Resonant Mirkovi\'{c}-Vilonen polytopes and formulas for highest-weight characters

TL;DR

This paper explores the geometric and combinatorial structures underlying formulas for highest-weight characters, introducing resonant Mirković-Vilonen polytopes and new crystal graph structures, especially in the context of the exceptional group G2.

Contribution

It introduces resonant Mirković-Vilonen polytopes as a new geometric framework and develops a Tokuyama-type formula incorporating these structures for the G2 group.

Findings

Resonant Mirković-Vilonen polytopes are identified as the geometric setting for resonance.

New crystal graph structures are constructed on these polytopes for G2.

A novel Tokuyama-type formula with a geometric error term is derived for G2.

Abstract

Formulas for the product of an irreducible character $\chi_\lambda$ of a complex Lie group and a deformation of the Weyl denominator as a sum over the crystal $\mathcal{B}(\lambda+\rho)$ go back to Tokuyama. We study the geometry underlying such formulas using the expansion of spherical Whittaker functions of $p$-adic groups as a sum over the canonical basis $\mathcal{B}(-\infty)$, which we show may be understood as arising from tropicalization of certain toric charts that appear in the theory of total positivity and cluster algebras. We use this to express the terms of the expansion in terms of the corresponding Mirkovi\'{c}-Vilonen polytope. In this non-archimedean setting, we identify resonance as the appropriate analogue of total positivity, and introduce \emph{resonant Mirkovi\'{c}-Vilonen polytopes} as the corresponding geometric context. Focusing on the exceptional group $G_2$, we show that these polytopes carry new crystal graph structures which we use to compute a new Tokuyama-type formula as a sum over $\mathcal{B}(\lambda+\rho)$ plus a geometric error term coming from finitely many crystals of resonant polytopes.