Vol. 286, No. 2, 2017

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ISSN: 0030-8730
Calculating Greene's function via root polytopes and subdivision algebras

Karola Mészáros

Vol. 286 (2017), No. 2, 385–400
Abstract

Greene’s rational function ΨP(x) is a sum of certain rational functions in x = (x1,,xn) 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 ΨP(x) equals a valuation on a cone and calculated ΨP(x) for several posets this way. In this paper we give an expression for ΨP(x) 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 ΨP(x) 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
Authors
Karola Mészáros
Department of Mathematics
Cornell University
212 Garden Ave.
Ithaca, NY 14853
United States