Abstract
The Rees product of partially ordered sets was introduced by Bjorner 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.
Highlights
The Rees product P ∗ Q of two partially ordered sets was introduced and studied by Björner and Welker [8] as a combinatorial analogue of the Rees construction in commutative algebra
The connection of the Rees product of posets to enumerative combinatorics was hinted in [8, Section 5], where it was conjectured that the dimension of the homology of the Rees product of the truncated Boolean algebra Bn {∅} 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 [17], using enumerative and representation theoretic methods, and in [12], using the theory of lexicographic shellability
Summary
Setting y = 0 to (10) yields another identity, recently proven by Shareshian and Wachs (see Proposition 3.3 and Theorem 3.4 in [20]) in order to establish the equivariant Gal phenomenon for the symmetric group action on the n-dimensional stellohedron and Section 6 combines Equation (6) with (10) to establish the same phenomenon for the hyperoctahedral group action on its associated Coxeter complex.
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