Calculating Greene's function via root polytopes and subdivision algebras

Karola Mészáros

Vol. 286 (2017), No. 2, 385–400

DOI: 10.2140/pjm.2017.286.385

Abstract

Greene’s rational function ${\Psi }_P\left(x\right)$ is a sum of certain rational functions in $x=\left(x_1,\dots ,x_n\right)$ over the linear extensions of the poset $P$ (which has $n$ elements), which he introduced in his study of the Murnaghan–Nakayama formula for the characters of the symmetric group. In recent work Boussicault, Féray, Lascoux and Reiner showed that ${\Psi }_P\left(x\right)$ equals a valuation on a cone and calculated ${\Psi }_P\left(x\right)$ for several posets this way. In this paper we give an expression for ${\Psi }_P\left(x\right)$ for any poset $P$. We obtain such a formula using dissections of root polytopes. Moreover, we use the subdivision algebra of root polytopes to show that in certain instances ${\Psi }_P\left(x\right)$ can be expressed as a product formula, thus giving a compact alternative proof of Greene’s original result and its generalizations.

Keywords

Greene's function, root polytope, subdivision algebra

Mathematical Subject Classification 2010

Primary: 05E10

Milestones

Received: 13 August 2015

Revised: 9 March 2016

Accepted: 9 March 2016

Published: 15 January 2017