Alcoved Polytopes II

TL;DR

This paper studies polytopes from affine Coxeter arrangements, providing volume formulas, combinatorial definitions for Weyl groups, and a Grobner basis for specific cases, advancing geometric and algebraic understanding.

Contribution

It introduces new formulas and combinatorial tools for alcoved polytopes associated with all Weyl groups, including a q-analogue of Weyl's formula.

Findings

Volume formulas for alcoved polytopes

Compatible definitions of hypersimplices, descent numbers, and major index for Weyl groups

Grobner basis and triangulation for specific types

Abstract

This is the second of two papers where we study polytopes arising from affine Coxeter arrangements. Our results include a formula for their volumes, and also compatible definitions of hypersimplices, descent numbers and major index for all Weyl groups. We give a q-analogue of Weyl's formula for the order of the Weyl group. For A_n, C_n and D_4, we give a Grobner basis which induces the triangulation of alcoved polytopes.