Equivariant Ehrhart theory

Abstract

Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of numerous classical results, and give applications to the Ehrhart theory of rational polytopes and centrally symmetric polytopes. We also recover a character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of a Weyl group on the cohomology of a toric variety associated to a root system.

Binomial Eulerian polynomials for colored permutations

“…Assume further that the affine span of P contains the origin and let L be its intersection with Z N . Stapledon [36] defines the formal power series ϕ * P (t), with coefficients in the representation ring of G, via the generating function formula (34) k≥0…”

“…where χ kP is the permutation representation defined by the G-action on the set of lattice points of kP and ρ : G → GL(L) is the induced representation. The series ϕ * P (t) is a G-equivariant analogue of h * (P, t) which, under additional assumptions (see [36,Section 7]) is a polynomial in t whose coefficients are non-virtual Grepresentations. For example, if P is the standard unit cube in R n on which the symmetric group S n acts by permuting coordinates, then h * (P, t) = A n (t) and ϕ * P (t) = n−1 j=0 ϕ n,j t j , in the notation of Section 1.…”

“…if n = 3 1 + 65t + 185t 2 + 65t 3 The Eulerian polynomial A n (t) affords an S n -equivariant analogue, meaning a graded S n -representation ϕ n = ⊕ n−1 j=0 ϕ n,j such that n−1 j=0 dim(ϕ n,j )t j = A n (t), which arises very often in mathematics: most importantly, in the contexts of equivariant cohomology of toric varieties [25] [30, p. 529] [39], group actions on face rings of simplicial complexes [39] and the homology of posets [27], equivariant Ehrhart theory [36,Section 9], as well as in purely enumerative contexts; see [28] for an overview. The representation ϕ n can also be determined by the generating function formula (6) 1 + n≥1 z n n−1 j=0 ch(ϕ n,j )(x)t j = (1 − t)H(x; z) H(x; tz) − tH(x; z) , where ch stands for Frobenius characteristic and H(x; z) = n≥0 h n (x)z n is the generating function for the complete homogeneous symmetric functions in x = (x 1 , x 2 , .…”

The equivariant Ehrhart theory of the permutahedron

“…[11, Effectiveness Conjecture 12.1]). Let P be a lattice polytope invariant under the action of a group G. The following conditions are equivalent.…”

“…[11, Conjecture 12.4] If H * [z] is a polynomial and the i th coefficient of the h * -polynomial of P is positive, then the trivial representation occurs with non-zero multiplicity in the virtual character H * i . Proposition 5.9.…”