Séminaire Lotharingien de Combinatoire, 95B.66 (2026), 12 pp.
Spencer Backman, Richard Danner, Hsin-Chieh Liao and Crow Stephenson
Combinatorics of Leray Models for Permutahedral Varieties and Beyond
Abstract.
We study Bibby, Denham, and Feichtner's Leray model of a matroid for the maximum building set. Using the monomial basis they constructed, we prove the projective Leray model of a matroid is equivariant γ-positive in each graded component. For the Boolean matroid, which corresponds to the permutahedral variety,
we show that the Hilbert series of its Leray model is governed by a
Catalan-Stieltjes matrix studied in (Cheng, Eu, and Hsu, 2023), which
naturally admits a combinatorial interpretation in terms of bicolored
permutations. We further construct a bijection between these bicolored
permutations and the monomial basis of the Leray model. Then we show
that this combinatorial interpretation lifts to
the Sn-module structure on the Leray model,
thereby recovering the bicolored Eulerian quasisymmetric functions
introduced by (Hyatt, 2012).
Finally, we reveal an intriguing connection between
the Sn-representation on the Leray model and
the natural Weyl group representation on the cohomology of type B/C
permutahedral variety.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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