#### Vol. 286, No. 2, 2017

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Calculating Greene's function via root polytopes and subdivision algebras

### Karola Mészáros

Vol. 286 (2017), No. 2, 385–400
##### 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
Primary: 05E10