Some applications of Rees products of posets to equivariant gamma-positivity
Christos A. Athanasiadis
math.CO 05E18, 05E45, 05E05, 06A07

Rees products of posets and equivariant 
gamma-positivity]
Some applications of Rees products of posets to 
equivariant gamma-positivity



Department of Mathematics 

National and Kapodistrian University of Athens

Panepistimioupolis

15784 Athens, Greece
caath@math.uoa.gr


 Key words and phrases. 
Rees product, poset homology, group action, 
Schur gamma-positivity, local face module.


The Rees product of partially ordered sets was 
introduced by Björner and Welker. Using the theory 
of lexicographic shellability, Linusson, Shareshian 
and Wachs proved formulas, of significance in the theory 
of gamma-positivity, for the dimension of the homology
of the Rees product of a graded poset P with a certain 
t-analogue of the chain of the same length as P. 
Equivariant generalizations of these formulas are 
proven in this paper, when a group of automorphisms
acts on P, and are applied to establish the 
Schur gamma-positivity of certain symmetric functions 
arising in algebraic and geometric combinatorics.


[
    Christos A. Athanasiadis
    May 1, 2019
============================





§ INTRODUCTION



The Rees product P ∗ Q of two partially ordered 
sets (posets, for short) was introduced and studied by 
Björner and Welker <cit.> as a combinatorial 
analogue of the Rees construction in commutative 
algebra (a precise definition of P ∗ Q can be 
found in Section <ref>). The connection of 
the Rees product of posets to enumerative 
combinatorics was hinted in <cit.>, 
where it was conjectured that the dimension of the 
homology of the Rees product of the truncated 
Boolean algebra B_n {∅} of rank 
n-1 with an n-element chain equals the number of 
permutations of [n] := {1, 2,…,n} without 
fixed points. This statement was generalized in 
several ways in <cit.>, using enumerative and 
representation theoretic methods, and 
in <cit.>, using the theory of lexicographic 
shellability. 

One of the results of <cit.> proves formulas
<cit.> for the dimension of the 
homology of the Rees product of an EL-shellable poset 
P with a contractible poset which generalizes 
the chain of the same rank as P. This paper provides 
an equivariant analogue of this result which seems 
to have enough applications on its own to be of 
independent interest. To state it, let P be a 
finite bounded poset, with minimum element 0̂ 
and maximum element 1̂, which is graded of 
rank n+1, with rank function ρ: P →{0, 1,…,n+1} (for basic terminology on posets, 
see <cit.>). Fix a field  and 
let G be a finite group which 
acts on P by order preserving bijections. Then, G 
defines a permutation representation α_P(S) 
over  for every S ⊆ [n], induced by 
the action of G on the set of maximal chains of 
the rank-selected subposet 


    P_S  =  { x ∈ P: ρ(x) ∈ S } ∪ {0̂, 1̂}


of P. Following Stanley's seminal work <cit.>, 
we may consider the virtual G-representation


    β_P(S)  =  ∑_T ⊆ S (-1)^|S-T| α_P(T),


defined equivalently by the equations


    α_P(T)  =  ∑_S ⊆ Tβ_P(S)


for T ⊆ [n]. The dimensions of α_P(S) 
and β_P(S) are important enumerative invariants 
of P, known as the entries of its flag f-vector and
flag h-vector, respectively. When P is Cohen–Macaulay 
over , β_P(S) is isomorphic to the 
non-virtual G-representation H_|S|-1 
(P̅_S; ) induced on the top homology group of 
P̅_S := P_S {0̂, 1̂}
<cit.>. As discussed and illustrated 
in various situations in <cit.>, the decomposition 
of β_P(S) as a direct sum of 
irreducible G-representations often leads to very 
interesting refinements of the flag h-vector of P.

As in references <cit.>, we write 
β(P̅) in place of β_P([n]) and note,
as just mentioned, that this G-representation is 
isomorphic to H_n-1 (P̅; ) if 
P is Cohen–Macaulay over . We 
denote by T_t,n the poset whose Hasse diagram 
is a complete t-ary tree of height n, rooted at 
the minimum element. We denote by P^-, P_- 
and P̅ the poset obtained from P by 
removing its maximum element, or minimum element, 
or both, respectively, and recall 
from <cit.> (see also Section <ref>) 
that the action of G on P induces actions on
P^- ∗ T_t,n and P̅∗ T_t,n-1 
as well. We also write [a, b] := {a, 
a+1,…,b} for integers a ≤ b and denote 
by (Θ) the set of all subsets, called 
stable, of Θ⊆ which do not 
contain two consecutive integers. The following 
result reduces to <cit.>, 
proven in <cit.> under additional 
shellability assumptions on P, in the special 
case of a trivial action.

 
Let G be a finite group acting on a finite bounded 
graded poset P of rank n+1 by order preserving 
bijections. Then, 
  
    β ((P^- ∗ T_t,n)_-) 
         ≅_G    ∑_S ∈ ([n-1])β_P ([n]  S)  t^|S| (1+t)^n-2|S|  + 
          
          ∑_S ∈ ([n-2])β_P 
        ([n-1]  S)  t^|S|+1 (1+t)^n-1-2|S|


and 
  
    β (P̅∗ T_t,n-1) 
         ≅_G    ∑_S ∈ ([2, n-2])β_P 
        ([n-1]  S)  t^|S|+1 (1+t)^n-2-2|S|  + 
          
          ∑_S ∈ ([2, n-1])β_P ([n]  S)  t^|S| (1+t)^n-1-2|S|


for every positive integer t, where ≅_G 
stands for isomorphism of G-representations.
If P is Cohen–Macaulay over , then the 
left-hand sides of (<ref>) and (<ref>) 
may be replaced by the G-representations 
H_n-1 ((P^- ∗ T_t,n)_-; ) 
and H_n-1 (P̅∗ T_t,n-1; 
), respectively, and all representations which 
appear in these formulas are non-virtual.


Several applications of <cit.> 
to γ-positivity appear in <cit.> 
<cit.> and are summarized in 
<cit.>. Theorem <ref> 
has non-trivial applications to Schur 
γ-positivity, which we now briefly discuss. 
A polynomial in t with real coefficients is said 
to be γ-positive if for some m ∈, it can be written as a nonnegative linear 
combination of the binomials t^i (1+t)^m-2i for 
0 ≤ i ≤ m/2. Clearly, all such polynomials 
have symmetric and unimodal coefficients. Two 
symmetric function identities due to Gessel 
(unpublished), stated without proof 
in <cit.> <cit.>, 
can be written in the form 


    1-t/E(; tz) - tE(; z) =  
    1   +  ∑_n ≥ 2 z^n 
    ∑_k=0^⌊ (n-2)/2 ⌋ξ_n,k()   t^k+1 (1 + t)^n-2k-2


and


    (1-t) E(; tz)/E(; tz) - tE(; z) =  1   +  ∑_n ≥ 1 z^n 
    ∑_k = 0^⌊ (n-1)/2 ⌋γ_n,k()   t^k+1 (1 + t)^n-1-2k,


where E(; z) = ∑_n ≥ 0 e_n() z^n 
is the generating function for the elementary 
symmetric functions in = (x_1, x_2,…) 
and the ξ_n,k() and γ_n,k() 
are Schur-positive symmetric functions, whose 
coefficients in the Schur basis can be explicitly
described (see Corollary <ref>). The 
coefficients of z^n in the right-hand sides of 
Equations (<ref>) and (<ref>) are 
Schur γ-positive symmetric functions, in 
the sense that their coefficients in the Schur 
basis are γ-positive polynomials in t
with the same center of symmetry. Their Schur 
γ-positivity refines the γ-positivity
of derangement and Eulerian polynomials, respectively;
see <cit.> for more information.

We will show (see Section <ref>) that 
Gessel's identities 
can in fact be derived from the special case of 
Theorem <ref> in which P^- is the 
Boolean algebra B_n, endowed with the natural 
symmetric group action. Moreover, applying the 
theorem when P^- is a natural signed analogue 
of B_n, endowed with a hyperoctahedral group 
action, we obtain new identities of the form 


    E(; tz) - t E(; z)/E(; tz) E(; tz) - tE(; z) E(; z) =  ∑_n ≥ 0 z^n  ∑_k=0^⌊ n/2 ⌋ξ^+_n,k(, )   t^k (1 + t)^n-2k,



    E(; z) - E(; tz)/E(; tz) E(; tz) - tE(; z) E(; z) =  ∑_n ≥ 1 z^n
    ∑_k=0^⌊ (n-1)/2 ⌋ξ^-_n,k(, )   t^k (1 + t)^n-1-2k,



    E(; z) E(; tz) 
    ( E(; tz) - t E(; z) )/E(; tz) E(; tz) - tE(; z) E(; z) =  ∑_n ≥ 0 z^n 
    ∑_k=0^⌊ n/2 ⌋γ^+_n,k
    (, )   t^k (1 + t)^n-2k


and


    t E(; z) E(; tz) 
    ( E(; z) - E(; tz) )/E(; tz) E(; tz) - tE(; z) E(; z) = ∑_n ≥ 1 z^n ∑_k=1^⌊ (n+1)/2 ⌋γ^-_n,k(, )   t^k (1 + t)^n+1-2k,



where the ξ^±_n,k(, ) and 
γ^±_n,k(, ) are Schur-positive 
symmetric functions in the sets of variables = 
(x_1, x_2,…) and = (y_1, y_2,…) 
separately. Their Schur positivity refines 
the γ-positivity of type B analogues or 
variants of derangement and Eulerian polynomials; 
this is explained and 
generalized in the sequel <cit.> to this 
paper. Note that for = 0, the left-hand side 
of Equation (<ref>) specializes to that 
of (<ref>) (with  replaced by ) 
and the sum of the left-hand sides of 
Equations (<ref>) and  (<ref>)
specializes to that of (<ref>) (again with 
 replaced by ). 

Various combinatorial and algebraic-geometric 
interpretations of the left-hand sides of 
Equations (<ref>) and (<ref>) are 
discussed in <cit.> 
<cit.> <cit.>. For instance,
by <cit.>, the coefficient
of z^n in the left-hand side of (<ref>) 
can be interpreted as the Frobenius characteristic 
of the symmetric group representation on the local 
face module of the barycentric subdivision of the 
(n-1)-dimensional simplex, twisted by the sign 
representation. Thus, the Schur γ-positivity
of this coefficient, manifested by 
Equation (<ref>), is an instance of the 
local equivariant Gal phenomenon, as discussed in 
<cit.>. Section <ref> 
shows that another instance of this phenomenon 
follows from the specialization = of 
Equation (<ref>). Similarly, setting 
=0 to (<ref>) yields another 
identity, recently proven by Shareshian and Wachs 
(see Proposition 3.3 and Theorem 3.4 in <cit.>) 
in order to establish the equivariant Gal phenomenon 
for the symmetric group action on the n-dimensional 
stellohedron and Section <ref> combines 
Equation (<ref>) with (<ref>) to 
establish the same phenomenon for the hyperoctahedral
group action on its associated Coxeter complex. Further 
applications of Theorem <ref> are given 
in <cit.>. It would be interesting to find 
direct combinatorial proofs of 
Equations (<ref>) – (<ref>) and 
to generalize other known interpretations of the 
left-hand sides of Equations (<ref>) 
and (<ref>) to those of (<ref>) – 
(<ref>).

Outline. The proof of Theorem <ref> 
is given in Section <ref>, after the 
relevant background and definitions are explained in 
Section <ref>. This proof is fairly 
elementary and different from that 
of <cit.>. Section <ref> 
derives Equations (<ref>) – (<ref>) 
from Theorem <ref> and provides explicit 
combinatorial interpretations, in terms of standard 
Young (bi)tableaux and their descents, for the 
Schur-positive symmetric functions which appear 
there. Sections <ref> and <ref> 
provide the promised applications of 
Equations (<ref>) and (<ref>) to 
the equivariant γ-positivity of the symmetric 
group representation on the local face module of 
a certain triangulation of the simplex and the 
hyperoctahedral group representation on the cohomology
of the projective toric variety associated to the 
Coxeter complex of type B.



§ PRELIMINARIES



This section briefly records definitions and 
background on posets, group representations and 
(quasi)symmetric functions. For basic notions and 
more information on these topics, the reader is 
referred to the sources
<cit.> <cit.> <cit.> 
<cit.> <cit.>. The 
symmetric group of permutations of the set [n] 
is denoted by _n and the cardinality of a 
finite set S by |S|.

Group actions on posets and Rees products.
All groups and posets considered here are assumed 
to be finite. Homological notions for posets always 
refer to those of their order complex; 
see <cit.>. A poset P has the 
structure of a G-poset 
if the group G acts on P by order preserving 
bijections. Then, G induces a representation on 
every reduced homology group H_i (P; 
), for every field . 

Suppose that P is a G-poset with minimum 
element 0̂ and maximum element 1̂. 
Sundaram <cit.> (see also 
<cit.>) established the 
isomorphism of G-representations


    ⊕_k ≥ 0 (-1)^k ⊕_x ∈ P/GH_k-2 ((0̂, x); )
       ↑^G_G_x  ≅_G  0.


Here P/G stands for a complete set of G-orbit 
representatives, (0̂, x) denotes the open 
interval of elements of P lying strictly between 
0̂ and x, G_x is the stabilizer of 
x and ↑ denotes induction. Moreover,
H_k-2 ((0̂, x); ) is 
understood to be the trivial representation 
1_G_x if x = 0̂ and k=0, or x 
covers 0̂ and k = 1. 

The Lefschetz character of a finite 
G-poset P (over the field ) is defined 
as the virtual G-representation


    L(P; G)  :=  ⊕_k ≥ 0  (-1)^k  H_k (P; ).


Note that L(P; G) = (-1)^r H_r 
(P; ), if P is Cohen–Macaulay over  
of rank r. 

Given finite graded posets P and Q with rank 
functions ρ_P and ρ_Q, respectively, 
their Rees product is defined
in <cit.> as P ∗ Q = { (p, q) ∈ P 
× Q: ρ_P (p) ≥ρ_Q (q) }, with 
partial order defined by setting (p_1, q_1) 
≼ (p_2, q_2) if all of the following 
conditions are satisfied:


=0pt

  ∙
p_1 ≼ p_2 holds in P,


  ∙
q_1 ≼ q_2 holds in Q and


  ∙
ρ_P (p_2) - ρ_P (p_1) ≥ρ_Q 
(q_2) - ρ_Q (q_1).


Equivalently, (p_1, q_1) is covered by (p_2, q_2) 
in P ∗ Q if and only if (a) p_1 is covered by 
p_2 in P; and (b) either q_1 = q_2, or q_1 is 
covered by q_2 in Q. We note that the Rees product
P ∗ Q is graded with rank function given by 
ρ(p,q) = ρ_P(p) for (p,q) ∈ P ∗ Q, and 
that if P is a G-poset, then so is P ∗ Q 
with the G-action defined by setting g · (p, 
q) = (g · p, q) for g ∈ G and (p, q) ∈ 
P ∗ Q.

A bounded graded G-poset P, with maximum element
1̂, is said to be G-uniform 
<cit.> if the following hold:


=0pt

  ∙
the intervals [x, 1̂] and [y, 1̂] in 
P are isomorphic for all x, y ∈ P of the same 
rank,


  ∙
the stabilizers G_x and G_y are isomorphic for 
all x, y ∈ P of the same rank, and


  ∙
there is an isomorphism between [x, 1̂] and 
[y, 1̂] that intertwines the actions of G_x 
and G_y, for all x, y ∈ P of the same rank.


Given a sequence of groups G = (G_0, 
G_1,…,G_n), a sequence of posets (P_0, 
P_1,…,P_n) is said to be G-uniform 
<cit.> if 


=0pt

  ∙
P_k is G_k-uniform of rank k for all k,


  ∙
G_k is isomorphic to the stabilizer (G_n)_x
for every x ∈ P_n of rank n-k and all k, 
and


  ∙
there is an isomorphism between P_k and the 
interval [x, 1̂] in P_n that intertwines 
the actions of G_k and (G_n)_x for every x 
∈ P_n of rank n-k and all k.


Under these assumptions, Shareshian and Wachs 
<cit.> established the 
isomorphism of G-representations


    1_G_n⊕ ⊕_k=0^n   
      W_k(P_n;G_n)   [k+1]_t   L((P_n-k∗ 
      T_t,n-k)_-; G_n-k) ↑^G_n_G_n-k  ≅_G  0


for every positive integer t, where W_k(P_n;G_n) 
is the number of G_n-orbits of elements of P_n 
of rank k and [k+1]_t := 1 + t + ⋯ + t^k.

  The Boolean algebra B_n consists of all subsets 
of [n], partially ordered by inclusion. When endowed 
with the standard _n-action, it becomes a 
prototypical example of an _n-uniform poset. 
Every element x ∈ B_n of rank k is a set of 
cardinality k; its stabilizer (_n)_x = {w ∈_n: w x = x} is isomorphic to the Young 
subgroup _k ×_n-k of _n, which 
can be defined as the stalilizer of [k]. The sequence 
(B_0, B_1,…,B_n) can easily be verified to be 
(G_0, G_1,…,G_n)-uniform for G_k := _k 
×_n-k. 


Permutations, Young tableaux and symmetric
functions. Our notation concerning these topics 
follows mostly that of <cit.> 
<cit.> <cit.>.
In particular, the set of standard Young tableaux 
of shape λ is denoted by (λ), 
the descent set { i ∈ [n-1]: w(i) > w(i+1) } 
of a permutation w ∈_n is denoted by (w) and that of a tableau Q ∈(λ), consisting of those entries i for 
which i+1 appears in Q in a lower row than i,
is denoted by (Q). We recall that the 
Robinson–Schensted correspondence is a 
bijection from the 
symmetric group _n to the set of pairs 
(, Q) of standard Young tableaux of the same 
shape and size n. This correspondence has the
property <cit.> that (w) 
= (Q(w)) and (w^-1) = ((w)), 
where ((w), Q(w)) is the pair of tableaux 
associated to w ∈_n.

We will consider symmetric functions in the 
indeterminates = (x_1, x_2,…) over the 
complex field . We denote by E(; z) := 
∑_n ≥ 0 e_n() z^n and H(; z) := 
∑_n ≥ 0 h_n() z^n the generating 
functions for the elementary and complete 
homogeneous symmetric functions, respectively, 
and recall the identity E(; z) H(; -z) = 1. 
The (Frobenius) characteristic map, a -linear 
isomorphism of fundamental importance from the 
space of virtual _n-representations to that 
of homogeneous symmetric functions of degree n,
will be denoted by . This map sends the 
irreducible _n-representation corresponding
to λ⊢ n to the Schur function 
s_λ() associated to λ and thus
it sends non-virtual _n-representations to 
Schur-positive symmetric functions.
The fundamental quasisymmetric function 
associated to S ⊆ [n-1] is defined as
  
    F_n, S ()  =
        ∑_1 ≤ i_1 ≤ i_2 ≤⋯≤ i_n 
     j ∈ S  ⇒  
        i_j < i_j+1
        x_i_1 x_i_2⋯ x_i_n.


The following well known expansion 
<cit.>
  
    s_λ()  =  ∑_Q ∈ (λ) 
        F_n, (Q) ()


will be used in Section <ref>.

For the applications given there,
we need the analogues of these concepts in the 
representation theory of the hyperoctahedral group 
of signed permutations of the set [n], denoted 
here by _n. We will keep this discussion rather
brief and refer to <cit.> for 
more information.

A bipartition of a positive integer n, 
written (λ, μ) ⊢ n, is any pair 
(λ, μ) of integer partitions of total 
sum n. A standard Young bitableau of 
shape (λ, μ) ⊢ n is any pair Q 
= (Q^+, Q^-) of Young tableaux such that 
Q^+ has shape λ, Q^- has shape
μ and every element of [n] appears
exactly once as an entry of Q^+ or Q^-.
The tableaux Q^+ and Q^- are called the 
parts of Q and the number n is its 
size. The Robinson–Schensted 
correspondence of type B, as described 
in <cit.> (see 
also <cit.>), is a 
bijection from the group _n to the set 
of pairs (, Q) of standard Young bitableaux 
of the same shape and size n.
 
The (Frobenius) characteristic map for the 
hyperoctahedral group, denoted by _, 
is a -linear isomorphism from the space of 
virtual _n-representations to that of 
homogeneous symmetric functions of degree n in 
the sets of indeterminates = (x_1, x_2,…) 
and = (y_1, y_2,…) separately; see, for 
instance, <cit.>. The map _
sends the irreducible _n-representation 
corresponding to (λ, μ) ⊢ n to the 
function s_λ() s_μ() and thus
it sends non-virtual _n-representations to 
Schur-positive functions, meaning nonnegative 
integer linear combinations of the functions 
s_λ() s_μ(). The following basic 
properties of _ will be useful in 
Section <ref>:


=1pt

  ∙
_(1__n) = h_n(), where 1__n
is the trivial _n-representation,


  ∙
_ (σ⊗τ↑^_n__k ×_n-k) = 
_(σ) ·_ (τ) for all 
representations σ and τ of _k 
and _n-k, respectively, where _k 
×_n-k is the Young subgroup of _n
consisting of all signed permutations which preserve
the set {± 1, ± 2,…,± k}, 


  ∙
_ (↑^_n__nρ) = 
(ρ) (, ) for every _n-representation 
ρ, where _n ⊂_n is the standard
embedding.


We denote by E(, ; z) := ∑_n ≥ 0 e_n
(, ) z^n and H(, ; z) := ∑_n ≥ 0 
h_n(, ) z^n the generating function for the 
elementary and complete homogeneous, respectively,
symmetric functions in the variables (, ) = 
(x_1, x_2,…,y_1, y_2,…) and note that 
E(, ; z) = E(; z) E(; z), since e_n
(, ) = ∑_k=0^n e_k() e_n-k(), 
and similarly that H(, ; z) = H(; z) 
H(; z).  

The signed descent set 
<cit.> <cit.> of w ∈_n, denoted (w), records the positions 
of the increasing (in absolute value) runs of 
constant sign in the sequence (w(1), 
w(2),…,w(n)). Formally, we may define 
(w) as the pair ((w), ε), 
where ε = (ε_1, 
ε_2,…,ε_n) ∈{-, +}^n 
is the sign vector with ith coordinate equal to 
the sign of w(i) and (w) consists of the 
indices i ∈ [n-1] for which either ε_i 
= + and ε_i+1 = -, or else  
ε_i = ε_i+1 and |w(i)| > 
|w(i+1)| (this definition is slightly different 
from, but equivalent to, the ones given in 
<cit.>). The fundamental 
quasisymmetric function associated to w,
introduced by Poirier <cit.> in a more 
general setting, is defined as 
 
    F_(w) (, )  =
        ∑_i_1 ≤ i_2 ≤⋯≤ i_n 
     j ∈(w)  ⇒  i_j < i_j+1
        z_i_1 z_i_2⋯ z_i_n,

where z_i_j = x_i_j if ε_j = +, and 
z_i_j = y_i_j if ε_j = -. Given a 
standard Young bitableau Q of size n, one defines 
the signed descent set (Q) as the pair 
((Q), ε), where ε = 
(ε_1, ε_2,…,ε_n) 
∈{-, +}^n is the sign vector with ith 
coordinate equal to the sign of the part of Q in 
which i appears and (Q) is the set of indices 
i ∈ [n-1] for which either ε_i = + 
and ε_i+1 = -, or else  ε_i 
= ε_i+1 and i+1 appears in Q in a 
lower row than i. The function F_s(Q) (, 
) is then defined by the right-hand side of 
Equation (<ref>), with w replaced by
Q; see <cit.>. The analogue 
  
    s_λ() s_μ()  = ∑_Q ∈ 
        (λ,μ) F_s(Q) (, )

of the expansion (<ref>) holds 
(<cit.>) and the 
Robinson–Schensted correspondence of type B has 
the properties that (w) = (Q^B(w)) and 
(w^-1) = (^B(w)), where (^B(w), 
Q^B(w)) is the pair of bitableaux associated to w 
∈_n; see <cit.>.




§ PROOF OF THEOREM <REF>



This section proves Theorem <ref> using 
only the definition of Rees product and the defining 
equation (<ref>) of the representations 
β_P(S). For S = {s_1, s_2,…,s_k}⊆ [n] with s_1 < s_2 < ⋯ < s_k we 
set 
  
    φ_t(S)     :=     [s_1 + 1]_t   [s_2 - s_1 + 1]_t 
                            ⋯ [s_k - s_k-1 + 1]_t 
    ψ_t(S)     :=     [s_1]_t   [s_2 - s_1 + 1]_t ⋯ 
                            [s_k - s_k-1 + 1]_t


where, as mentioned already, [m]_t = 1 + t + ⋯ + 
t^m-1 for positive integers m.

 
Let G be a finite group, P be a finite bounded 
graded G-poset of rank n+1, as in 
Theorem <ref>, and Q, R be the posets 
defined by Q̅ = (P^- ∗ T_t,n)_- and 
R̅ = P̅∗ T_t,n-1. Then, 
  
    α_Q (S)    ≅_G    φ_t(S)  α_P (S) 
    α_R (S)    ≅_G    ψ_t(S)  α_P (S)


for all positive integers t and S ⊆ [n].



Let S = {s_1, s_2,…,s_k}⊆ [n] with 
s_1 < s_2 < ⋯ < s_k and let ρ: T_t,n→ be the rank function of T_t,n. By the 
definition of Rees product, the maximal chains in 
Q_S have the form


    0̂ ≺  (p_1, τ_1)  ≺  
       (p_2, τ_2)  ≺ ⋯ ≺  
       (p_k, τ_k)  ≺ 1̂


where 0̂≺ p_1 ≺ p_2 ≺⋯≺ p_k ≺1̂ is a maximal chain in 
P_S and τ_1 ≼τ_2 ≼⋯≼τ_k is a multichain in T_t,n such 
that 


=0pt

  ∙
0 ≤ρ(τ_1) ≤ s_1 and


  ∙
ρ(τ_j) - ρ(τ_j-1) ≤ s_j - s_j-1 
for 2 ≤ j ≤ k.


Let m_t(S) be the number of these multichains. 
Since the elements of G act on (<ref>) by 
fixing the τ_j and acting on the corresponding 
maximal chain of P_S, we have α_Q(S) 
≅_G m_t(S)  α_P(S). To choose such a 
multichain τ_1 ≼τ_2 ≼⋯≼τ_k, we need to specify the sequence 
i_1 ≤ i_2 ≤⋯≤ i_k of ranks of its 
elements so that i_j - i_j-1≤ s_j - s_j-1 
for 1 ≤ j ≤ k, where i_0 := s_0 := 0, and 
choose its maximum element τ_k in t^i_k 
possible ways. Summing over all such sequences, 
we get


    m_t (S)  =  ∑_(i_1, i_2,…,i_k) 
        t^i_k =  ∑_0 ≤ a_j ≤ s_j - 
        s_j-1 t^a_1 + a_2 + ⋯ + a_k =  φ_t (S)
 

and the result for α_Q(S) follows. The same
argument applies to α_R(S); one simply has to 
switch the condition for the rank of τ_1 to 0 
≤ρ(τ_1) ≤ s_1 - 1.  



The proof of the following technical lemma will be 
given after that of Theorem <ref>.

 
For every S ⊆ [n] we have


    ∑_S ⊆ T ⊆ [n] (-1)^n-|T| φ_t(T)  =   
         0,    if [n]  S is not stable, 
    
         t^r(1+t)^n-2r,    if [n]  S is 
                            stable and n ∈ S, 
    
         t^r(1+t)^n+1-2r,    if [n]  S is 
                            stable and n ∉S


and


    ∑_S ⊆ T ⊆ [n] (-1)^n-|T| ψ_t(T)  =   
         0,    if 1 ∉S, 
    
         0,    if [n]  S is not stable, 
    
         t^r(1+t)^n-1-2r,    if [n]  S is 
                            stable and 1, n ∈ S, 
    
         t^r(1+t)^n-2r,    if [n]  S is 
                    stable, 1 ∈ S and n ∉S,


where r = n - |S|.


Proof of Theorem <ref>. Using 
Equations (<ref>) and (<ref>), 
as well as Lemma <ref>, we compute that

  
    β_Q ([n])     =    ∑_T ⊆ [n] 
        (-1)^n-|T| α_Q (T)  ≅_G  ∑_T ⊆ [n] (-1)^n-|T| φ_t(T)  α_P (T) 
          
        =    ∑_T ⊆ [n] (-1)^n-|T| φ_t(T)  ∑_S ⊆ Tβ_P(S)
      
          
        =    ∑_S ⊆ [n]β_P(S) 
        ∑_S ⊆ T ⊆ [n] (-1)^n-|T| φ_t(T)


and find similarly that


    β_R ([n])  ≅_G  ∑_S ⊆ [n]β_P(S) ∑_S ⊆ T ⊆ [n] 
       (-1)^n-|T| ψ_t(T).


The proof follows from these formulas and 
Lemma <ref>. 
For the last statement of the theorem one has to note 
that, as a consequence of <cit.>, 
if P is Cohen–Macaulay over , then so are the 
Rees products P^-∗ T_t,n and P̅∗ 
T_t,n-1.


Proof of Lemma <ref>. Let us denote
by χ_t(S) the left-hand side of (<ref>)
and write S = {s_1, s_2,…,s_k}, with 1 ≤ 
s_1 < s_2 < ⋯ < s_k ≤ n. By definition, we 
have 


    χ_t(S)  =  χ_t(s_1)  χ_t(s_2 - s_1) 
       ⋯χ_t(s_k - s_k-1)  ω_t(n - s_k),


where
  
    χ_t(n)     :=    ∑_n ∈ T ⊆ [n] 
                   (-1)^n-|T| φ_t(T) 
          
    ω_t(n)     :=    ∑_T ⊆ [n] 
                   (-1)^n-|T| φ_t(T)


for n ≥ 1 and ω_t(0) := 1. We claim that 


    χ_t(n)  =   
         1+t,    if n = 1, 
    
         t,    if n = 2, 
    
         0,    if n ≥ 3     and     ω_t(n)  =   
         1,    if n = 0, 
    
         t,    if n = 1, 
    
         0,    if n ≥ 2.


Equation (<ref>) is a direct consequence 
of (<ref>) and this claim. To verify the 
claim, note that the defining equation for 
χ_t(n) can be rewritten as 


    χ_t(n)  =  ∑_(a_1, a_2,…,a_k) 
        n (-1)^n-k  [a_1 + 1]_t   [a_2 + 1]_t 
       ⋯ [a_k + 1]_t,


where the sum ranges over all sequences 
(compositions) (a_1, a_2,…,a_k) of positive 
integers summing to n. We leave it as a simple 
combinatorial exercise for the interested reader
to show (for instance, by standard generating 
function methods) that χ_t(n) = 0 for 
n ≥ 3. The claim follows from this fact and 
the obvious recurrence ω_t(n) = χ_t(n) - 
ω_t(n-1), valid for n ≥ 1.

Finally, note that Equation (<ref>) is 
equivalent to (<ref>) in the case 1 ∈ 
S. Otherwise, the terms in the left-hand side 
can be partitioned into pairs of terms, 
corresponding to pairs {T, T ∪{1}} of 
subsets with 1 ∉T, canceling each other.
This shows that the left-hand side vanishes.




§ SYMMETRIC FUNCTION IDENTITIES



This section derives Equations (<ref>) 
– (<ref>) from Theorem <ref> 
(Corollaries <ref>, <ref> 
and <ref>) and interprets combinatorially
the Schur-positive symmetric functions which 
appear there. We first explain why Gessel's 
identities are special cases of this theorem. 
The set of ascents of a permutation w ∈_n is defined as (w) := [n-1] (w) and, similarly, we have () 
:= [n-1] () for every standard 
Young tableau  of size n. Let us recall 
the fact (used in the following proof) that 
the reduced homology groups of a poset with a 
minimum or maximum element vanish. 

 
Equations (<ref>) and (<ref>) are 
valid for the functions 


    ξ_n,k ()  =  ∑_λ⊢ n 
        c_λ,k· s_λ ()  =  ∑_w F_n,(w)()


and


    γ_n,k ()  =  ∑_λ⊢ n 
        d_λ,k· s_λ ()  =  ∑_w F_n,(w)(),


where c_λ,k (respectively, 
d_λ,k ) stands for the number of 
tableaux ∈(λ) for which () 
∈([2,n-2]) (respectively, 
() ∈([n-2]) ) has k 
elements and, similarly, w ∈_n runs through 
all permutations for which (w^-1) ∈([2,n-2]) (respectively, (w^-1) 
∈([n-2]) ) has k elements.



We will apply Theorem <ref> when P^- is the 
Boolean lattice B_n, considered as an _n-poset 
as in Example <ref>. On the one hand, we have 
the equality


    1   +  ∑_n ≥ 2 ( H_n-1 
       ((B_n {∅}) ∗ T_t,n-1; 
       ) ) z^n 
        =  1-t/E(; tz) - tE(; z)


which, although not explicitly stated in <cit.>,
follows as in the proof of its special case t=1 
<cit.>. On the other hand,
since B_n has a maximum element, the second summand
in the right-hand side of Equation (<ref>)
vanishes and hence this equation gives


    ( H_n-1 
       ((B_n {∅}) ∗ T_t,n-1; ) 
       )  = ∑_S ∈ ([2,n-2])( β_B_n ([n-1]  S) )  
       t^|S|+1  (1+t)^n-2-2|S|


for n ≥ 2. The representations β_B_n (S)
for S ⊆ [n-1] are known to satisfy (see, 
for instance, <cit.>) 


    ( β_B_n (S) )  =  ∑_λ⊢ n c_λ,S· 
       s_λ (),


where c_λ,S is the number of standard 
Young tableaux of shape λ and descent set 
equal to S. Combining the previous three 
equalities yields the first equality in 
Equation (<ref>). The second equality 
follows from the first by expanding s_λ() 
according to Equation (<ref>) to get


    ξ_n,k ()  =  ∑_λ⊢ n∑_∑_Q ∈ (λ) 
       F_n, (Q) ()


where, in the inner sum,  runs through all 
tableaux in (λ) for which () ∈_([2,n-2]) has k elements, and then using 
the Robinson–Schensted correspondence and its 
standard properties (w) = (Q(w)) and 
(w^-1) = ((w)) to replace the 
summations with one running over elements of _n,
as in the statement of the corollary. 

The proof of (<ref>) is entirely 
similar; one has to use Equation (<ref>) 
instead of (<ref>), as well as the equality


    1   +  ∑_n ≥ 1 ( H_n-1 
       ((B_n ∗ T_t,n)_-; ) ) z^n  =  (1-t) E(; tz)/E(; tz) - tE(; z).


The latter follows from the proof of Equation (3.3) 
in <cit.>, where the left-hand side 
is equal to -F_t(-z), in the notation used in that 
proof. 


 The coefficient of z^4 in the left-hand sides 
of Equations (<ref>) and (<ref>) 
equals


=3pt

  ∙
e_4()   (t + t^2 + t^3) + e_2()^2   t^2, 
and


  ∙
e_4()   (t + t^2 + t^3 + t^4) + e_1() 
e_3()   (t^2 + t^3) + e_2()^2   (t^2 + t^3),


respectively. These expressions may be rewritten as


=3pt

  ∙
s_(1,1,1,1) ()   t(1+t)^2 + s_(2,1,1) () 
  t^2 + s_(2,2) ()   t^2, and


  ∙
s_(1,1,1,1) ()   t(1+t)^3 + 2 s_(2,1,1) 
()   t^2(1+t) + s_(2,2) ()   t^2(1+t),


respectively, and hence ξ_4,0 () = 
s_(1,1,1,1) (), ξ_4,1 () = s_(2,1,1) 
() + s_(2,2) (), γ_4,0 () = 
s_(1,1,1,1) () and γ_4,1 () = 2 
s_(2,1,1) () + s_(2,2) (). We leave it to 
the reader to verify that these formulas agree with 
Corollary <ref>.


We now focus on the identities (<ref>)
– (<ref>). We will apply 
Theorem <ref> to the collection sB_n 
of all subsets of {1, 2,…,n}∪{ -1, 
-2,…,-n} which do not contain {i, -i} 
for any index i, partially ordered by 
inclusion. This signed analogue of the Boolean 
algebra B_n is a graded poset of rank n, 
having the empty set as its minimum element, on 
which the hyperoctahedral group _n acts in
the obvious way <cit.>, turning 
it into a _n-poset. It is isomorphic to the 
poset of faces (including the empty one) of the 
boundary complex of the n-dimensional 
cross-polytope and hence it is Cohen–Macaulay 
over  and any field. The left-hand sides of 
Equations (<ref>) and (<ref>) 
for P^- = sB_n will be computed using the 
methods of <cit.>. 

Consider the n-element chain C_n = {0, 
1,…,n-1}, with the usual total order. 
Following <cit.>, we denote by I_j(B_n) the 
order ideal of elements of (B_n {∅})
∗ C_n which are strictly less than ([n], j). 
Then I_j(B_n) is an _n-poset for every j ∈ 
C_n and one of the main results of <cit.> (see
<cit.> <cit.>) 
states that


    1   +  ∑_n ≥ 1  z^n  ∑_j=0^n-1  t^j  (
       H_n-2 (I_j(B_n); ) ) 
        =  (1-t) E(; z)/E(; tz) - tE(; z).


 
For the _n-poset sB_n we have


    1   +  ∑_n ≥ 1 _( H_n-1 ((sB_n {∅}) 
      ∗ T_t,n-1; ) ) z^n 
       =  (1-t) E(; z)/E(; tz) E(; tz) - tE(; z) E(; z).




Following the reasoning in the proof 
of <cit.>, we apply (<ref>)
to the Cohen–Macaulay _n-poset obtained from 
(sB_n {∅}) ∗ T_t,n-1 by adding
a minimum and a maximum element. For 0 ≤ j < k ≤ 
n, there are exactly t^j _n-orbits of elements 
x of rank k in this poset with second coordinate 
of rank j in T_t,n-1 and for each one of these, 
the open interval (0̂, x) is isomorphic to I_j(B_k) 
and the stabilizer of x is isomorphic to _k 
×_n-k. We conclude that


    H_n-1 ((sB_n {∅}) 
       ∗ T_t,n-1; )  ≅__n ⊕_k=0^n   (-1)^n-k ⊕_j=0^k-1  t^j (
       H_k-2 (I_j(B_k); ) ⊗ 
       1__n-k)
       ↑^_n__k ×_n-k.


Applying the map _ and 
using the transitivity ↑^_n__k 
×_n-k ≅__n ↑^_k 
×_n-k__k ×_n-k ↑^_n__k ×_n-k of 
induction and properties of _ discussed in 
Section <ref>, the right-hand side becomes


    ∑_k=0^n   (-1)^n-k ∑_j=0^k-1  t^j  (
       H_k-2 (I_j(B_k); ) )
       (, ) · h_n-k().


Thus, the left-hand side of Equation (<ref>)
is equal to


    H(; -z) ·(   1   +  ∑_n ≥ 1  z^n  ∑_j=0^n-1  t^j  (
       H_n-2 (I_j(B_n); ) )
       (, )  )


and the result follows from Equation (<ref>) 
and the identities E(; z) H(; -z) = 1 and 
E(, ; z) = E(; z) E(; z).


Recall the definition of the sets (w) and 
() for a signed permutation w ∈_n 
and standard Young bitableau  of size n, 
respectively, from Section <ref>. Following
<cit.>, we define the type 
B descent set of = (^+, ^-) as 
_B() = () ∪{n}, if n appears
in ^+, and _B() = () otherwise.
The complement of _B() in the set [n] is 
called the type B ascent set of  and 
is denoted by _B(). Similarly, we define 
the type B descent set of w ∈_n 
as _B(w) = (w) ∪{n}, if w(n) is
positive, and _B(w) = (w) otherwise. The 
complement of _B(w) in the set [n] is 
called the type B ascent set of w and 
is denoted by _B(w). The sets _B(w) and
_B() depend only on the signed descent sets 
_B(w) and _B(), respectively, and  
<cit.>, mentioned at the 
end of Section <ref>, implies that _B(w) 
= _B(Q^B(w)) and _B(w^-1) = _B
(^B(w)) for every w ∈_n.

 
Equations (<ref>) and (<ref>) are 
valid for the functions 


    ξ^+_n,k (, )  =  ∑_(λ, μ) ⊢ n 
      c^+_(λ,μ),k· s_λ () s_μ 
      ()  =  ∑_w F_(w)(, )


and


    ξ^-_n,k (, )  =  ∑_(λ, μ) ⊢ n 
      c^-_(λ,μ),k· s_λ () s_μ 
      ()  =  ∑_w F_(w)(, ),


where c^+_(λ,μ),k (respectively, 
c^-_(λ,μ),k ) stands for the number 
of bitableaux ∈(λ,μ) for which 
_B() ∈([2,n]) has k elements and 
contains (respectively, does not 
contain) n and where, similarly, w ∈_n runs through all signed permutations for which 
_B(w^-1) ∈([2,n]) has k 
elements and contains (respectively, does not 
contain) n.



We apply the second part of Theorem <ref> 
for P^- = sB_n, thought of as a _n-poset. 
The representations β_P (S) for S ⊆ 
[n] were computed in this case 
in <cit.>, which implies that 


    _( β_B_n (S) )  =  ∑_(λ,μ) ⊢ n c_(λ,μ),S· s_λ () s_μ()


for S ⊆ [n], where c_(λ,μ),S 
is the number of standard Young bitableaux  of 
shape (λ,μ) such that _B() = S. 
Switching the roles of  and  and 
combining this result with the second part of 
Theorem <ref> and 
Proposition <ref> we get
  
    (1-t) E(; z)/E(; tz) E(; tz) - tE(; z) E(; z)    =    ∑_n ≥ 0 z^n  ∑_k=0^⌊ n/2 ⌋ξ^+_n,k(, )   t^k (1 + t)^n-2k  + 
          
          ∑_n ≥ 1 z^n
        ∑_k=0^⌊ (n-1)/2 ⌋ξ^-_n,k(, )   t^k (1 + t)^n-1-2k,


where the ξ^±_n,k(, ) are given 
by the first equalities in (<ref>) 
and (<ref>). We now note that the 
left-hand side of Equation (<ref>) is equal
to the sum of the left-hand sides, say
Ξ^+(, , t; z) and Ξ^-(, , t; 
z), of Equations (<ref>) 
and (<ref>). Since, as one can readily 
verify, Ξ^+(, , t; z) is left invariant
under replacing t with 1/t and z with tz,
while Ξ^-(, , t; z) is multiplied by
t after these substitutions, the coefficient of
z^n in Ξ^+(, , t; z) (respectively,
Ξ^-(, , t; z)) is a symmetric 
polynomial in t with center of symmetry n/2 
(respectively, (n-1)/2) for every n ∈.
Since the corresponding properties are clear for
the coefficient of z^n in the two summands in
the right-hand side of Equation (<ref>) 
and because of the uniqueness of the decomposition
of a polynomial f(t) as a sum of two symmetric
polynomials with centers of symmetry n/2 and 
(n-1)/2 (see <cit.>), we 
conclude that (<ref>) 
and (<ref>) follow from the single 
equation (<ref>). 

The second equalities in (<ref>) 
and (<ref>) follow by expanding 
s_λ() s_μ() according to 
Equation (<ref>) and then using the 
Robinson–Schensted correspondence of type B and 
its properties (w) = (Q^B(w)) and 
_B(w^-1) = _B(^B(w)), exactly as 
in the proof of Corollary <ref>.


 The coefficient of z^2 in the left-hand side 
of Equations (<ref>) and (<ref>) 
equals


=3pt

  ∙
e_1() e_1()   t + e_2()   t = s_(1)
() s_(1)()   t + s_(1,1)()   t, and


  ∙
e_2()   (1 + t) = s_(1,1)()   (1 + t),


respectively and hence ξ^+_2,0 (, ) = 
0, ξ^+_2,1 (, ) = s_(1)() s_(1)() 
+ s_(1,1)() and ξ^-_2,0 (, ) = 
s_(1,1)(), in agreement with 
Corollary <ref>.


 
For the _n-poset sB_n we have


    1   +  ∑_n ≥ 1 _( H_n-1 
      ((sB_n ∗ T_t,n)_-; ) ) z^n  =  (1-t) E(; z) E(; tz) E(; tz)/E(; tz) E(; tz) - tE(; z) E(; z).




Following the reasoning in the proof 
of <cit.>, we set 


    L_n (, ; t)  :=  _(
       L((sB_n ∗ T_t,n)_-; _n) ),


where L(P; G) denotes the Lefschetz character of 
the G-poset P over  (see 
Section <ref>). Since (sB_n ∗ 
T_t,n)_- is Cohen–Macaulay over  of rank 
n-1, we have 


    _( H_n-1 ((sB_n ∗ 
       T_t,n)_-; ) )  =  (-1)^n-1  
       L_n(, ; t).


Thus, the left-hand side of (<ref>) is 
equal to -∑_n ≥ 0 L_n(, ; t) (-z)^n. 
The sequence of posets (sB_0, sB_1,…,sB_n) can 
easily be verified to be (_0 ×_n, _1 
×_n-1,…,_n ×_0)-uniform 
(see Section <ref>). Moreover, there
is a single _n-orbit of elements of sB_n of
rank k for each k ∈{0, 1,…,n}.
Thus, applying (<ref>) to this 
sequence gives


    1__n⊕ ⊕_k=0^n   [k + 1]_t 
         L((sB_n-k∗ T_t,n-k)_-; _n-k×_k) ↑^_n__n-k×_k  ≅__n 0.


Applying the characteristic map _, as in 
the proof of Proposition <ref>, gives


    ∑_k=0^n   [k + 1]_t   h_k(, )   
       L_n-k (, ; t)  =  - h_n ().


Standard manipulation with generating functions, 
as in the proof of <cit.>, 
results in the formula


    ∑_n ≥ 0 L_n(, ; t) z^n  =  - H(; z)/∑_n ≥ 0  [n+1]_t  
       h_n(, ) z^n =  -  (1-t) H(; z)/H(, ; z) - t H(, ; tz).



The proof now follows by switching z to -z and 
using the identities E(; z) H(; -z) = 1 and 
E(, ; z) = E(; z) E(; z).


 
Equations (<ref>) and (<ref>) 
are valid for the functions 


    γ^+_n,k (, )  =  ∑_(λ, μ) ⊢ n 
      d^+_(λ,μ),k· s_λ () s_μ 
      ()  =  ∑_w F_(w)(, )


and


    γ^-_n,k (, )  =  ∑_(λ, μ) ⊢ n 
      d^-_(λ,μ),k· s_λ () s_μ 
      ()  =  ∑_w F_(w)(, ),


where d^+_(λ,μ),k (respectively, 
d^-_(λ,μ),k ) is the 
number of bitableaux ∈(λ,μ) for 
which _B() ∈([n]) has k 
elements and does not contain (respectively,
contains) n and, similarly, w ∈_n 
runs through all signed permutations for which 
_B(w^-1) ∈([n]) has k elements 
and does not contain (respectively,  
contains) n.



This statement follows by the same reasoning as in 
the proof of Corollary <ref>, provided one 
appeals to the first part of Theorem <ref> 
and Proposition <ref> instead. 


 The coefficient of z^2 in the left-hand side 
of Equations (<ref>) and (<ref>) 
equals


=3pt

  ∙
e_2()   (1 + t + t^2) + e_1()^2   t + 
e_1() e_1()   t = s_(1,1)()   (1 + t)^2
+ s_(2) ()   t + s_(1)() s_(1)()   
t,


  ∙
e_1() e_1()   (t + t^2) + e_2()   
(t + t^2) = s_(1) () s_(1)()   t(1+t) + 
s_(1,1)()   t(1+t), 


respectively and hence we have γ^+_2,0 (, 
) = s_(1,1)(), γ^+_2,1 (, ) = 
s_(2)() + s_(1)() s_(1)() and 
γ^-_2,1 (, ) = s_(1)() s_(1)
() + s_(2)(), in agreement with 
Corollary <ref>.





§ AN INSTANCE OF THE LOCAL EQUIVARIANT GAL 
PHENOMENON
 

This section uses Equation (<ref>) to 
verify an equivariant analogue of Gal's 
conjecture <cit.> for the local face module 
of a certain triangulation of the simplex with 
interesting combinatorial properties. Background 
and any undefined terminology on simplicial 
complexes can be found in <cit.>.

To explain the setup, let V_n = {ε_1, 
ε_2,…,ε_n } be the set
of unit coordinate vectors in ^n and 
Σ_n be the geometric simplex on the vertex 
set V_n. Consider a triangulation Γ of 
Σ_n (meaning, a geometric simplicial 
complex which subdivides Σ_n) with 
vertex set V_Γ and the polynomial ring 
S = [x_v: v ∈ V_Γ] in indeterminates 
which are in one-to-one correspondence with the 
vertices of Γ. The face ring 
<cit.> of Γ is defined 
as the quotient ring [Γ] = S/I_Γ,
where I_Γ is the ideal of S generated by 
the square-free monomials which correspond to the
non-faces of Γ. Following 
<cit.>, we consider the linear forms


    θ_i  =  ∑_v ∈ V_Γ⟨ 
       v, ε_i ⟩ x_v


for i ∈ [n], where ⟨ ,  ⟩ 
is the standard inner product on ^n, and 
denote by Θ the ideal in [Γ] 
generated by θ_1,θ_2,…,θ_n. 
This sequence is a special linear system of 
parameters for [Γ], in the sense of 
<cit.>. As a result, the 
quotient ring (Γ) = [Γ] / Θ 
is a finite dimensional, graded -vector space 
and so is the local face module L_V_n 
(Γ), defined 
<cit.> as the image in 
(Γ) of the ideal of [Γ] 
generated by the square-free monomials which 
correspond to the faces of Γ lying in the
interior of Σ_n. The Hilbert polynomials 
∑_i=0^n _ ((Γ))_i t^i and
∑_i=0^n _ (L_V_n(Γ))_i t^i 
of (Γ) and L_V_n (Γ) are two 
important enumerative invariants of Γ,
namely the h-polynomial <cit.> and the local h-polynomial 
<cit.> <cit.>, respectively.  

Suppose that G is a subgroup of the 
automorphism group _n of Σ_n which
acts simplicially on Γ. Then, G acts on 
the polynomial ring S and (as discussed on 
<cit.>) leaves the -linear 
span of θ_1, θ_2,…,θ_n 
invariant. Therefore, G acts on the 
graded -vector spaces (Γ) 
and L_V_n (Γ) as well and the 
polynomials ∑_i=0^n ((Γ))_i t^i 
and ∑_i=0^n (L_V_n(Γ))_i t^i, whose 
coefficients lie in the representation ring of
G, are equivariant generalizations of the 
h-polynomial and local h-polynomial of 
Γ, respectively. The pair (Γ, G)
is said (see also <cit.>) to 
satisfy the local equivariant Gal 
phenomenon if 


    ∑_i=0^n   (L_V_n(Γ))_i   t^i  =  ∑_k=0^⌊ n/2 ⌋ M_k   t^k 
      (1+t)^n-2k


for some non-virtual G-representations M_k.
This is an analogue for local face modules of 
the equivariant Gal phenomenon, formulated by 
Shareshian and Wachs <cit.> for
group actions on (flag) triangulations of spheres
as an equivariant version of Gal's conjecture
<cit.>. For trivial
actions on flag triangulations of simplices, 
the validity of the local equivariant Gal 
phenomenon was conjectured in <cit.> and 
has been verified in many special cases; see 
<cit.> <cit.>
and references therein. 

Although it would be too optimistic to expect
that the local equivariant Gal phenomenon holds 
for all group actions on flag triangulations of 
Σ_n, the case G = _n deserves special 
attention. We then use the notation


    ( (Γ), t )     :=    ∑_i=0^n  ( (Γ) )_i 
      t^i, 
    ( L_V_n(Γ), t )     :=    ∑_i=0^n  ( L_V_n(Γ) )_i 
      t^i.


For the (first) barycentric subdivision of 
Σ_n we have the following result of Stanley.

 
(<cit.>) 
For the _n-action on the barycentric 
subdivision Γ_n of the simplex Σ_n, 
we have


    1   +  ∑_n ≥ 1 ( L_V_n(Γ_n), t ) z^n  =  1 - t/H(; tz) - tH(; z).




Combining this result with Gessel's identity 
(<ref>) gives 


    ( L_V_n(Γ_n), t )  =  ∑_k=0^⌊ (n-2)/2 ⌋ω ξ_n,k()   t^k+1 (1 + t)^n-2k-2,


where ω is the standard involution on 
symmetric functions exchanging e_λ() 
and h_λ() for every λ, whence 
it follows that (Γ_n, _n) satisfies the 
local equivariant Gal phenomenon for every n.

The combinatorics of the barycentric subdivision 
Γ_n is related to the symmetric group _n.
We now consider a triangulation K_n of the simplex 
Σ_n, studied in <cit.> (see 
also <cit.> 
<cit.>) and shown on the right of 
Figure <ref> for n=3, the combinatorics of 
which is related to the hyperoctahedral group _n.
The triangulation K_n can be defined as the 
barycentric subdivision of the standard cubical 
subdivision of Σ_n, shown on the left of 
Figure <ref> for n=3, whose faces are in 
inclusion-preserving bijection with the nonempty 
closed intervals in the truncated Boolean lattice 
B_n {∅}. Thus, the faces of 
K_n correspond bijectively to chains of nonempty 
closed intervals in B_n {∅} 
and _n acts simplicially on K_n in the 
obvious way. As a simplicial complex, K_n can be 
thought of as a `half Coxeter complex' for _n.

  

 
For the _n-action on K_n we have


    1   +  ∑_n ≥ 1 ( (K_n), t ) z^n  =  H(; z)
      ( H(; tz) - tH(; z) )/H(; tz)^2 - tH(; z)^2


and


    1   +  ∑_n ≥ 1 ( L_V_n(K_n), t ) z^n  =  H(; tz) - tH(; z)/H(; tz)^2 - tH(; z)^2.


Moreover, the pair (K_n, _n) satisfies the 
local equivariant Gal phenomenon for every n.


The proof relies on methods developed by 
Stembridge <cit.> to study representations of 
Weyl groups on the cohomology of the toric varieties
associated to Coxeter complexes. To prepare for it, 
we recall that the h-polynomial of a
simplicial complex Δ of dimension n-1
is defined as 


    h(Δ, t)  =  ∑_i=0^n f_i-1 
      (Δ)   t^i (1-t)^n-i,


where f_i(Δ) stands for the number of 
i-dimensional faces of Δ. Consider  
a pair (Γ, G), consisting of a 
triangulation Γ of Σ_n and a 
subgroup G of _n acting on Γ, as 
discussed earlier. Following <cit.>, we call the action of G 
on Γ proper if w fixes all 
vertices of every face F ∈Δ which is 
fixed by w, for every w ∈ G. Note that 
group actions, such as the _n-actions on 
Γ_n and K_n, on the order complex 
(simplicial complex of chains) of a poset which 
are induced by an action on the poset itself, 
are always proper. Under this assumption, the 
set Γ^w of faces of Γ which are 
fixed by w forms an induced subcomplex of 
Γ, for every w ∈ G.

Although Stembridge <cit.> deals with 
triangulations of spheres, rather than simplices, 
his methods apply to our setting and his Theorem 1.4,
combined with the considerations of Section 6 in
<cit.>, imply that  


    ( (Γ), t )  =  1/n!∑_w ∈_nh(Γ^w, t)/(1 - t)^1+(Γ^w) ∏_i ≥ 1  (1 - t^λ_i(w))  
      p_λ_i(w) ()


for every proper _n-action on Γ, where 
λ_1(w) ≥λ_2(w) ≥⋯ are the 
sizes of the cycles of w ∈_n and p_k()
is a power sum symmetric function.

Proof of Proposition <ref>. To prove 
Equation (<ref>), we follow the analogous 
computation in <cit.> for the 
barycentric subdivision of the boundary complex of 
the simplex. We first note that
(K_n)^w is combinatorially isomorphic to 
K_c(w) for every w ∈_n, where c(w) is 
the number of cycles of w. Furthermore, it was 
shown in <cit.> that, in the 
notation of Section <ref>, h(K_n, t)
is the `half _n-Eulerian polynomial'


    B^+_n(t)  =  ∑_w ∈^+_n 
       t^|_B(w)|,
 

where ^+_n consists of the signed permutations
w ∈_n with negative first coordinate.
These remarks and Equation (<ref>) imply 
that 


    ( (K_n), t ) z^n  = 
       ∑_λ = (λ_1, λ_2,…) 
       ⊢ n m^-1_λ B^+_ℓ(λ)(t)/(1 - t)^ℓ(λ) ∏_i ≥ 1  
       (1 - t^λ_i)   p_λ_i ()
       z^λ_i,


where n!/m_λ is the cardinality of the 
conjugacy class of _n which corresponds to 
λ⊢ n and ℓ(λ) is the number 
of parts of λ. The polynomials B^+_n(t)
are known (see, for instance, 
<cit.>) to satisfy 


    B^+_n(t)/(1 - t)^n =  ∑_k ≥ 0( (2k+1)^n - (2k)^n ) t^k
 

and hence, we may rewrite the previous formula as 


    ( (K_n), t ) z^n  = 
       ∑_k ≥ 0  t^k 
       ∑_λ = (λ_1, λ_2,…) 
       ⊢ n m^-1_λ( (2k+1)^ℓ(λ) - 
       (2k)^ℓ(λ)) ∏_i ≥ 1  
       (1 - t^λ_i)   p_λ_i ()
       z^λ_i .


Summing over all n ≥ 1 and using the standard 
identities


    H(; z)  =  ∑_λ m^-1_λ 
       p_λ () z^|λ| =  exp( 
        ∑_n ≥ 1 p_n() z^n / n )


just as in the proof of <cit.>
(one considers the p_n as algebraically 
independent indeterminates and replaces first each
p_n with (2k+1) (1 - t^n)p_n, then with (2k) 
(1 - t^n)p_n), we conclude that 


    1   +  ∑_n ≥ 1 ( (K_n), t ) z^n     =     1   +  ∑_k ≥ 0  t^k 
      ( H(; z)^2k+1/H(; tz)^2k+1 
      - H(; z)^2k/H(; tz)^2k) 
      
          
        =     1   +  ( H(; z)/H(; tz) 
      - 1 ) ( 1 - t  H(; z)^2/H(; tz)^2)^-1
          
        =    H(; z)
      ( H(; tz) - tH(; z) )/H(; tz)^2 - tH(; z)^2


and the proof of (<ref>) follows. To prove
(<ref>), it suffices to observe that


    1   +  ∑_n ≥ 1 ( 
       L_V_n(K_n), t ) z^n  =  E(, -z) 
       ( 1   +  ∑_n ≥ 1 ( 
       (K_n), t ) z^n ).


The latter follows exactly as the corresponding 
identity for the barycentric subdivision 
Γ_n, shown in the proof of 
<cit.>. Finally, from
Equations (<ref>) and (<ref>)
we deduce that


    ( L_V_n(K_n), t )  =  ∑_k=0^⌊ n/2 ⌋ω ξ^+_n,k (, )   t^k (1 + t)^n-2k.


This expression, Corollary <ref> and the 
well known fact that s_λ() s_μ() 
is Schur-positive for all partitions λ, μ
imply that ( L_V_n(K_n), t ) is 
Schur γ-positive for every n, as claimed
in the last statement of the proposition.




§ AN INSTANCE OF THE EQUIVARIANT GAL 
PHENOMENON
 

A very interesting group action on a simplicial 
complex is that of a finite Coxeter group W on 
its Coxeter complex <cit.>. When W 
is crystallographic, this action induces a graded 
W-representation on the (even degree) cohomology of 
the associated projective toric variety which has been 
studied by Procesi <cit.>, 
Stanley <cit.>, Dolgachev and 
Lunts <cit.>, Stembridge <cit.> and
Lehrer <cit.>, among others. The graded dimension
of this representation is equal to the W-Eulerian 
polynomial. Its equivariant γ-positivity is
a consequence of a variant of Equation (<ref>)
in the case of the symmetric group _n; see 
<cit.> <cit.>. In the 
case of the hyperoctahedral group _n, by 
<cit.> or <cit.>, 
the Frobenius characteristic of this graded 
_n-representation is equal to the coefficient of 
z^n in 


    (1-t) H(; z) H(; tz)/H(, ; tz) - tH(, ; z) .


The following statement (and its proof) shows that the 
equivariant γ-positivity of this graded 
representation is a consequence of the results of 
Section <ref> and confirms another instance 
of the equivariant Gal phenomenon of Shareshian and 
Wachs. As discussed in <cit.>, it is 
reasonable to expect that the same holds for the 
action of any finite crystallographic Coxeter group 
W on its Coxeter complex; that would provide a 
natural equivariant analogue to the γ-positivity 
of W-Eulerian polynomials <cit.>. 


The coefficient of z^n in (<ref>) 
is Schur γ-positive for every n ∈.



Using Equations  (<ref>) with (<ref>)
we find that 


    (1-t) E(; z) E(; tz)/E(, ; tz) - tE(, ; tz)    =    E(; z) E(; tz)(E(; tz) - tE(; z))/E(; tz)E(; tz) - tE(; z)E(; tz)·1-t/E(; tz) - tE(; z)
          
        =    (  ∑_n ≥ 0 z^n 
    ∑_i=0^⌊ n/2 ⌋γ^+_n,i
    (, )   t^i (1 + t)^n-2i) ·
          
          (   1   +  ∑_n ≥ 2 z^n 
    ∑_j=1^⌊ n/2 ⌋ξ_n,j-1()   t^j (1 + t)^n-2j) 
          
        =    ∑_n ≥ 0 z^n ∑_k+ℓ=n∑_i, j γ^+_k,i (, ) 
     ξ_ℓ,j-1()    t^i+j (1+t)^n-2i-2j ,


where we have set ξ_0, -1() := 1 and ξ_1, -1
() := 0. Since Schur-positivity is preserved 
by sums, products and the standard involution on symmetric 
functions, this computation implies that the coefficient of 
z^n in (<ref>) is Schur γ-positive 
for every n ∈ and the proof follows.


 We have shown that


    (1-t) H(; z) H(; tz)/H(, ; tz) - tH(, ; z) =  1  +  ∑_n ≥ 1 z^n ∑_i=0^⌊ n/2 ⌋γ^B_n,i(,)   t^i (1 + t)^n-2i


for some Schur-positive symmetric functions γ^B_n,i
(, ). It is an open problem to find an explicit
combinatorial interpretation of the coefficient
c^B_(λ, μ), i of s_λ() s_μ() in 
γ^B_n,i(, ), for (λ, μ) ⊢ n.
Comparing the graded dimensions of the 
_n-representations whose Frobenius characteristic is
given by the two sides of Equation (<ref>) we 
get


    B_n(t)  =  ∑_(λ,μ) ⊢ nn  |λ| f^λ f^μ∑_i=0^⌊ n/2 ⌋ c^B_(λ,μ), i  
    	 t^i (1 + t)^n-2i ,


where B_n(t) := ∑_w ∈_n t^|_B(w)| is the 
_n-Eulerian polynomial and f^λ stands for the 
number of standard Young tableaux of shape λ. Thus, 
a solution to this problem would provide a refinement of the 
known γ-positivity of _n-Eulerian polynomials; 
see <cit.>.


Acknowledgements. 
The author wishes to thank Eric Katz and Kalle Karu
for some helpful e-discussions about the local 
equivariant Gal phenomenon and the anonymous referees
for their useful comments.

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