}}\cdots
x_{k}^{a_{k}}]f$ for every sequence $(a_{1},\ldots,a_{k})$ and every
increasing sequence $i_{1}<i_{2}<\cdots<i_{k}.$ The most common basis for the
ring of quasisymmetric functions is the basis of monomial quasisymmetric
functions. We define our quasisymmetric functions in terms of this basis.

For $\mathfrak{g}\in\mathfrak{G}$, we define the _flag quasisymmetric class
function_ of $(\Phi,\mathfrak{G})$ to be

$\operatorname{Hilb}(\Phi,\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum_{\sigma\in\Phi:\mathfrak{g}\sigma=\sigma}M_{\kappa(\sigma),d+1}.$

This is a quasisymmetric function. As we vary $\mathfrak{g}\in\mathfrak{G}$,
we obtain a class function on $\mathfrak{G}$ whose values are quasisymmetric
functions. Equivalently, we can write
$\operatorname{Hilb}(\Phi,\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum_{S\subseteq[d]}f_{S}(\Phi,\mathfrak{G};\mathfrak{g})M_{S,d+1}$
for certain constants $f_{S}(\Phi,\mathfrak{G};\mathfrak{g})$. Then
$f_{S}(\Phi,\mathfrak{G})$ is also a character. We refer to the character
$f_{S}(\Phi,\mathfrak{G})$ as the _equivariant flag $f$-vector_ of
$(\Phi,\mathfrak{G})$. Our goal will be to prove inequalities about the
equivariant flag $f$-vector. To that end, given two characters $\chi$ and
$\rho$ of a group $\mathfrak{G}$, we write $\chi\geq_{\mathfrak{G}}\rho$ if
$\chi-\rho$ is also a character. The equivariant flag $h$-vector is defined by

$h_{S}(\Phi,\mathfrak{G})=\sum_{T:S\subseteq T}(-1)^{|T\setminus
S|}f_{S}(\Phi,\mathfrak{G}).$

######  Theorem 3.

Let $\Phi$ be a balanced relative simplicial complex of dimension $d-1$ and
let $\mathfrak{G}$ act on $\Phi$. Suppose $\Phi$ satisfies condition
$(S_{\ell})$. Given $S\subseteq T\subseteq[d]$ with $|S|\leq\ell$, then

$\sum_{R:S\subseteq R\subseteq
T}h_{R}(\Phi,\mathfrak{G})\geq_{\mathfrak{G}}0.$

In particular, if $\Phi$ is Cohen-Macaulay, then
$h_{T}(\Phi,\mathfrak{G})\geq_{\mathfrak{G}}0$ for all $T\subseteq[d]$.

There is also an equivariant $f$-vector, given by
$f_{i-1}(\Phi,\mathfrak{G})=\sum_{S\subseteq[d]:|S|=i}f_{S}(\Phi,\mathfrak{G})$.
The equivariant $h$-vector is given by
$h_{i}(\Phi,\mathfrak{G})=\sum_{S\subseteq[d]:|S|=i}h_{S}(\Phi,\mathfrak{G})$.
First, we state some new inequalities for the equivariant $h$-vector of a
balanced relative simplicial complex which satisfies condition $(S_{\ell})$.

######  Theorem 4.

Let $\Phi$ be a balanced relative simplicial complex of dimension $d-1$. Let
$\mathfrak{G}$ act on $\Phi$, and suppose that $\Phi$ satisfies $(S_{\ell})$.
Then $h_{i}(\Phi,\mathfrak{G})\geq_{\mathfrak{G}}0$ for $i\leq\ell$. For
$0\leq i\leq\ell\leq j\leq d$, we have

$\sum_{k=\ell}^{j}\binom{d-k}{j-k}\binom{k-\ell+i}{i}h_{k}(\Phi,\mathfrak{G})\geq_{\mathfrak{G}}0.$

For pure simplicial complexes which satisfy $(S_{\ell})$, and with $j=d$,
these inequalities were previously shown in [MT09] and [GPSFY12] for ordinary
$h$-vectors.

We also give some inequalities for the equivariant $f$-vector.

######  Theorem 5.

Let $\Phi$ be a balanced relative simplicial complex of dimension $d-1$ and
let $\mathfrak{G}$ act on $\Phi$. Suppose $\Phi$ satisfies condition
$(S_{\ell})$, and that $f_{i-1}(\Phi,\mathfrak{G})=0$ for $i<\ell$.

  1. (1)

For $i\geq\ell$, we have

$(d-i)f_{i-1}(\Phi,\mathfrak{G})\leq_{\mathfrak{G}}(i-\ell+2)f_{i}(\Phi,\mathfrak{G}).$

  2. (2)

For $\ell\leq i\leq\lfloor(d+\ell-1)/2\rfloor$, we have
$f_{i-1}(\Phi,\mathfrak{G})\leq_{\mathfrak{G}}f_{i}(\Phi,\mathfrak{G})$.

  3. (3)

For $\ell\leq i\leq\lfloor(d+\ell)/2\rfloor$, we have
$f_{i-1}(\Phi,\mathfrak{G})\leq_{\mathfrak{G}}f_{d+\ell-i-1}(\Phi,\mathfrak{G})$.

If we assume that the group action is trivial, and we define
$a_{i}(\Phi)=f_{i+\ell-1}(\Phi,\mathfrak{G})$, then the latter two conditions
become:

  1. (1)

For $i\leq\lfloor(d-\ell-1)/2\rfloor$, we have $a_{i}\leq a_{i+1}$.

  2. (2)

For $i\leq\lfloor(d-\ell)/2\rfloor$, we have $a_{i}\leq a_{d-\ell-i}$.

In the literature, the sequence $(a_{0},\ldots,a_{d-\ell})$ is called
_strongly flawless_. Thus, our Theorem states that if the first several
entries of the sequence are $0$, then deleting them results in a strongly
flawless sequence.

Our primary application is to double posets and mixed graphs. A mixed graph is
a generalization of a graph where we allow directed edges. A coloring of a
mixed graph $G$ satisfies $f(u)\neq f(v)$ when $uv$ is an undirected edge, and
$f(u)\leq f(v)$ when $(u,v)$ is a directed edge. Given an automorphism
$\mathfrak{g}$ of $G$, we let

$\chi(G,\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum_{f:\mathfrak{g}f=f}\prod_{v\in
V}x_{f(v)}.$

Then $\chi(G,\mathfrak{G},\mathbf{x})$ is a quasisymmetric class function,
generalizing the chromatic symmetric function. We wished to determine when
$[F_{S,n}]\chi(G,\mathfrak{G},\mathbf{x})$ is an effective character. There is
also a corresponding chromatic polynomial class function
$\chi(G,\mathfrak{G},x).$

A mixed graph is _acyclic_ if it does not contain a directed cycle. A _mixed
cycle_ in $G$ is a cycle in the underlying undirected graph that has at least
one directed edge. A _near cycle_ is a mixed cycle of length three that has
exactly one undirected edge. We prove the following:

######  Theorem 6.

Let $G$ be an acyclic mixed graph on $n$ vertices with no near cycles. Let
$m(G)$ be the minimum number of undirected edges on a mixed cycle of $G$, with
$m(G)=|G|$ if there are no mixed cycles.

Let $\mathfrak{G}$ act on $G$. Then for $S\subseteq[n-1]$ with $|S|\leq m(G)$,
we have

$[F_{S,n}]\chi(G,\mathfrak{G},\mathbf{x})\mbox{ is an effective character}.$

We write
$\chi(G,\mathfrak{G},x)=\sum_{i=0}^{n}f_{i-2}\binom{x}{i}=\sum_{i=0}^{n}h_{i-1}\binom{x+n-i}{n}.$

  1. (1)

We have $h_{i}\geq_{\mathfrak{G}}0$ for $i\leq\ell$.

  2. (2)

For $i\leq d-\ell$, we have
$\sum_{j=\ell}^{d}\binom{i+j-\ell}{i}h_{j}\geq_{\mathfrak{G}}0.$

Suppose that $m(G)\geq\chi(G)$.

  1. (1)

For $i\geq\chi(G)$, we have

$(d-i)f_{i-1}\leq_{\mathfrak{G}}(i-\chi(G)+2)f_{i}.$

  2. (2)

For $\chi(G)\leq i\leq\lfloor(d+\chi(G)-1)/2\rfloor$, we have
$f_{i-1}\leq_{\mathfrak{G}}f_{i}$.

  3. (3)

For $\chi(G)\leq i\leq\lfloor(d+\chi(G))/2\rfloor$, we have
$f_{i-1}\leq_{\mathfrak{G}}f_{d+\ell-i-1}$.

We show that there is a balanced relative simplicial complex $\Phi(G)$ such
that

$\operatorname{Hilb}(\Phi(G),\mathfrak{G},\mathbf{x})=\chi(G,\mathfrak{G},\mathbf{x}).$

Under this identification, $f_{-2}=h_{-1}=0$, and for $i\geq-1$ we have
$f_{i}=f_{i}(\Phi(G))$, which explains our choice of notation in Theorem 6.
The complex $\Phi(G)$ is a generalization of Steingrímsson’s coloring complex,
and was previously introduced in [Whi]. However, we give a self-contained
introduction to this complex. We then show that this complex satisfies Serre’s
condition $(S_{m(G)}).$ Thus, we obtain a natural collection of balanced
relative simplicial complexes which satisfy Serre’s condition $(S_{k})$ for
any $k$.

A double poset $D$ is a set equipped with two partial orders. A double poset
is _tertispecial_ if, whenever $m\prec_{1}m^{\prime}$, then $m$ and
$m^{\prime}$ are $\leq_{2}$-comparable. There is a natural quasisymmetric
generating function $\Omega(D,\mathbf{x})$ associated to $D$, which was
introduced by Grinberg [Gri17] and enumerates $D$-partitions.

A function $f:N\rightarrow\mathbb{N}$ is a $D$-partition if and only if it
satisfies the following two properties:

  1. (1)

For $i\leq_{1}j$ in $D$, we have $f(i)\leq f(j)$.

  2. (2)

For $i\leq_{1}j$ and $j\leq_{2}i$ in $D$, we have $f(i)<f(j)$.

When a group $\mathfrak{G}$ acts on $D$, then it also acts on the set of
$D$-partitions of a given weight, and we can define a quasisymmetric class
function $\Omega(D,\mathfrak{G},\mathbf{x})$. If we write
$\Omega(D,\mathfrak{G},\mathbf{x})=\sum_{S\subseteq{[n-1]}}\chi_{D,\mathfrak{G},S}F_{S,n}$,
then the $\chi_{D,\mathfrak{G},S}$ are virtual characters. We proved the
following:

######  Theorem 7 ([Whi21]).

Let $D$ be a tertispecial double poset on $n$ elements, and let $\mathfrak{G}$
act on $D$. Given $S\subseteq[n-1]$, we have
$[F_{S,n}]\Omega(D,\mathfrak{G},\mathbf{x})$ is an effective character.

This paper gives an alternate proof. We show that, given a double poset $P$,
there is a mixed graph $G$ such that
$\Omega(P,\mathfrak{G},\mathbf{x})=\chi(G(P),\mathfrak{G},\mathbf{x}).$ Thus
we are able to determine results about double posets from corresponding
results on mixed graphs.

The paper is organized as follows. First, we give an overview of
quasisymmetric class functions in Section 2. In Section 3, we review the
definition of balanced relative simplicial complexes, and discuss the
equivariant flag $f$-vector. Then we discuss the Stanley-Reisner module and
the equivariant $f$-vector. Then we connect groups actions on balanced
relative simplicial complexes to characters on cohomology groups. In Section
4, we study Serre’s condition, and then prove the various results mentioned in
the introduction. In Section 5, we study mixed graphs and their chromatic
quasisymmetric class functions. Finally, in Section 6 we study double posets.

##  2\. Preliminaries

Given a basis $B$ for a vector space $V$ over $\mathbb{C}$, and
$\vec{\beta}\in B,\vec{v}\in V$, we let $[\vec{\beta}]\vec{v}$ denote the
coefficient of $\vec{\beta}$ when we expand $\vec{v}$ in the basis $B$.

Let $\mathbf{x}=x_{1},x_{2},\ldots$ be a sequence of commuting indeterminates.
Recall that an _integer composition_ $\alpha$ of a positive integer $n$ is a
sequence $(\alpha_{1},\ldots,\alpha_{k})$ of positive integers such that
$\alpha_{1}+\cdots+\alpha_{k}=n$. We write $\ell(\alpha)=k$, and
$\alpha\models n$. Let $n\in\mathbb{N}$ and let
$F(\mathbf{x})\in\mathbb{C}[[\mathbf{x}]]$ be a homogeneous formal power
series in $\mathbf{x}$, where the degree of every monomial in $F(\mathbf{x})$
is $n$. Then $F(\mathbf{x})$ is a _quasisymmetric function_ if it satisfies
the following property: for every $i_{1}<i_{2}<\cdots<i_{k}$, and every
integer composition $\alpha\models n$ with $\ell(\alpha)=k$, we have
$[\prod_{j=1}^{k}x_{i_{j}}^{\alpha_{j}}]F(\mathbf{x})=[\prod_{j=1}^{k}x_{j}^{\alpha_{j}}]F(\mathbf{x})$.
Often, we will define quasisymmetric functions that are generating functions
over a collection of functions. Given a function $w:S\rightarrow\mathbb{N}$,
we define

(1) $\mathbf{x}^{w}=\prod_{v\in S}x_{w(v)}.$

For example, the chromatic symmetric function of a graph $G$ is defined as
$\chi(G,\mathbf{x})=\sum\limits_{f:V\rightarrow\mathbb{N}}\mathbf{x}^{f}$
where the sum is over all proper colorings of $G$.

Given a subset $\\{s_{1},\ldots,s_{k}\\}\subseteq[n-1]$, with
$s_{1}<s_{2}<\cdots<s_{k},$ we let

$M_{\\{s_{1},\ldots,s_{k}\\},n}=\sum\limits_{i_{1}<\cdots<i_{k+1}}\prod_{j=1}^{k+1}x_{i_{j}}^{s_{j}-s_{j-1}}$

where we define $s_{0}=0$ and $s_{k+1}=n$. These are the _monomial
quasisymmetric functions_ , which form a basis for the vector space of
quasisymmetric functions of degree $n$. This basis is partially ordered, by
saying $M_{S,n}\leq M_{T,n}$ if $S\subseteq T.$

The second basis we focus on is the basis of fundamental quasisymmetric
functions, first introduced by Gessel [Ges84]. The _fundamental quasisymmetric
functions_ $F_{S,d}$ are defined by:

$F_{S,d}=\sum\limits_{T:S\subseteq T}M_{T,d}.$

###  2.1. Group actions and class functions

Given a group action $\mathfrak{G}$ on a set $X$, we let $X/\mathfrak{G}$
denote the set of orbits. For $x\in X$, $\mathfrak{G}_{x}$ is the stabilizer
subgroup, and $\mathfrak{G}(x)$ is the orbit of $x$. Finally, for
$\mathfrak{g}\in\mathfrak{G}$, we let
$\operatorname{Fix}_{\mathfrak{g}}(X)=\\{x\in X:\mathfrak{g}x=x\\}$.

We assume familiarity with the theory of complex representations of finite
groups - see [FH91] for basic definitions. Recall that, given any group action
of $\mathfrak{G}$ on a finite set $X$, there is a group action on
$\mathbb{C}^{X}$ as well, which gives rise to a representation. The resulting
representations are called _permutation representations_. Given any
$\mathfrak{G}$-set $X$, or $\mathfrak{G}$-module $V$, we let
$\chi_{X,\mathfrak{G}}$ and $\chi_{V,\mathfrak{G}}$ denote the corresponding
characters. Let $R$ be a $\mathbb{C}$-algebra. Then an _$R$ -valued class
function_ is a function $\chi:\mathfrak{G}\rightarrow R$ such that, for every
$\mathfrak{g},\mathfrak{h}\in\mathfrak{G}$, and $\chi\in C(\mathfrak{G},R)$,
we have $\chi(\mathfrak{hg}\mathfrak{h}^{-1})=\chi(\mathfrak{g}).$ Let
$C(\mathfrak{G},R)$ be the set of $R$-valued class functions from
$\mathfrak{G}$ to $R$. For our paper, $R$ is usually $\operatorname{QSym}$ or
$\mathbb{C}[x]$. We refer to $\chi\in C(\mathfrak{G},\mathbb{C})$ as class
functions when no confusion arises.

There is an orthonormal basis of $C(\mathfrak{G},\mathbb{C})$ given by the
characters of the irreducible representations of $\mathfrak{G}$. We refer to
elements $\chi\in C(\mathfrak{G},\mathbb{C})$ that are integer combinations of
irreducible characters as _virtual characters_ , and elements that are
nonnegative integer linear combinations as _effective characters_. Let
$E(\mathfrak{G},\mathbb{C})$ be the set of effective characters. Finally, we
say $\chi$ is a _permutation character_ if it is the character of a
permutation representation. We partially order $E(\mathfrak{G},\mathbb{C})$ by
saying $\chi\leq_{\mathfrak{G}}\psi$ if $\psi-\chi$ is an effective character.

Let $\mathbf{B}$ be a basis for $R$. For $b\in\mathbf{B}$,
$\mathfrak{g}\in\mathfrak{G}$, and $\chi\in C(\mathfrak{G},R)$, let
$\chi_{b}(\mathfrak{g})=[b]\chi(\mathfrak{g})$. Then $\chi_{b}$ is a
$\mathfrak{C}$-valued class function. Thus we can write
$\chi=\sum\limits_{b\in\mathbf{B}}\chi_{b}b$. Conversely, given a family
$\chi_{b}$ of $\mathfrak{C}$-valued class functions, one for each
$b\in\mathbf{B}$, the function $\chi$ defined by
$\chi(\mathfrak{g})=\sum\limits_{b\in\mathbf{B}}\chi_{b}(\mathfrak{g})b$ is an
$R$-valued class function in $C(\mathfrak{G},R)$.

Let $\chi$ be an $R$-valued class function. We say that $\chi$ is _B
-realizable_ if $\chi_{b}$ is a permutation character for all $b$.

A _quasisymmetric class function_ is a $\operatorname{QSym}$-valued class
function. Given a quasisymmetric class function $F(\mathfrak{G},\mathbf{x})$,
we write
$F(\mathfrak{G},\mathbf{x})=\sum_{S\subseteq[n-1]}f_{S,\mathfrak{G}}M_{S,n}$,
where $f_{S,\mathfrak{G}}\in C(\mathfrak{G},\mathbb{C})$.

Finally, we define a function
$\langle\cdot,\cdot\rangle:C(\mathfrak{G},R)\times
C(\mathfrak{G},R)\rightarrow R$ by
$\langle\chi,\psi\rangle=\frac{1}{|\mathfrak{G}|}\overline{\chi}(\mathfrak{g})\psi(\mathfrak{g})$
where $\overline{x}$ is the complex conjugate. In the case where
$R=\mathbb{C}$, this is the usual inner product on class functions.

######  Proposition 8.

Let $\mathfrak{G}$ be a finite group, let $R$ be a $\mathbb{C}$-algebra with
basis $\mathbf{B}$. Fix $\chi\in C(\mathfrak{G},R)$.

  1. (1)

Given an irreducible character $\psi$, we have
$\langle\chi,\psi\rangle=\sum\limits_{b,c\in B}\langle\chi_{b},\psi_{c}\rangle
b\cdot c$.

  2. (2)

Let $\psi\in C(\mathfrak{G},\mathbb{C})$. If $\chi$ is $B$-realizable, then
for all $b\in\textbf{B}$ we have $[b]\langle\psi,\chi\rangle\geq 0.$

###  2.2. Polynomial class functions and Principal specialization

Given a polynomial $p(x)$ of degree $d$, define the $f$-vector
$(f_{-1},\ldots,f_{d-1})$ via $p(x)=\sum_{i=0}^{d}f_{i-1}\binom{x}{i}$. We say
that $p(x)$ is _strongly flawless_ if the following inequalities are
satisfied:

  1. (1)

for $0\leq i\leq\frac{d-1}{2}$, we have $f_{i-1}\leq f_{i}$.

  2. (2)

For $0\leq i\leq\frac{d}{2}$, we have $f_{i-1}\leq f_{d-i-1}$.

There is a lot of interest in log-concave and unimodal sequences in
combinatorics. We consider strongly flawless sequences to be interesting, as
strongly flawless unimodal sequences can be seen as a generalization of
symmetric unimodal sequences. Examples of results with strongly flawless
sequences include the work of Hibi [Hib89] and Juhnke-Kubitzke and Van Le
[JKL18]. We denote the entries of the $f$-vector as $f_{i}(p(x))$ when
discussing multiple polynomials.

There is a generalization of $f$-vector for polynomial class functions, which
we call the _equivariant $f$-vector_. Given a group $\mathfrak{G}$, and a
polynomial class function $p(\mathfrak{G},x)$, we write
$p(\mathfrak{G},x)=\sum_{i=1}^{d}f_{i-1}\binom{x}{i}$, where the $f_{i}$ are
characters. We say that $p(\mathfrak{G},x)$ is _effectively flawless_ if we
have the following system of inequalities:

  1. (1)

For $0\leq i\leq\frac{d-1}{2}$, we have $f_{i-1}\leq_{\mathfrak{G}}f_{i}$.

  2. (2)

For $0\leq i\leq\frac{d}{2}$, we have $f_{i-1}\leq_{\mathfrak{G}}f_{d-i-1}$.

Given a quasisymmetric function $F(\mathbf{x})$ of degree $d$, there is an
associated polynomial $\operatorname{ps}(F)(x)$ given by principal
specialization. For $x\in\mathbb{N}$, we set

$x_{i}=\begin{cases}1&i\leq x\\\ 0&i>x\end{cases}$

The resulting sequence is a polynomial function in $x$ of degree $d$, which we
denote by $\operatorname{ps}(F)(x)$. If we write
$F(\mathbf{x})=\sum\limits_{S\subseteq[d-1]}c_{S,d}M_{S,d}$, then
$f_{i}(\operatorname{ps}(F(\mathbf{x})))=\sum\limits_{S\subseteq[d-1]:|S|=i+1}c_{S,d}$.

Let $F(\mathfrak{G},\mathbf{x})$ be a quasisymmetric class function of degree
$d$, and $\mathfrak{g}\in\mathfrak{G}$. Define $\operatorname{ps}(F)\in
C(\mathfrak{G},\mathbb{C}[x])$ by
$\operatorname{ps}(F)(x;\mathfrak{g})=\operatorname{ps}(F(\mathbf{x};\mathfrak{g}))(x).$
We refer to $\operatorname{ps}(F)$ as the principal specialization, which
results in an polynomial class function.

The following results were obtained in [Whi21].

######  Proposition 9.

Let $F(\mathfrak{G},\mathbf{x})$ be a quasisymmetric class function be of
degree $d$, and $\mathfrak{g}\in\mathfrak{G}$.

  1. (1)

If we write
$F(\mathfrak{G},\mathbf{x})=\sum\limits_{S\subseteq\models[d-1]}\chi_{S,d}M_{S,d}$,
then $\operatorname{ps}(F)=\sum\limits_{i=0}^{d}\chi_{S,d}\binom{x}{|S|+1}$.

  2. (2)

If $F(\mathfrak{G},\mathbf{x})$ is $M$-realizable and $M$-increasing, then
$\operatorname{ps}(F)$ is effectively flawless.

######  Proposition 10.

Let $(f_{-1},\ldots,f_{d-1})$ be a sequence of characters for $\mathfrak{G}$.
Suppose that there exists an integer $0\leq r\leq d$ such that, for all $i$,
we have

$(d-i)f_{i-1}\leq_{\mathfrak{G}}(i-r)f_{i}.$

Then the sequence $(f_{r-1},\ldots,f_{d-1})$ is effectively flawless.

###### Proof.

We prove the result by induction on $r$. First, suppose that $r=0$.

We see that $(i+1)\leq(d-i)$. Thus we have
$(i+1)f_{i}\leq_{\mathfrak{G}}(d-i)f_{i-1}\leq_{\mathfrak{G}}(i+1)f_{i}$.
Hence $(i+1)(f_{i}-f_{i-1})$ is an effective character. This implies that
$(i+1)c_{j}$ is a non-negative integer for all $j$, which means $c_{j}$ is a
non-negative rational number. Therefore $f_{i}-f_{i-1}$ is an effective
character.

Now let $i\leq\lfloor d/2\rfloor$. We can generalize the inequality
$(d-i)f_{i-1}\leq_{\mathfrak{G}}if_{i}$ to obtain
$\binom{d-i}{j-i}f_{i-1}\leq_{\mathfrak{G}}\binom{j}{j-i}f_{j-1}$ whenever
$i\leq j$. When we set $j=d-i$, then $\binom{d-i}{j-i}=\binom{j}{j-i}$, and
thus we get $f_{i-1}\leq_{\mathfrak{G}}f_{d-i-1}$.

Now suppose that $r>0$. Let $g_{i}=f_{i+1}$. Then we see that

$(d-1-i)g_{i-1}=(d-(i+1))f_{i}\leq_{\mathfrak{G}}(i+1-r)f_{i+1}=(i+(r-1))g_{i}.$

Thus, by the induction hypothesis, we have $(g_{r-2},\ldots,g_{d-1})$ is
effectively flawless. However,
$(g_{r-2},\ldots,g_{d-2})=(f_{r-1},\ldots,f_{d-1})$. ∎

##  3\. Balanced Relative Simplicial Complexes

Now we discuss balanced relative simplicial complexes, and their flag
quasisymmetric class functions.

$d$$c$$b$$a$$e$ | $3$$1$$3$$1$$2$
---|---
Figure 1. A coloring complex $\Phi$. Dashed lines correspond to faces that are
not in $\Phi$.

######  Definition 11.

A _balanced relative simplicial complex_ of dimension $d-1$ on a vertex set
$V$ is a non-empty collection $\Phi$ of subsets of $V$, along with a function
$\kappa:V\rightarrow[d]$ with the following properties:

  1. (1)

For every $\rho\subseteq\sigma\subseteq\tau$, if $\rho,\tau\in\Phi$, then
$\sigma\in\Phi$.

  2. (2)

For every $\rho\in\Phi$, there exists $\sigma\in\Phi$ such that
$\rho\subseteq\sigma$ and $|\sigma|=d$,

  3. (3)

For every $\rho\in\Phi$, we have $\\{\kappa(v):v\in\rho\\}$ has size $|\rho|$.

The name comes from the fact that there exists simplicial complexes
$(\Delta,\Gamma)$ with $\Gamma\subseteq\Delta$, and
$\Phi=\Delta\setminus\Gamma$. Given $\sigma\in\Phi$, we let
$\kappa(\sigma)=\\{\kappa(v):v\in\sigma\\}$.

Given $S\subseteq[d]$, we let $f_{S}(\Phi)$ denote the number of faces
$\sigma$ such that $\kappa(\sigma)=S$. This is the _flag $f$-vector_ of
$\Phi$. We encode the flag $f$-vector with a quasisymmetric function

$\operatorname{Hilb}(\Phi,\mathbf{x})=\sum_{S\subseteq[d]}f_{S}(\Phi)M_{S,d+1}.$

This is the _flag quasisymmetric function_ associated to $\Phi$. It has degree
$d+1$.

Let $V(\Phi)$ be the vertex set of $\Phi$. A bijection
$\mathfrak{g}:V\rightarrow V$ is an _automorphism_ of $\Phi$ if it satisfies
the following two properties:

  1. (1)

For every $v\in V$, we have $\kappa(\mathfrak{g}v)=\kappa(v)$.

  2. (2)

For every $\\{v_{1},\ldots,v_{k}\\}\in\Phi$, we have
$\\{\mathfrak{g}(v_{1}),\ldots,\mathfrak{g}(v_{k})\\}\in\Phi$.

Let $\mathfrak{G}$ be a group which acts on $V(\Phi)$. Then $\mathfrak{G}$
also acts on $2^{V(\Phi)}$ by $\mathfrak{g}(S)=\\{\mathfrak{g}v:v\in S\\}$.
Suppose that, for each $\mathfrak{g}\in\mathfrak{G}$, the resulting action on
$2^{V(\Phi)}$ is an automorphism of $\Phi$. Then we say $\mathfrak{G}$ acts on
$\Phi$. For $S\subseteq[d]$, let
$\kappa^{-1}(S)=\\{\sigma\in\Phi:\kappa(\sigma)=S\\}$. Then $\mathfrak{G}$
acts on $\kappa^{-1}(S)$ as well. Given $\mathfrak{g}\in\mathfrak{G}$, we
define

$\operatorname{Hilb}(\Phi,\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum_{\sigma\in\Phi:\mathfrak{g}\sigma=\sigma}M_{\kappa(\sigma),d+1}.$

As we vary $\mathfrak{g}$, we obtain a quasisymmetric class function, which we
call the _flag quasisymmetric class function_ of $(\Phi,\mathfrak{G})$, and
denote $\operatorname{Hilb}(\Phi,\mathfrak{G},\mathbf{x})$.

As an example, consider the balanced relative simplicial complex $\Phi$ with
vertex set $\\{a,b,c,d,e\\}$ appearing on the left in Figure 1. Here we note
that

$\Phi=\\{\\{e\\},\\{a,e\\},\\{b,e\\},\\{c,e\\},\\{d,e\\},\\{a,b,e\\},\\{b,c,e\\},\\{c,d,e\\},\\{a,d,e\\}\\}.$

The coloring $\kappa$ is given on the right in Figure 1. Then $\Phi$ is a
balanced relative simplicial complex. We see that $\operatorname{Aut}(\Phi)$
is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$. We let
$\mathbb{Z}/2\mathbb{Z}$ act as the permutation $(ac)(bd)(e).$ Then

$\operatorname{Hilb}(\Phi,\mathbb{Z}/2\mathbb{Z},\mathbf{x})=M_{\\{2\\},4}+\rho(M_{\\{1,2\\},4}+M_{\\{2,3\\},4}+2M_{\\{1,2,3\\},4}).$

We denote the coefficients of
$[M_{S,d}]\operatorname{Hilb}(\Phi,\mathfrak{G}),$ by
$f_{S}(\Phi,\mathfrak{G})$ and refer to them as the _equivariant flag
$f$-vector_ of $\Phi$. These coefficients are permutation characters.
Similarly, we can write

$\operatorname{Hilb}(\Phi,\mathfrak{G},\mathbf{x})=\sum_{S\subseteq[d]}h_{S}(\Phi,\mathfrak{G})F_{S,d+1},$

where the $h_{S}(\Phi,\mathfrak{G})$ are virtual characters, which we call the
_equivariant flag $h$-vector_.

Since $\mathfrak{G}$ acts on $\Phi$, we can say two faces $\sigma$ and $\tau$
are $\mathfrak{G}$-equivalent if $\sigma=\mathfrak{g}\tau$ for
$\mathfrak{g}\in\mathfrak{G}$. We see that $\kappa(\tau)=\kappa(\sigma)$. For
$C\in\Phi/\mathfrak{G}$, we define $\kappa(C)=\kappa(\sigma)$ for any
$\sigma\in C$. We see that $\kappa$ is well-defined on $\Phi/\mathfrak{G}$. We
let

$\operatorname{Hilb}^{O}(\Phi,\mathfrak{G},\mathbf{x})=\sum_{C\in\Phi/\mathfrak{G}}M_{\kappa(C),d+1}$

be the _orbital flag quasisymmetric function_.

######  Proposition 12.

Let $\Phi$ be a balanced relative simplicial complex of dimension $d-1$.
Suppose that $\mathfrak{G}$ acts on $\Phi$.

  1. (1)

Let $S\subseteq[d]$. Then
$f_{S}(\Phi,\mathfrak{G})=\chi_{\kappa^{-1}(S),\mathfrak{G}}$.

  2. (2)

For $S\subseteq[d]$, we have
$f^{O}_{S}(\Phi,\mathfrak{G})=|\kappa^{-1}(S)/\mathfrak{G}|.$

  3. (3)

We have

$\operatorname{Hilb}^{O}(\Phi,\mathfrak{G},\mathbf{x})=\frac{1}{|\mathfrak{G}|}\sum_{\mathfrak{g}\in\mathfrak{G}}\operatorname{Hilb}(\Phi,\mathfrak{G},\mathbf{x};\mathfrak{g}).$

###### Proof.

For the first part, let $S\subseteq[d]$. Given $\mathfrak{g}\in\mathfrak{G}$,
we see that
$[M_{S,d+1}]\operatorname{Hilb}(\Phi,\mathfrak{G},\mathbf{x};\mathfrak{g})=|\\{\sigma\in\Phi:\kappa(\sigma)=S,\mathfrak{g}\sigma=\sigma\\}|=|\operatorname{Fix}_{\mathfrak{g}}(\kappa^{-1}(S))|$.
The first result follows.

For the second result, we see that
$[M_{S,d+1}]\operatorname{Hilb}^{O}(\Phi,\mathfrak{G},\mathbf{x})=|\\{C\in\Phi/\mathfrak{G}:\kappa(C)=S\\}|=|\kappa^{-1}(S)/\mathfrak{G}|$.

For the third result, we have

$\displaystyle\frac{1}{|\mathfrak{G}|}\sum_{\mathfrak{g}\in\mathfrak{G}}\operatorname{Hilb}(\Phi,\mathfrak{G},\mathbf{x};\mathfrak{g})$
$\displaystyle=\frac{1}{|\mathfrak{G}|}\sum_{\mathfrak{g}\in\mathfrak{G}}\sum_{S\subseteq[d]}|\operatorname{Fix}_{\mathfrak{g}}(\kappa^{-1}(S))|M_{S,d+1}$
$\displaystyle=\sum_{S\subseteq[d]}\frac{1}{|\mathfrak{G}|}\sum_{\mathfrak{g}\in\mathfrak{G}}|\operatorname{Fix}_{\mathfrak{g}}(\kappa^{-1}(S))|M_{S,d+1}$
$\displaystyle=\sum_{S\subseteq[d]}|\kappa^{-1}(S)/\mathfrak{G}|M_{S,d+1}$
$\displaystyle=\operatorname{Hilb}^{O}(\Phi,\mathfrak{G},\mathbf{x})$

where the first equality is our first identity, the third equality comes from
Burnside’s Lemma, and the last equality comes from our second identity. ∎

###  3.1. Group action on homology

Let $\Phi$ be a balanced simplicial complex of dimension $d-1$, with
presentation $(\Delta,\Gamma)$. Suppose that $\mathfrak{G}$ acts on $\Phi$. We
discuss various group actions on chain groups and homology groups. Here we let
$\widetilde{C}_{i}(\Phi)=\widetilde{C}_{i}(\Delta,\Gamma)$, the relative chain
group of the pair $(\Delta,\Gamma)$. Similarly, we let
$\widetilde{H}_{i}(\Phi)=\widetilde{H}_{i}(\Delta,\Gamma)$.

Let $S\subseteq[d]$ We write $\Delta|_{S}=\\{\sigma\subseteq
V(\Phi):\sigma\subseteq\tau\mbox{ for some }\tau\in\Phi|_{S}\\}$, and let
$\Gamma|_{S}=\Phi|_{S}\setminus\Delta|_{S}$. We see that $\mathfrak{G}$ acts
on both $\Delta|_{S}$ and $\Gamma|_{S}$ as simplicial complexes. We let
$\widetilde{C}_{i}(\Phi|_{S})=\widetilde{C}_{i}(\Delta|_{S})/\widetilde{C}_{i}(\Gamma|_{S})$
denote the $i$th reduced chain group, which is generated by faces of dimension
$i$. We see that
$\partial_{i}:\widetilde{C}_{i}(\Delta|_{S})\rightarrow\widetilde{C}_{i-1}(\Delta|_{S})$
and
$\partial_{i}:\widetilde{C}_{i}(\Gamma|_{S})\rightarrow\widetilde{C}_{i-1}(\Gamma|_{S})$
are both $\mathfrak{G}$-invariant. Thus
$\partial_{i}:\widetilde{C}_{i}(\Phi|_{S})\rightarrow\widetilde{C}_{i-1}(\Phi|_{S})$
is also $\mathfrak{G}$-invariant. Hence $\mathfrak{G}$ acts on
$\widetilde{H}_{i}(\Phi|_{S})$. We let
$\chi_{\widetilde{H}_{i-1}(\Phi|_{S}),\mathfrak{G}}$ denote the resulting
character.

######  Theorem 13.

Let $\Phi$ be a balanced relative simplicial complex, and let $\mathfrak{G}$
act on $\Phi$. Then

$\operatorname{Hilb}(\Phi,\mathfrak{G},\mathbf{x})=\sum_{S\subseteq[d]}\left(\sum_{i=0}^{|S|}(-1)^{|S|-i}\chi_{\widetilde{H}_{i-1}(\Phi|_{S}),\mathfrak{G}}\right)F_{S,d}.$

###### Proof.

Let $S\subseteq[d]$, and let $\mathfrak{g}\in\mathfrak{G}$. Then

$\displaystyle[F_{S,d}]\operatorname{Hilb}(\Phi,\mathfrak{G},\mathbf{x};\mathfrak{g})$
$\displaystyle=\sum_{T\subseteq S}(-1)^{|S\setminus
T|}[M_{T,d}]\operatorname{Hilb}(\Phi,\mathfrak{G},\mathbf{x};\mathfrak{g})$
$\displaystyle=\sum_{T\subseteq S}(-1)^{|S\setminus
T|}|\\{\sigma\in\kappa^{-1}(T):\mathfrak{g}\sigma=\sigma\\}|$
$\displaystyle=\sum_{i=0}^{|S|}(-1)^{|S|-i}\sum_{T\subseteq
S:|T|=i}|\\{\sigma\in F_{T}(\Phi|_{S}):\mathfrak{g}\sigma=\sigma\\}|$
$\displaystyle=\sum_{i=0}^{|S|}(-1)^{|S|-i}|\\{\sigma\in\Phi|_{S}:|\kappa(\sigma)|=i,\mathfrak{g}\sigma=\sigma\\}|$
$\displaystyle=(-1)^{|S|}\sum_{i=0}^{|S|}(-1)^{i}\operatorname{tr}_{\mathfrak{g}}(\widetilde{C}_{i-1}(\Phi|_{S}))$
$\displaystyle=(-1)^{|S|}\sum_{i=0}^{|S|}(-1)^{i}\operatorname{tr}_{\mathfrak{g}}(\widetilde{H}_{i-1}(\Phi|_{S}))$
$\displaystyle=\sum_{i=0}^{|S|}(-1)^{|S|-i}\chi_{\widetilde{H}_{i-1}(\Phi|_{S}),\mathfrak{G}}(\mathfrak{g}).$

The first equality comes from the definition of fundamental quasisymmetric
functions. The second equality uses the definition of the flag quasisymmetric
class function. The third equality comes from choosing to sum over subsets by
their size, and recognizing that if $\sigma\in\kappa^{-1}(T)$ and $T\subseteq
S$, then $\sigma\in F_{T}(\Phi|_{S})$. The fourth equality comes from
simplifying the inner summation. The fifth and last equalities come from the
definition of the related group actions. The sixth equality is the Hopf trace
formula.

∎

##  4\. Complexes Satisfying Serre’s Condition

Given a balanced relative simplicial complex $\Phi$ and $v\in V(\Phi)$, we
define
$\operatorname{link}_{\Phi}(v)=(\operatorname{link}_{\Delta}(v),\operatorname{link}_{\Gamma}(v))$.
Similarly, given $S\subseteq V(\Phi)$, we let $\Phi\setminus
S=(\Delta\setminus S,\Gamma\setminus S).$

A relative complex $\Phi$ with presentation $(\Delta,\Gamma)$ satisfies
_Serre’s condition_ $(S_{\ell})$ if

$\widetilde{H}_{i-1}(\operatorname{link}_{\Delta}(\sigma),\operatorname{link}_{\Gamma}(\sigma))=0$

for all $\sigma\in\Delta$ with
$i\leq\min(\dim\operatorname{link}_{\Phi}(\sigma),\ell-1).$ We say that $\Phi$
is _relatively Cohen-Macaulay_ if it satisfies $(S_{\dim\Phi})$.

Our first goal in this section is to prove Theorem 2. In the case $\Phi$ is a
balanced simplicial complex, this result was shown by Holmes and Lyle [HL21].
Our proof is a generalization of their proof.

We prove several lemmas first.

######  Lemma 14.

Let $\Phi$ be a relative simplicial complex with presentation
$(\Delta,\Gamma)$ which satisfies $(S_{\ell})$, and let $\sigma\in\Delta$.
Then $\operatorname{link}_{\Phi}(\sigma)$ also satisfies $(S_{\ell}).$

###### Proof.

Let $\tau\in\operatorname{link}_{\Phi}(\sigma)$. Then
$\operatorname{link}_{\operatorname{link}_{\Phi}(\sigma)}(\tau)=\operatorname{link}_{\Phi}(\sigma\cup\tau)$.
Since $\Phi$ satisfies $(S_{\ell})$, we have
$\widetilde{H}_{i-1}(\operatorname{link}_{\Phi}(\sigma\cup\tau))=0$ for
$i\leq\min(\dim(\operatorname{link}_{\Phi}(\sigma\cup\tau)),\ell-1)$. Hence
$\widetilde{H}_{i-1}(\operatorname{link}_{\operatorname{link}_{\Phi}(\sigma)}(\tau))=0$
for
$i\leq\min(\dim(\operatorname{link}_{\operatorname{link}_{\Phi}(\sigma)}(\tau)),\ell-1).$
Thus $\operatorname{link}_{\Phi}(\sigma)$ satisfies $(S_{\ell}).$ ∎

A subset $J\subseteq V(\Phi)$ is _independent_ if, for all $\sigma\in\Delta$,
we have $|\sigma\cap J|\leq 1$. The set $J$ is _excellent_ if for every facet
$\sigma\in\Delta$, we have $|\sigma\cap J|=1$.

######  Lemma 15.

Let $\Phi$ be a relative simplicial complex with presentation
$(\Delta,\Gamma)$. Let $I$ be an independent set of $\Phi$ such that
$\Delta\setminus I\neq\emptyset$. Suppose that $\Phi$ satisfies $(S_{\ell})$.
Then $\widetilde{H}_{i-1}(\Phi\setminus I)=0$ for
$i\leq\min(\dim(\Phi\setminus I),\ell-1).$

###### Proof.

We prove the result by induction on $\ell$, where the case $\ell=1$ is
trivial. So suppose $\ell\geq 2$. Since $\Phi$ satisfies $(S_{\ell})$, it also
satisfies $(S_{\ell-1})$, and thus by the induction hypothesis, we have
$\widetilde{H}_{i-1}(\Phi\setminus I)=0$ for $i\leq\min(\dim(\Phi\setminus
I),\ell-2)$. Thus we only need to show that

(2) $\widetilde{H}_{\ell-2}(\Phi\setminus I)=0\mbox{ when
}\ell-1\leq\dim(\Phi\setminus I).$

We prove Equation (2) by induction on $|I|.$ Suppose that $I=\\{x\\}$. Let
$\Delta_{1}=\\{\sigma\in\Delta:\sigma\cup\\{x\\}\in\Delta\\}$, and
$\Gamma_{1}=\\{\sigma\in\Gamma:\sigma\cup\\{x\\}\in\Gamma\\}$. Then
$\Delta=\Delta_{1}\cup\Delta\setminus\\{x\\}$, and
$\Gamma=\Gamma_{1}\cup\Gamma\setminus\\{x\\}$. Also,
$\operatorname{link}_{\Delta}(x)=\Delta_{1}\cap\Delta\setminus\\{x\\}$ and
$\operatorname{link}_{\Gamma}(x)=\Gamma_{1}\cap\Gamma\setminus\\{x\\}$. We let
$\Phi_{1}=(\Delta_{1},\Gamma_{1}).$

Consider the Mayer-Vietoris long exact sequence

$\widetilde{H}_{\ell-1}(\Phi_{1})\oplus\widetilde{H}_{\ell-1}(\Phi\setminus\\{x\\})\xrightarrow{i^{\ast}_{x}}\widetilde{H}_{\ell-1}(\Phi)\rightarrow\widetilde{H}_{\ell-1}(\operatorname{link}_{\Phi}(x))\rightarrow\widetilde{H}_{\ell-2}(\Phi_{1})\oplus\widetilde{H}_{\ell-2}(\Phi\setminus\\{x\\})\rightarrow\widetilde{H}_{\ell-2}(\Phi)$

We claim that $\widetilde{H}_{i}(\Phi_{1})=0$ for all $i$. We see that $x$ is
a cone point of $\Delta_{1}$. If $x\in\Gamma$, then $x$ is a cone point of
$\Gamma_{1}$. Otherwise, $\Gamma_{1}=\emptyset$. Thus,
$\widetilde{H}_{i}(\Delta_{1})=\widetilde{H}_{i}(\Gamma_{1})=0$ for all $i$.
Applying the long exact sequence for a relative pair, it follows that
$\widetilde{H}_{i}(\Phi_{1})=0$ for all $i$.

Since $\ell-1\leq\dim(\Phi\setminus\\{x\\})\leq\dim(\Phi)$, we see that
$\ell-2\leq\dim(\operatorname{link}_{\Phi}(x))$. Since $\Phi$ satisfies
$(S_{\ell})$, it follows that
$\widetilde{H}_{\ell-2}(\operatorname{link}_{\Phi}(x))=0=\widetilde{H}_{\ell-2}(\Phi).$
By exactness, we see that $\widetilde{H}_{\ell-2}(\Phi\setminus\\{x\\})=0$.
Also, if we let $i_{x}$ denote the inclusion $\Phi\setminus\\{x\\}$ into
$\Phi$, then $i^{\ast}_{x}$ is the induced map on homology. By exactness,
$i^{\ast}_{x}$ is surjective.

Now suppose that $|I|>2$, and let $x\in I$. Define
$I^{\prime}=I\setminus\\{x\\}$. Then $\Delta=\Delta\setminus
I^{\prime}\cup\Delta\setminus\\{x\\}$, and $\Delta\setminus I=\Delta\setminus
I^{\prime}\cap\Delta\setminus\\{x\\}$. Similarly, $\Gamma=\Gamma\setminus
I^{\prime}\cup\Gamma\setminus\\{x\\}$, and $\Gamma\setminus I=\Gamma\setminus
I^{\prime}\cap\Gamma\setminus\\{x\\}$.

Consider the Mayer-Vietoris long exact sequence

$\widetilde{H}_{\ell-1}(\Phi\setminus
I^{\prime})\oplus\widetilde{H}_{\ell-1}(\Phi\setminus\\{x\\})\xrightarrow{j^{\ast}_{I^{\prime}}-i^{\ast}_{x}}\widetilde{H}_{\ell-1}(\Phi)\rightarrow\widetilde{H}_{\ell-2}(\Phi\setminus
I)\rightarrow\widetilde{H}_{\ell-2}(\Phi\setminus
I^{\prime})\oplus\widetilde{H}_{\ell-2}(\Phi\setminus\\{x\\}).$

Here $j^{\ast}_{I^{\prime}}$ is induced by the inclusion map
$j_{I^{\prime}}:\Phi\setminus I^{\prime}\to\Phi$. By induction, we see that
$\widetilde{H}_{\ell-2}(\Phi\setminus
I^{\prime})=0=\widetilde{H}_{\ell-2}(\Phi\setminus\\{x\\})$. Also,
$j_{I^{\prime}}$ is the inclusion map of $\Phi\setminus I^{\prime}$ into
$\Phi$. Since $i_{x}^{\ast}$ is surjective, it follows from exactness that
$\widetilde{H}_{\ell-2}(\Phi\setminus I)=0$.

∎

######  Lemma 16.

Let $\Phi$ be a relative simplicial complex with presentation
$(\Delta,\Gamma)$. Let $J$ be an excellent set of $\Phi$. Suppose that $\Phi$
satisfies $(S_{\ell}).$ Then $\Phi\setminus J$ satisfies $(S_{\ell}).$

###### Proof.

We prove the result by induction on $\dim\Phi$. Let $\sigma\in\Delta\setminus
J$ with $|\sigma|>0$. By Lemma 14, we know that
$\operatorname{link}_{\Phi}(\sigma)$ satisfies $(S_{\ell})$. Define
$J^{\prime}=V(\operatorname{link}_{\Phi}(\sigma))\cap J$. We claim that
$J^{\prime}$ is excellent. If $J^{\prime}$ is not independent as a subset of
$\operatorname{link}_{\Phi}(\sigma)$, then $J^{\prime}$ is not independent in
$\Phi$, and thus $J$ is not independent. Therefore, $J^{\prime}$ is an
independent set. Let $\tau$ be a facet of
$\operatorname{link}_{\Phi}(\sigma)$. Then $\tau\cup\sigma$ is a facet of
$\Phi$. Hence $(\tau\cup\sigma)\cap J=\\{x\\}$ for some unique $x\in J$. Since
$\sigma\in\Delta\setminus J$, we see that $x\in\tau$, and thus $|\tau\cap
J^{\prime}|\leq 1$. If $|\tau\cap J^{\prime}|\geq 2$, then
$|(\sigma\cup\tau)\cap J|\geq 2$, a contradiction. Therefore, $J^{\prime}$ is
excellent.

Since $\dim(\operatorname{link}_{\Phi}(\sigma))<\dim(\Phi)$, by induction
$\operatorname{link}_{\Phi}(\sigma)\setminus J^{\prime}$ satisfies
$(S_{\ell}).$ Observe that $\operatorname{link}_{\Phi\setminus
J}(\sigma)=\operatorname{link}_{\Phi}(\sigma)\setminus J^{\prime}$. Thus
$\widetilde{H}_{i-1}(\operatorname{link}_{\Phi\setminus J}(\sigma))=0$ for
$i\leq\min(\dim(\operatorname{link}_{\Phi\setminus J}(\sigma)),\ell-1).$

It remains to show that $\widetilde{H}_{i-1}(\Phi\setminus J)=0$ for
$i\leq\min(\dim(\Phi\setminus J),\ell-1)$, which follows from Lemma 15. ∎

###### Proof of Theorem 2.

Let $\Phi$ be a balanced relative simplicial complex of dimension $d-1$ which
satisfies $(S_{\ell})$. Let $S\subseteq[d]$. We prove the result by induction
on $|[d]\setminus S|.$ The base case where $S=[d]$ is true by assumption.

Let $x\in S$, and let $S^{\prime}=S\setminus\\{x\\}$, and let
$J=\kappa^{-1}(x)$. By induction, $\Phi|_{S^{\prime}}$ satisfies $(S_{\ell})$.
We see that $J$ is excellent in $\Phi|_{S^{\prime}}$, and that
$\Phi|_{S}=\Phi|_{S^{\prime}}\setminus J$. Thus the result follows from Lemma
16. ∎

We state a few properties about $(S_{\ell})$ that we need.

######  Proposition 17.

Let $\Phi$ be a balanced simplicial complex of dimension $d-1$ with
presentation $(\Delta,\Gamma)$. Suppose that $\Delta$ satisfies $(S_{\ell})$
and $\Gamma$ satisfies $(S_{\ell-1})$. If $\Gamma$ has dimension $d-2$, or is
void, then $\Phi$ satisfies $(S_{\ell})$.

###### Proof.

Let $\sigma\in\Delta$. Consider the long exact sequence

$\widetilde{H}_{i-1}(\operatorname{link}_{\Delta}(\sigma))\rightarrow\widetilde{H}_{i-1}(\operatorname{link}_{\Phi}(\sigma))\rightarrow\widetilde{H}_{i-2}(\operatorname{link}_{\Gamma}(\sigma))$

We see that, when
$i\leq\min(\dim(\operatorname{link}_{\Phi}(\sigma)),\ell-1)$, then
$\widetilde{H}_{i}(\operatorname{link}_{\Delta}(\sigma))=0$. If
$\operatorname{link}_{\Gamma}(\Phi)=\emptyset$, then
$\widetilde{H}_{i-1}(\operatorname{link}_{\Gamma}(\sigma))=0$. Otherwise,
$\dim(\operatorname{link}_{\Gamma}(\sigma))=\dim(\operatorname{link}_{\Delta}(\sigma))-1$.
Then since $\Gamma$ satisfies $(S_{\ell-1})$, we have
$\widetilde{H}_{i-2}(\operatorname{link}_{\Gamma}(\sigma))=0$. By exactness,
$\widetilde{H}_{i-1}(\operatorname{link}_{\Phi}(\sigma))=0.$ ∎

######  Proposition 18.

Let $\Delta_{1}$ and $\Delta_{2}$ be two $d$-dimensional simplicial complexes
such that $\Delta_{1}\cap\Delta_{2}$ is $(d-1)$-dimensional. If $\Delta_{1}$
and $\Delta_{2}$ both satisfy $(S_{\ell})$, and $\Delta_{1}\cap\Delta_{2}$
satisfies $(S_{\ell-1})$, then $\Delta_{1}\cup\Delta_{2}$ satisfies
$(S_{\ell})$.

###### Proof.

Let $\sigma\in\Delta_{1}\cup\Delta_{2}$, and $i\leq\min(d-|\sigma|,\ell-1)$.
If $\sigma\in\Delta_{1}\setminus\Delta_{2}$, then
$\operatorname{link}_{\Delta_{1}\cup\Delta_{2}}(\sigma)=\operatorname{link}_{\Delta_{1}}(\sigma)$,
and
$\widetilde{H}_{i-1}(\operatorname{link}_{\Delta_{1}\cup\Delta_{2}}(\sigma))=0$,
since $\Delta_{1}$ satisfies $(S_{\ell})$. A similar result holds if
$\sigma\in\Delta_{2}\setminus\Delta_{1}$.

So we may assume $\sigma\in\Delta_{1}\cap\Delta_{2}.$ Let
$\Gamma_{i}=\operatorname{link}_{\Delta_{i}}(\sigma)$, and let
$\Gamma=\operatorname{link}_{\Delta_{1}\cup\Delta_{2}}(\sigma)$. Then
$\Gamma=\Gamma_{1}\cup\Gamma_{2}$, and
$\Gamma_{1}\cap\Gamma_{2}=\operatorname{link}_{\Delta_{1}\cap\Delta_{2}}(\sigma)$.
We consider the Mayer-Vietoris sequence

$\widetilde{H}_{i-1}(\Gamma_{1})\oplus\widetilde{H}_{i-1}(\Gamma_{2})\rightarrow\widetilde{H}_{i-1}(\Gamma)\rightarrow\widetilde{H}_{i-2}(\Gamma_{1}\cap\Gamma_{2}).$

If $0\leq i\leq\min(d-|\sigma|,\ell-1)$, then since $\Delta_{j}$ satisfies
$(S_{\ell})$, we see that $\widetilde{H}_{i-1}(\Gamma_{j})=0$. Similarly,
since $\Delta_{1}\cap\Delta_{2}$ satisfies $(S_{\ell-1})$, we see that
$\widetilde{H}_{i-2}(\Gamma_{1}\cap\Gamma_{2})=0$. Thus, by exactness,
$\widetilde{H}_{i-1}(\Gamma)=0$, and $\Delta_{1}\cup\Delta_{2}$ satisfies
$(S_{\ell}).$ ∎

###  4.1. Inequalities

Now we discuss inequalities that hold for various equivariant flag $f$\- and
$h$-vectors for balanced relative simplicial complexes which satisfy
$(S_{\ell})$. First, we introduce a generalization
$h_{S,T}(\Phi,\mathfrak{G})$ of the flag $h$-vector and prove some identities
about this generalization. A lot of inequalities come from expressing some
summation of flag $h$-vectors in terms of $h_{S,T}(\Phi,\mathfrak{G}).$

######  Theorem 19.

Let $\Phi$ be a balanced relative simplicial complex of dimension $d-1$. Let
$\mathfrak{G}$ act on $\Phi$. Given $S\subseteq T\subseteq[d]$, define

$h_{S,T}(\Phi,\mathfrak{G})=\sum_{R:S\subseteq R\subseteq
T}h_{R}(\Phi|_{T},\mathfrak{G}).$

Let $\mathcal{T}_{T\setminus S}(\Phi)$ be a transversal for $\mathfrak{G}$
acting on $F_{T\setminus S}(\Phi).$ Then

$\displaystyle h_{S,T}(\Phi,\mathfrak{G})$ $\displaystyle=\sum_{Q:T\setminus
S\subseteq Q\subseteq T}(-1)^{|T\setminus Q|}f_{Q}(\Phi|_{T},\mathfrak{G})$
$\displaystyle=\sum_{\tau\in\mathcal{T}_{T\setminus
S}(\Phi)}h_{S}((\operatorname{link}_{\Phi}(\tau))|_{S},\mathfrak{G}_{\tau})\uparrow_{\mathfrak{G}_{\tau}}^{\mathfrak{G}}.$

###### Proof.

First, we show that

$\sum_{Q:T\setminus S\subseteq Q\subseteq T}(-1)^{|T\setminus
Q|}f_{Q}(\Phi|_{T},\mathfrak{G})=\sum_{R:S\subseteq R\subseteq
T}h_{R}(\Phi|_{T},\mathfrak{G}).$

We write

$\displaystyle\sum_{Q:T\setminus S\subseteq Q\subseteq T}(-1)^{|T\setminus
Q|}f_{Q}(\Phi|_{T},\mathfrak{G})$ $\displaystyle=\sum_{Q:T\setminus S\subseteq
Q\subseteq T}(-1)^{|T\setminus Q|}\sum_{R\subseteq
Q}h_{R}(\Phi|_{T},\mathfrak{G})$ $\displaystyle=\sum_{R\subseteq
T}\left(\sum_{Q:R\cup(T\setminus S)\subseteq Q\subseteq T}(-1)^{|T\setminus
Q|}\right)h_{R}(\Phi|_{T},\mathfrak{G}).$

We see that the coefficient of $h_{R}(\Phi|_{T},\mathfrak{G})$ is $0$ unless
$R\cup(T\setminus S)=T$, in which case it is $1$. Thus

$\displaystyle\sum_{Q:(T\setminus S)\subseteq Q\subseteq T}(-1)^{|T\setminus
Q|}f_{Q}(\Phi|_{T},\mathfrak{G})$ $\displaystyle=\sum_{R:R\cup(T\setminus
S)=T}h_{R}(\Phi,\mathfrak{G})$ $\displaystyle=\sum_{R:S\subseteq R\subseteq
T}h_{R}(\Phi,\mathfrak{G}).$

Let $\mathfrak{g}\in\mathfrak{G}$. We prove that

(3) $\sum_{Q:(T\setminus S)\subseteq Q\subseteq T}(-1)^{|T\setminus
Q|}f_{Q}(\Phi|_{T},\mathfrak{G};\mathfrak{g})=\sum_{\tau\in\operatorname{Fix}_{\mathfrak{g}}(\Phi|_{T\setminus
S})}h_{S}((\operatorname{link}_{\Phi}(\tau))|_{S},\mathfrak{G}_{\tau},\mathfrak{g}).$

First, observe that

$\displaystyle\sum_{Q:(T\setminus S)\subseteq Q\subseteq T}(-1)^{|T\setminus
Q|}f_{Q}(\Phi|_{T},\mathfrak{G};\mathfrak{g})$
$\displaystyle=\sum_{\sigma\in\operatorname{Fix}_{\mathfrak{g}}(\Phi|_{T}):(T\setminus
S)\subseteq\kappa(\sigma)}(-1)^{|T\setminus\kappa(\sigma)|}$
$\displaystyle=\sum_{\tau\in\operatorname{Fix}_{\mathfrak{g}}(\Phi|_{T\setminus
S})}\sum_{\sigma\in\operatorname{Fix}_{\mathfrak{g}}(\Phi|_{T}):\tau\subseteq\sigma}(-1)^{|T\setminus\kappa(\sigma)|}$
$\displaystyle=\sum_{\tau\in\operatorname{Fix}_{\mathfrak{g}}(\Phi|_{T\setminus
S})}\sum_{\sigma\in\operatorname{Fix}_{\mathfrak{g}}(\operatorname{link}_{\Phi|_{T}}(\tau))}(-1)^{|S\setminus\kappa(\sigma))|}.$

For a fixed $\tau$, the inner summation becomes

$\displaystyle\sum_{\sigma\in\operatorname{Fix}_{\mathfrak{g}}(\operatorname{link}_{\Phi|_{T}}(\tau))}(-1)^{|S\setminus\kappa(\sigma)|}$
$\displaystyle=\sum_{R:R\subseteq S}(-1)^{|(S\setminus
R)|}f_{R}(\operatorname{link}_{\Phi}(\tau)|_{S},\mathfrak{G}_{\tau};\mathfrak{g})$
$\displaystyle=h_{S}(\operatorname{link}_{\Phi}(\tau)|_{S},\mathfrak{G}_{\tau};\mathfrak{g})$

Hence we have shown Equation (3).

Let $\mathcal{T}_{T\setminus S}(\Phi)$ be a traversal for $\mathfrak{G}$
acting on $\kappa^{-1}(T\setminus S)$. Fix $\tau\in\mathcal{T}_{T\setminus
S}(\Phi)$. Then

$\displaystyle
h_{S}((\operatorname{link}_{\Phi}(\tau))|_{S},\mathfrak{G}_{\tau})\uparrow_{\mathfrak{G}_{\tau}}^{\mathfrak{G}}(\mathfrak{g})$
$\displaystyle=\frac{1}{|\mathfrak{G}_{\tau}|}\sum_{\begin{subarray}{c}\mathfrak{h}\in\mathfrak{G}\\\
\mathfrak{h}^{-1}\mathfrak{gh}\in\mathfrak{G}_{\tau}\end{subarray}}h_{S}((\operatorname{link}_{\Phi}(\tau))|_{S},\mathfrak{G}_{\tau};\mathfrak{h}^{-1}\mathfrak{gh})$
$\displaystyle=\frac{1}{|\mathfrak{G}_{\tau}|}\sum_{\begin{subarray}{c}\mathfrak{h}\in\mathfrak{G}\\\
\mathfrak{g}\in\mathfrak{G}_{\mathfrak{h}\tau}\end{subarray}}h_{S}((\operatorname{link}_{\Phi}(\mathfrak{h}\tau))|_{S},\mathfrak{G}_{\mathfrak{h}\tau};\mathfrak{g})$
$\displaystyle=\frac{1}{|\mathfrak{G}_{\tau}|}\sum_{\begin{subarray}{c}\rho\simeq\tau\\\
\mathfrak{g}\in\mathfrak{G}_{\rho}\end{subarray}}\sum_{\begin{subarray}{c}\mathfrak{h}\in\mathfrak{G}\\\
\mathfrak{h}\tau=\rho\end{subarray}}h_{S}((\operatorname{link}_{\Phi}(\rho))|_{S},\mathfrak{G}_{\rho};\mathfrak{g})$
$\displaystyle=\sum_{\begin{subarray}{c}\rho\simeq\tau\\\
\mathfrak{g}\in\mathfrak{G}_{\rho}\end{subarray}}h_{S}((\operatorname{link}_{\Phi}(\rho))|_{S},\mathfrak{G}_{\rho};\mathfrak{g})\left(\sum_{\begin{subarray}{c}\mathfrak{h}\in\mathfrak{G}\\\
\mathfrak{h}\tau=\rho\end{subarray}}\frac{1}{|\mathfrak{G}_{\tau}|}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}\rho\simeq\tau\\\
\mathfrak{g}\in\mathfrak{G}_{\rho}\end{subarray}}h_{S}((\operatorname{link}_{\Phi}(\rho))|_{S},\mathfrak{G}_{\rho};\mathfrak{g})$

where the first equality is a formula for induced class functions. The second
equality comes from applying conjugation. The third equality is the result of
replacing one summation with two summmations, the outer summation being over
faces $\rho$ that are equivalent to $\tau$, and then over group elements
$\mathfrak{h}$ such that $\mathfrak{h}\tau=\rho$. Thus, summing over all
$\tau\in\mathcal{T}_{T\setminus S}(\Phi)$, we obtain

$\displaystyle\sum_{\tau\in\mathcal{T}_{T\setminus
S}(\Phi)}h_{S}((\operatorname{link}_{\Phi}(\tau))|_{S},\mathfrak{G}_{\tau})\uparrow_{\mathfrak{G}_{\tau}}^{\mathfrak{G}}(\mathfrak{g})$
$\displaystyle=\sum_{\tau\in\mathcal{T}_{T\setminus
S}(\Phi)}\sum_{\begin{subarray}{c}\rho\simeq\tau\\\
\mathfrak{g}\in\mathfrak{G}_{\rho}\end{subarray}}h_{S}((\operatorname{link}_{\Phi}(\rho))|_{S},\mathfrak{G}_{\rho};\mathfrak{g})$
$\displaystyle=\sum_{\begin{subarray}{c}\tau\in\operatorname{Fix}_{\mathfrak{g}}(\Phi)\\\
\kappa(\tau)=T\setminus
S\end{subarray}}h_{S}((\operatorname{link}_{\Phi}(\tau))|_{S},\mathfrak{G}_{\tau};\mathfrak{g})$
$\displaystyle=\sum_{Q:(T\setminus S)\subseteq Q\subseteq T}(-1)^{|T\setminus
Q|}f_{Q}(\Phi|_{T},\mathfrak{G};\mathfrak{g}).$

∎

######  Theorem 20.

Let $\Phi$ be a balanced relative simplicial complex of dimension $d$-1.
Suppose that $\mathfrak{G}$ acts on $\Phi$, and that $\Phi$ satisfies
$(S_{\ell}).$

Give $S\subseteq T\subseteq[d]$ with $|S|\leq\ell$, let $\mathcal{T}$ be a
transversal for $\mathfrak{G}$ acting on $\kappa^{-1}(T\setminus S)$. Then we
have

$\displaystyle h_{S,T}(\Phi,\mathfrak{G})=\sum_{\tau\in\mathcal{T}_{T\setminus
S}(\Phi)}\chi_{\widetilde{H}_{|S|-2}(\operatorname{link}_{\Phi}(\tau)|_{S}),\mathfrak{G}_{\tau}}\uparrow_{\mathfrak{G}_{\tau}}^{\mathfrak{G}}.$

In particular, $h_{S,T}(\Phi,\mathfrak{G})\geq_{\mathfrak{G}}0.$

Observe that this result implies Theorem 3.

###### Proof.

Now suppose that $\Phi$ satisfies $(S_{\ell})$ and $|S|\leq\ell$. Let
$\tau\in\mathcal{T}_{T\setminus S}(\Phi)$. By Theorem 13,we have

(4)
$h_{S}((\operatorname{link}_{\Phi}(\tau))|_{S},\mathfrak{G}_{\tau})=\sum_{i=0}^{|S|}(-1)^{|S|-i}\chi_{\widetilde{H}_{i-1}(\operatorname{link}_{\Phi}(\tau)|_{S}),\mathfrak{G}_{\tau}}.$

Since $\Phi$ satisfies $(S_{\ell})$, we know $\Phi|_{T}$ satisfies
$(S_{\ell})$ as well, and hence
$\widetilde{H}_{i-1}(\operatorname{link}_{\Phi|_{T}}(\tau))=0$ for
$i\leq\min(\dim(\operatorname{link}_{\Phi|_{T}}(\tau)),\ell-1)$. Since
$\dim(\operatorname{link}_{\Phi|_{T}}(\tau))=|S|-1$, and $|S|\leq\ell$, we see
that the homology is concentrated in the top dimension. Moreover,
$\operatorname{link}_{\Phi|_{T}}(\tau)=\operatorname{link}_{\Phi}(\tau)|_{S}$.
Thus all the terms on the right hand side of Equation (4) are $0$, except when
$i=|S|$. Hence

$h_{S}((\operatorname{link}_{\Phi}(\tau))|_{S},\mathfrak{G}_{\tau})=\chi_{\widetilde{H}_{|S|-2}(\operatorname{link}_{\Phi}(\tau)|_{S}),\mathfrak{G}_{\tau}}.$

Hence $h_{S}((\operatorname{link}_{\Phi}(\tau))|_{S},\mathfrak{G}_{\tau})$ is
an effective character, and thus $h_{S,T}(\Phi,\mathfrak{G})$ is also an
effective character. ∎

We now prove a Lemma that will be used to prove Theorems 4 and 5.

######  Lemma 21.

Let $\Phi$ be a balanced relative simplicial complex of dimension $d-1$. Let
$\mathfrak{G}$ act on $\Phi$. Suppose that $\Phi$ satisfies $(S_{\ell})$, and
let $S\subseteq[d]$. Let $i\leq\ell$. Then

$\sum_{T\subseteq
S:|T|\geq\ell}\binom{|T|-(\ell-i)}{i}h_{T}(\Phi,\mathfrak{G})\geq_{\mathfrak{G}}0.$

###### Proof.

For a set $S\subseteq[d]$, and $i<d$, we let $m_{i}(S)$ be the $(\ell-i)$th
smallest element, and let

$M_{i}(S)=\\{x\in S:x\leq m_{i}(S)\\}.$

So if we write $S=\\{s_{1},\ldots,s_{k}\\}$ with $s_{1}<s_{2}<\cdots<s_{k}$,
then $m_{i}(S)=s_{\ell-i}$ and $M_{i}(S)=\\{s_{1},\ldots,s_{\ell-i}\\}$.

Given $i\leq\ell$, and $|T|\geq\ell$, we see that $\binom{|T|-(\ell-i)}{i}$
counts the number of subsets $R\subseteq T$ with $|R|=\ell$ and
$M_{i}(R)=M_{i}(T)$.

We see that

$\displaystyle\sum_{T\subseteq
S:|T|\geq\ell}\binom{|T|-(\ell-i)}{i}h_{T}(\Phi,\mathfrak{G})$
$\displaystyle=\sum_{T\subseteq
S:|T|\geq\ell}\sum_{\begin{subarray}{c}R\subseteq T:|R|=\ell\\\
M_{i}(R)=M_{i}(T)\end{subarray}}h_{T}(\Phi,\mathfrak{G})$
$\displaystyle=\sum_{R\subseteq[d]:|R|=\ell}\sum_{\begin{subarray}{c}T:R\subseteq
T\subseteq S\\\ M_{i}(R)=M_{i}(T)\end{subarray}}h_{T}(\Phi,\mathfrak{G})$
$\displaystyle=\sum_{R\subseteq[d]:|R|=\ell}\sum_{\begin{subarray}{c}T:R\subseteq
T\\\ T\setminus M_{i}(R)\subseteq
S\setminus[m_{i}(R)]\end{subarray}}h_{T}(\Phi,\mathfrak{G}).$

The first equality comes from our interpretation of the binomial coefficient,
while the second equality comes from rearranging the order of summation. The
third equality comes from recognizing that $R\subseteq T\subseteq S$ with
$M_{i}(R)=M_{i}(T)$ holds if and only if $T\setminus M_{i}(R)\subseteq
S\setminus[m_{i}(R)].$ For each $R$, the inner sum becomes

$\sum_{T:R\subseteq T\subseteq
M_{i}(R)\cup(S\setminus[m_{i}(R)])}h_{T}(\Phi,\mathfrak{G})=h_{R,M_{i}(R)\cup(S\setminus[m_{i}(R)])}(\Phi,\mathfrak{G})$

which is an effective character by Lemma 19, since $|R|=\ell$. Thus we have a
sum of effective characters, which is effective. Hence

$\sum_{T\subseteq
S:|T|\geq\ell}\binom{|T|-(\ell-i)}{i}h_{T}(\Phi,\mathfrak{G})\geq_{\mathfrak{G}}0.$

∎

###### Proof of Theorem 5.

Let $\Phi$, $d$, $\mathfrak{G}$ be as in the statement of the Theorem. Suppose
$f_{i-1}(\Phi,\mathfrak{G})=0$ for all $i<\ell$, and that $\Phi$ satisfies
$(S_{\ell})$.

Let $i\geq\ell$. Then we have

$\displaystyle(i-\ell+2)f_{i}(\Phi,\mathfrak{G})$ $\displaystyle-$
$\displaystyle(d-i)f_{i-1}(\Phi,\mathfrak{G})$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}S\subseteq[d]\\\
|S|=i+1\end{subarray}}(|S|-(\ell-1))f_{S}(\Phi,\mathfrak{G})$ $\displaystyle-$
$\displaystyle\sum_{R\subseteq[d]:|R|=i}(d-|R|)f_{R}(\Phi,\mathfrak{G})$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}S\subseteq[d]\\\
|S|=i+1\end{subarray}}(|S|-(\ell-1))f_{S}(\Phi,\mathfrak{G})$ $\displaystyle-$
$\displaystyle\sum_{R\subseteq[d]:|R|=i}\sum_{x\in[d]\setminus
R}f_{R}(\Phi,\mathfrak{G})$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}S\subseteq[d]\\\
|S|=i+1\end{subarray}}(|S|-(\ell-1))f_{S}(\Phi,\mathfrak{G})$ $\displaystyle-$
$\displaystyle\sum_{S\subseteq[d]:|S|=i+1}\sum_{x\in
S}f_{S\setminus\\{x\\}}(\Phi,\mathfrak{G})$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}S\subseteq[d]\\\
|S|=i+1\end{subarray}}\sum_{T\subseteq
S}(|S|-(\ell-1))h_{T}(\Phi,\mathfrak{G})$ $\displaystyle-$
$\displaystyle\sum_{\begin{subarray}{c}S\subseteq[d]\\\
|S|=i+1\end{subarray}}\sum_{x\in S}\sum_{T\subseteq
S\setminus\\{x\\}}h_{T}(\Phi,\mathfrak{G})$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}S\subseteq[d]\\\
|S|=i+1\end{subarray}}\left(\sum_{T\subseteq
S}(|S|-(\ell-1))h_{T}(\Phi,\mathfrak{G})\right.$ $\displaystyle-$
$\displaystyle\left.\sum_{x\in S}\sum_{T\subseteq
S\setminus\\{x\\}}h_{T}(\Phi,\mathfrak{G})\right)$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}S\subseteq[d]\\\
|S|=i+1\end{subarray}}\left(\sum_{T\subseteq
S}(|S|-(\ell-1))h_{T}(\Phi,\mathfrak{G})\right.$ $\displaystyle-$
$\displaystyle\left.\sum_{T\subseteq S}|S\setminus
T|h_{T}(\Phi,\mathfrak{G})\right)$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}S\subseteq[d]\\\
|S|=i+1\end{subarray}}\sum_{T\subseteq
S}(|T|-(\ell-1))h_{T}(\Phi,\mathfrak{G})$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}S\subseteq[d]\\\
|S|=i+1\end{subarray}}\sum_{\begin{subarray}{c}T\subseteq S\\\
|T|\geq\ell\end{subarray}}\binom{|T|-(\ell-1)}{1}h_{T}(\Phi,\mathfrak{G})$
$\displaystyle\geq_{\mathfrak{G}}$ $\displaystyle 0.$

The first equality comes from rewriting the $f$-vector in terms of the flag
$f$-vector. The second equality comes from rewriting the second summation to
replace $(d-|R|)$ with a sum over elements of $[d]\setminus R$. The third
equality comes from reindexing the summations by setting $S=R\cup\\{x\\}$. The
fourth equality involves combining the summations, and then expressing the
flag $f$-vector in terms of the flag $h$-vector. The fifth equality follows
from simplifying the second summations by computing the coefficient of a given
$h_{T}(\Phi,\mathfrak{G})$. Combining like terms, and using the fact that
$h_{T}(\Phi,\mathfrak{G})=0$ when $|T|<\ell$, we arrive at the final
expression. However, then we can apply Lemma 21. Thus
$(d-i)f_{i-1}(\Phi,\mathfrak{G})\leq(i-\ell+2)f_{i}(\Phi,\mathfrak{G})$ for
all $i\geq\ell$.

Applying Proposition 10, we see that
$(f_{\ell-1}(\Phi,\mathfrak{G}),\ldots,f_{d-1}(\Phi,\mathfrak{G}))$ is
equivariantly flawless. ∎

###### Proof of Theorem 4.

We see that

$\displaystyle\sum_{k=\ell}^{j}\binom{d-k}{j-k}\binom{k-(\ell-i)}{i}h_{k}(\Phi,\mathfrak{G})$
$\displaystyle=\sum_{k=\ell}^{j}\binom{d-k}{j-k}\binom{k-(\ell-i)}{i}\sum_{T\subseteq
S:|T|=k}h_{T}(\Phi,\mathfrak{G})$
$\displaystyle=\sum_{k=\ell}^{j}\sum_{T\subseteq
S:|T|=k}\binom{d-k}{j-k}\binom{|T|-(\ell-i)}{i}h_{T}(\Phi,\mathfrak{G})$
$\displaystyle=\sum_{k=\ell}^{j}\sum_{T\subseteq
S:|T|=k}\sum_{S\subseteq[d]:|S|=j,T\subseteq
S}\binom{|T|-(\ell-i)}{i}h_{T}(\Phi,\mathfrak{G})$
$\displaystyle=\sum_{T\subseteq
S:|T|\geq\ell}\sum_{S\subseteq[d]:|S|=j,T\subseteq
S}\binom{|T|-(\ell-i)}{i}h_{T}(\Phi,\mathfrak{G})$
$\displaystyle=\sum_{S\subseteq[d]:|S|=j}\sum_{T\subseteq
S:|T|\geq\ell}\binom{|T|-(\ell-i)}{i}h_{T}(\Phi,\mathfrak{G})$
$\displaystyle\geq_{\mathfrak{G}}0$

where the last inequality comes from Lemma 21, applied to $S$. ∎

##  5\. Mixed Graphs

Given a finite set $V$, a _mixed graph_ is a triple $(V,U,D)$, where $U$ is a
set of undirected edges, and $D$ is a set of directed edges. A mixed graph is
_acyclic_ if it does not contain a directed cycle. A _mixed cycle_ in $G$ is a
cycle in the underlying undirected graph that has at least one directed edge.
A _near cycle_ is a mixed cycle of length three that has exactly one
undirected edge.

There are two polynomial invariants associated to acyclic mixed graphs: the
weak and strong chromatic polynomial, both introduced in [BBC+15], motivated
by work in [BBP12]. Given an acyclic mixed graph $\mathbf{g},$ the _weak
chromatic polynomial_ $\chi(\mathbf{g},k)$ counts the number of functions
$f:V\to[k]$ subject to:

  1. (1)

For every $uv\in U$, we have $f(u)\neq f(v)$.

  2. (2)

For every $(u,v)\in D,$ we have $f(u)\leq f(v)$.

Let $F(G)$ be the set of all weak graph colorings. The _strong_ chromatic
polynomial $\bar{\chi}(\mathbf{g},k)$ counts similar functions, only with
strict inequalities for the second condition instead of the weak inequality.
We introduced quasisymmetric function generalizations of both polynomials in
[Whi], by showing how both polynomial invariants come from characters on
$\mathbf{MG}$.

An automorphism of a mixed graph is a bijection on the vertices which preserve
directed edges and undirected edges. Let $\operatorname{Aut}(G)$ be the group
of automorphisms of a mixed graph $G$.

Let $\mathfrak{G}\subseteq\operatorname{Aut}(G).$ Then $\mathfrak{G}$ acts on
$F(G)$, via $\mathfrak{g}f=f\circ\mathfrak{g}^{-1}$ for $f\in F(G)$ and
$\mathfrak{g}\in\mathfrak{G}$. Given $\mathfrak{g}\in\mathfrak{G}$, we define

$\chi(G,\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum_{f\in\operatorname{Fix}_{\mathfrak{g}}(F(G))}\mathbf{x}^{f}.$

We show that $\chi(G,\mathfrak{G},\mathbf{x})$ is a Hilbert quasisymmetric
class function for a relative simplicial complex.

Given a mixed graph $G$, we let $P$ be the transitive closure of the relation
given by the directed edges. When $G$ has no directed cycles, $P$ is a poset.
Let $I\subseteq J$ be two order ideals. We say that the pair $(I,J)$ is
_stable_ if there is no undirected edge in $J\setminus I$. Let $J(G)$ be the
set of order ideals of $P$, and let

$\Phi(G)=\\{I_{1}\subset I_{2}\subset\cdots\subset I_{k}:(I_{j},I_{j+1})\mbox{
is stable for all }0\leq j\leq k\\}$

We define $I_{0}=\emptyset$, and $I_{k+1}=N$. Then $\Phi(G)$ is a balanced
relative simplicial complex, with coloring $\kappa(I)=|I|.$

######  Proposition 22.

Let $G$ be a mixed graph with no directed cycles, and let
$\mathfrak{G}\subseteq\operatorname{Aut}(G)$. Then
$\chi(G,\mathfrak{G},\mathbf{x})=\operatorname{Hilb}(\Phi(G),\mathfrak{G},\mathbf{x})$.
Thus
$\chi(G,\mathfrak{G},x)=\operatorname{ps}\operatorname{Hilb}(\Phi(G),\mathfrak{G},\mathbf{x}).$

###### Proof.

Let $V$ be the vertex set of $G.$ Let $\mathfrak{g}\in\mathfrak{G}$, and write

$\operatorname{Hilb}(\Phi(G),\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum_{I_{\cdot}\in\Phi}\sum_{i_{0}<i_{1}<\cdots<i_{k}}x_{i_{0}}^{|I_{1}|}x_{i_{1}}^{|I_{2}\setminus
I_{1}|}\cdots x_{i_{k}}^{|N\setminus I_{k}|}.$

Given $I_{\cdot}$ and $(i_{0},\ldots,i_{k})$, define a function
$f:V\to\mathbb{N}$ by $f(n)=i_{j}$ if $n\in I_{j+1}\setminus I_{j},$ where
$I_{k+1}=N$. We claim that $f$ is a weak coloring. Let $uv$ be an undirected
edge, and suppose $f(u)=i_{j}$. If $f(v)=i_{j}$, then $I_{j+1}\setminus I_{j}$
contains an edge, which is a contradiction. Thus $f(v)\neq i_{j}$. Let $(u,v)$
be a directed edge, and let $f(u)=i_{j}$. Then $u\geq v$ in $P$. Since $I_{j}$
is an ideal, $v\in I_{j}$, so $f(v)\leq i_{j}$. Hence $f$ is a weak coloring.

Moreover, every coloring arises from this construction. Let $f$ be a proper
coloring of $G$ that is fixed by $\mathfrak{g}$. Let $I_{j}=f^{-1}([j])$. Then
$I_{j}\subseteq I_{k}$ whenever $j\leq k$. If we remove the repeats, we get a
sequence $I_{1}\subset I_{2}\subset\cdots\subset I_{k}$, which we denote by
$I_{\cdot}(f)$. Moreover, we claim each $I_{j}$ must be an order ideal of $P$.
If not, then there exists $x\in I_{j}$, $y\leq x$ with $y\not\in I_{j}$. This
implies that $f(y)>f(x)$, and that there is a directed path in $G$ from $x$ to
$y$. Hence there is an edge $(u,v)$ on the path where $f(u)>f(v)$, a
contradiction.

Therefore $I_{j}$ is an order ideal for all $j$. Similarly, $I_{j}\setminus
I_{j-1}$ is stable for all $j.$ Finally, the chain $I_{1}\subset
I_{2}\subset\cdots\subset I_{k}$ is fixed by $\mathfrak{g}$. Thus
$I_{\cdot}(f)\in\Phi(G)$. We also write $f(N)=\\{i_{1},\ldots,i_{k}\\}$ where
$i_{1}<i_{2}<\cdots,i_{k}$. Then the map
$f\mapsto(I_{\cdot}(f),(i_{1},\cdots,i_{k}))$ defines a bijection between
colorings and terms of
$\operatorname{Hilb}(\Phi(G),\mathfrak{G},\mathbf{x};\mathfrak{g}).$ ∎

###### Proof of Theorem 6.

Let $P_{G}$ be the partial order given by the transitive closure of the
directed edges of $G$. Let $\Delta(P)$ be the order complex of $J(P)$. We
write $\Phi(G)=(\Delta(P),\Gamma(G))$, where $\Gamma(P)$ consists of chains of
ideals $I_{1}\subset I_{2}\subset\cdots\subset I_{k}$ such that
$I_{j}\setminus I_{j-1}$ contains an undirected edge of $G$ for some $j$.
Since $\Delta(J(P))$ is shellable, it is Cohen-Macaulay. If
$\Gamma=\emptyset$, then we are done.

Otherwise, it suffices by Proposition 17 to show that
$\dim\Gamma(G)=\dim\Delta(P)-1$, and that $\Gamma(G)$ satisfies
$(S_{m(G)-1})$. We first show that $\dim\Gamma=\dim(\Delta(P))-1$. To see
this, consider any undirected edge $e$ of $G$. Since $G$ has no near-cycles,
$P/e$ is still a partial order, so we can consider any linear extension $\ell$
of $P/e$. Then we can view $\ell$ as a set composition $C_{1}|\cdots|C_{k}$ of
$V$ where $C_{i}=e$ for some $i$, and the other blocks are singletons. Then
$C_{1}\subset C_{1}\cup C_{2}\subset\cdots\subset C_{1}\cup\cdots\cup C_{k}$
is an element of $\Gamma(G)$ of size $\dim(\Delta(P))$.

Suppose that there is only one undirected edge $e$, and that $G$ has no near
cycles. Consider the function $f:V(\Gamma(M))\rightarrow V(J(P/e))$ given by

$f(I)=\begin{cases}I/e&e\subseteq I\\\ I&\mbox{ otherwise}\end{cases}$

Then $f$ is an isomorphism between $\Gamma(G)$ and $\Delta(G/e)$. Hence
$\Gamma(G)$ is relatively Cohen-Macaulay.

Suppose that $m(G)=2$. It suffices to show that $\Gamma(G)$ is non-empty.
However, that follows from the fact that it has dimension $\dim(\Delta(P))-1.$

So suppose that $m(G)>2$. Then there are at least two undirected edges. Let
$e$ be an undirected edge. Let $U^{\prime}=U\setminus\\{e\\}$. Then
$\Gamma(G)=\Gamma(G-e)\cup\Gamma(G\setminus U^{\prime})$. We see that
$m(G-e)\geq m(G)$ and $m(G\setminus U^{\prime})\geq m(G)$. By induction,
$\Gamma(G-e)$ and $\Gamma(G\setminus U^{\prime})$ both satisfy $(S_{m(G)-1})$.

We claim that $\Gamma(G-e)\cap\Gamma(G\setminus U^{\prime})$ is isomorphic to
$\Gamma(G/e)$. Let $I\in V(\Gamma(G-e)\cap\Gamma(G\setminus U^{\prime})$. Then
there exists $f\in U^{\prime}$ such that $I$ is an ideal of $P$ and $e\cup
f\subseteq I$. Then $f\subseteq I/e$, so $I/e\in V(\Gamma(G/e))$. If we define
$F:V(\Gamma(G-e)\cap\Gamma(G\setminus U^{\prime}))\rightarrow V(\Gamma(G/e))$
by $F(I)=I/e$, then $F$ is an isomorphism of simplicial complexes.

We see that $m(G)-1\leq m(G/e)\leq m(G)$. Moreover, if $G/e$ has a near cycle,
then $m(G)=2.$ Since we assume $m(G)>2$, it follows that $G/e$ has no near
cycle, and hence $\dim(\Gamma(G/e))=\dim(\Gamma(G))-1$. By induction,
$\Gamma(G/e)$ satisfies $(S_{m(G)-2})$. By Proposition 18, it follows that
$\Gamma(G)$ satisfies $(S_{m(G)-1})$. Thus By Proposition 17, $\Phi(G)$
satisfies $(S_{m(G)}).$

Now let $\mathfrak{G}\subseteq\operatorname{Aut}(G)$. By Proposition 22, we
see that
$\chi(G,\mathfrak{G},\mathbf{x})=\operatorname{Hilb}(\Phi(G),\mathfrak{G},\mathbf{x})$.
Hence
$\chi(G,\mathfrak{G},x)=\operatorname{ps}\operatorname{Hilb}(\Phi(G),\mathfrak{G},\mathbf{x}).$
We write

$\chi(G,\mathfrak{G},x)=\sum_{i=0}^{n}f_{i-2}\binom{x}{i}=\sum_{i=0}^{n}h_{i-1}\binom{x+n-i}{n}.$

From principal specialization, it follows that

$\displaystyle f_{i-1}$
$\displaystyle=\sum_{S\subseteq[n-1]:|S|=i}[M_{S,n}]\chi(G,\mathfrak{G},\mathbf{x})$
$\displaystyle=\sum_{S\subseteq[n-1]:|S|=i}[M_{S,n}]\operatorname{Hilb}(\Phi(G),\mathfrak{G},\mathbf{x})$
$\displaystyle=f_{i-1}(\Phi(G)).$

Similarly, $h_{i}=h_{i}(\Phi(G)).$ The other results follow from Theorem 5 and
from Theorem 20.∎

##  6\. Double Posets

Now we will discuss double posets, and marked posets. The Hopf algebra of
double posets was introduced by Malvenuto and Reutenauer [MR11]. Grinberg
associated a quasisymmetric function to any double poset, which is a
generalization of Gessel’s $P$-partition enumerator. This quasisymmetric
function is studied extensively by Grinberg [Gri17], who proved a
combinatorial reciprocity theorem. We studied a quasisymmetric class function
associated to a double poset [Whi21].

Given a finite set $N$, a _double poset_ on $N$ is a triple
$(N,\leq_{1},\leq_{2})$ where $\leq_{1}$ and $\leq_{2}$ are both partial
orders on $N$. Often for standard poset terminology, we will use $\leq_{i}$ as
a prefix to specify which of the two partial orders is being referred to. For
instance, a $\leq_{1}$-order ideal is a subset that is an order ideal with
respect to the first partial order, and a $\leq_{1}$-covering relation refers
to a pair $(x,y)$ such that $x\prec_{1}y$.

$b$$a$$c$$d$ | $c$$b$$d$$a$
---|---
Figure 2. A double poset, with $\leq_{1}$ on the left, and $\leq_{2}$ on the
right.

Given a double poset $D$, a pair $(m,m^{\prime})\in M$ is an _inversion_ if
$m<_{1}m^{\prime}$ and $m^{\prime}<_{2}m$. We refer to an inversion
$(m,m^{\prime})$ as a _descent_ if $m\prec_{1}m^{\prime}$. We say that $D$ is
_inversion-reducible_ if for every inversion $(m,m^{\prime})$, either
$(m,m^{\prime})$ is a descent, or there exists $m^{\prime\prime}$ with
$m<_{1}m^{\prime\prime}<_{1}m^{\prime}$ such that $(m,m^{\prime\prime})$ or
$(m^{\prime\prime},m^{\prime})$ is an inversion. Finally, a double poset is
_tertispecial_ if, whenever $m\prec_{1}m^{\prime}$, then $m$ and $m^{\prime}$
are $\leq_{2}$-comparable.

Let $D$ be a double poset on a finite set $N$, and let
$f:N\rightarrow\mathbb{N}$. Then $f$ is a $D$-partition if and only if it
satisfies the following two properties:

  1. (1)

For $i\leq_{1}j$ in $D$, we have $f(i)\leq f(j)$.

  2. (2)

For $i\leq_{1}j$ and $j\leq_{2}i$ in $D$, we have $f(i)<f(j)$.

We let $P_{D}$ be the set of $D$-partitions.

Given a double poset $D$, and a permutation $\mathfrak{g}\in\mathfrak{S}_{N}$,
we say $\mathfrak{g}$ is an _automorphism_ of $D$ if for all $i,j\in N$, and
$k\in\\{1,2\\}$, if $i\leq_{k}j$, then
$\mathfrak{g}(i)\leq_{k}\mathfrak{g}(j)$. We let $\operatorname{Aut}(D)$ be
the automorphism group of $D$. For instance, for the double poset in Figure 2,
the permutation $(ac)(bd)$ is the only nontrivial automorphism. Similarly the
only nontrivial automorphism of the double poset in Figure 3 is the
permutation $(a)(bd)(c)$.

Let $\mathfrak{G}\subseteq\operatorname{Aut}(D)$. For
$\mathfrak{g}\in\mathfrak{G}$ and $f:N\rightarrow\mathbb{N}$, let
$\mathfrak{g}\cdot f$ be defined by $(\mathfrak{g}\cdot
f)(v)=f(\mathfrak{g}^{-1}\cdot v)$ for all $v\in N$. This defines an action of
$\mathfrak{G}$ on $P_{D}$. Moreover, we see that
$\mathbf{x}^{f}=\mathbf{x}^{\mathfrak{g}\cdot f}$.

For a double poset $D$ on $N$, $\mathfrak{G}\subseteq\operatorname{Aut}(D)$,
and $\mathfrak{g}\in\mathfrak{G}$, let

$\Omega(D,\mathfrak{G},\mathbf{x};\mathfrak{g})=\sum_{f\in
P_{D}:\mathfrak{g}f=f}\mathbf{x}^{f}.$

We call $\Omega(D,\mathfrak{G},\mathbf{x})$ the _$D$ -partition quasisymmetric
class function_. We introduced this invariant in [Whi21].

As an example, consider the double poset $D$ in Figure 2, and let
$\mathfrak{G}=\operatorname{Aut}(D)$. Let $\rho$ denote the regular
representation. Then

$\displaystyle\Omega(D,\mathfrak{G},\mathbf{x})$
$\displaystyle=M_{\\{2\\},4}+\rho(M_{\\{2,3\\},4}+M_{\\{1,3\\},4}+M_{\\{1,2\\},4}+2M_{\\{1,2,3\\},4})$
$\displaystyle=F_{\\{2\\},4}+\operatorname{sgn}(F_{\\{2,3\\},4}+F_{\\{1,2\\},4}-F_{\\{1,2,3\\},4})+\rho
F_{\\{1,3\\},4}.$

As another example, consider the double poset $D$ in Figure 3, and let
$\mathfrak{G}=\operatorname{Aut}(D)$. Let $\rho$ denote the regular
representation. Then

$\displaystyle\Omega(D,\mathfrak{G},\mathbf{x})$
$\displaystyle=M_{\\{1\\},4}+M_{\\{1,3\\},4}+\rho(M_{\\{1,2\\},4}+M_{\\{1,2,3\\},4})$
$\displaystyle=F_{\\{1\\},4}+\operatorname{sgn}F_{\\{1,2\\},4}.$

abcd | $a$$b$$d$$c$
---|---
Figure 3. A double poset, with $\leq_{1}$ on the left, and $\leq_{2}$ on the
right.

###  6.1. From Double Posets to Mixed graphs

Let $D$ be a double poset on a finite set $N$. We can associate a mixed graph
$G(D)$ to an inversion-reducible double poset $D$. We let
$D_{1}=\\{(u,v):v\mbox{ $\leq_{1}$-covers }u\\}.$ We let $U=\\{uv:v\mbox{
$\leq_{1}$-covers }u,v\leq_{2}v\\}.$ Then $G(D)=(V(P),U,D_{1}).$ We see that,
if $\mathfrak{g}$ is an automorphism of $D$, then it is also an automorphism
of $G(D).$ Thus, $\operatorname{Aut}(D)\subseteq\operatorname{Aut}(G(D)).$

Moreover, we see that a function $f:N\to\mathbb{N}$ is a $D$-partition if and
only if $f$ is a weak coloring of $G(D).$ Thus
$\Omega(D,\mathfrak{G},\mathbf{x})=\chi(G(D),\mathfrak{G},\mathbf{x}).$

Thus, we can transfer the results from the previous section to derive results
about double posets. First, $G(D)$ has no near cycles. If $G(D)$ had a near
cycle $C$ with undirected edge $uv$, and there is a directed path from $u$ to
$v$. By definition of $G(D)$, since $uv\in E(G(P))$, we know $v$
$\leq_{1}$-covers $u$ and $v\leq_{2}u$. However, there is also a directed path
of length at least two from $v$ to $u$, which implies that $u\leq_{1}v$ but
that $v$ does not cover $u$. Hence there is no near cycle. From Theorem 6 we
conclude the following:

######  Theorem 23.

Let $D$ be a double poset on $n$ elements. Suppose that $D$ is inversion-
reducible, and let $m(D)=m(G(D)).$ Let $\mathfrak{G}$ act on $D$. Then
$\mathfrak{G}$ acts on $G(D)$, and
$\Omega(D,\mathfrak{G},\mathbf{x})=\chi(G(D),\mathfrak{G},\mathbf{x}).$

For all $S\subseteq[n-1]$, if $|S|\leq m(D)$, we have
$[F_{S,n}]\Omega(D,\mathfrak{G},\mathbf{x})\geq 0.$

Now we apply our results to tertispecial posets.

######  Proposition 24.

Suppose that $D$ is a tertispecial poset on $n$ elements. Then $D$ is
inversion reducible. Moreover $G(D)$ has no mixed cycles. Thus, for every
$\mathfrak{G}\subseteq\operatorname{Aut}(D)$ and every $S\subseteq[n-1]$, we
have $[F_{S,n}]\Omega(D,\mathfrak{G},\mathbf{x})\geq 0.$

###### Proof.

We prove the first condition by induction on $n$. Suppose that $(x,y)$ is an
inversion pair. if $n=2$, then the result is immediate. So suppose $n>2.$

Choose $t$ such that $x\prec_{1}t\leq_{1}y$. Since $D$ is tertispecial, we
have $x\leq_{2}t$ or $t\leq_{2}x$. In the latter case, we have found a descent
pair $(x,t)$. In the former case the pair $(t,y)$ forms an inversion pair. We
observe that the interval $[t,y]$ is equal to $(y)\setminus(t)$, where $(a)$
is the principal order ideal generated by $a$. Since $|[t,y]|<|N|$, by
induction there is a descent pair $(w,z)$ in $[t,y]$, and hence $[x,y]$ also
has a descent.

To see that $G(D)$ has no mixed cycles, let $(u,v)$ be a directed edge with
$v\prec_{1}u$. Then we have $u\not\leq_{2}v$. However, by the definition of
tertispecial, $u$ and $v$ are $\leq_{2}$-comparable, so we have $v\leq_{2}u$.
We see then that for every directed edge $(u,v)$, we have $v\leq_{2}u$. Hence
$D(P)$ must be acyclic. The rest follows from the previous Theorem. ∎

Thus we have shown Theorem 7.

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