Calculating Greene's function via root polytopes and subdivision algebras

Abstract

Greene's rational function ${\Psi }_P(\mathbf{x})$ is a sum of certain rational functions in $\mathbf{x}=(x_1,\dots ,x_n)$ 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 ${\Psi }_P(\mathbf{x})$ equals a valuation on a cone and calculated ${\Psi }_P(\mathbf{x})$ for several posets this way. In this paper we give an expression for ${\Psi }_P(\mathbf{x})$ 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 ${\Psi }_P(\mathbf{x})$ can be expressed as a product formula, thus giving a compact alternative proof of Greene's original result and its generalizations.

References in corpus (2)

• Flow polytopes of signed graphs and the Kostant partition function
• h-Polynomials of Reduction Trees