Mixed Eulerian Numbers and Peterson Schubert Calculus

DOI10.1093/IMRN/RNAD030arXiv2104.14083OpenAlexW3159045262MaRDI QIDQ6183073

Tatsuya Horiguchi

Publication date: 26 January 2024

Published in: Unnamed Author (Search for Journal in Brave)

Abstract: Let $Phi$ be a root system. Postnikov introduced and studied the mixed $Phi$-Eulerian numbers. These numbers indicate the mixed volumes of $Phi$-hypersimplices. As specializations of these numbers, one can obtain the usual Eulerian numbers, the Catalan numbers, and the binomial coefficients. Recent work of Berget-Spink-Tseng gave a simple computation for the mixed $Phi$-Eulerian numbers when $Phi$ is of type $A$. In this paper we connect a relation between mixed $Phi$-Eulerian numbers and Peterson Schubert calculus. By using the connection, we provide a combinatorial model for the computation of Berget-Spink-Tseng in terms of left-right diagrams which were introduced by Abe-Horiguchi-Kuwata-Zeng for the purpose of Peterson Schubert calculus. We also derive a simple computation for the mixed $Phi$-Eulerian numbers in arbitrary Lie types from Peterson Schubert calculus.

Full work available at URL: https://arxiv.org/abs/2104.14083

Mathematics Subject Classification ID

Bernoulli and Euler numbers and polynomials (11B68) Classical problems, Schubert calculus (14N15)

Related Items (3)

Forest polynomials and the class of the permutahedral variety ⋮ Positivity of Peterson Schubert calculus ⋮ Matroidal mixed Eulerian numbers

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