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Affine descents and the Steinberg torus

Abstract : Let $W \ltimes L$ be an irreducible affine Weyl group with Coxeter complex $\Sigma$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $\Sigma$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $\gamma$-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the $h$-polynomials of Steinberg tori.
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Kevin Dilks, T. Kyle Petersen, John R. Stembridge. Affine descents and the Steinberg torus. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.25-38. ⟨hal-01185166⟩

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