Ideal Polytopes for Representations of $GL_n(\mathbb{C})$

TL;DR

This paper constructs polytopes associated with $GL_n(C)$ representations using superpotentials and ideal coordinates, revealing a universal tropical critical point linked to ideal fillings and total positivity.

Contribution

It introduces a new class of polytopes for $GL_n(C)$ representations, showing their relation to tropical critical points and ideal fillings, independent of reduced expressions.

Findings

Polytope vertices encode bases of $V_\lambda$

Tropical critical point is unique and expression-independent

Connections between total positivity and ideal fillings

Abstract

In this paper we use the superpotential for the flag variety $GL_n/B$ and particular coordinate systems that we call ideal coordinates for $\mathbf{i}$, to construct polytopes $\mathcal{P}^{\mathbf{i}}_\lambda$ inside $\mathbb{R}^{R_+}$, associated to highest weight representations $V_\lambda$ of $GL_n(\mathbb{C})$. Here $\mathbf{i}$ is a reduced expression of the longest element of the Weyl group and $R_+$ is the set of positive roots of $GL_n$. The lattice points of $\mathcal{P}^{\mathbf{i}}_\lambda$ can be used to encode a basis of the representation $V_\lambda$. In particular, for a specific choice of $\mathbf{i}$, the polytope $\mathcal{P}^{\mathbf{i}}_\lambda$ is unimodularly equivalent to a Gelfand-Tsetlin polytope. The construction of the polytopes involves tropicalisation of the superpotential. Using work of Judd (arXiv:1606.06883) we have that there is a unique positive critical point of the superpotential $\mathcal{W}_{t^\lambda}$ over the field of Puiseux series. Its coordinates, in terms of the ideal coordinates for $\mathbf{i}$, are positive in the sense of having positive leading term. The remarkable property of our new polytopes relates to the tropical version of this critical point, which, for every choice of $\mathbf{i}$, gives a point in $\mathbb{R}^{R_+}$ that lies in the interior of the polytope $\mathcal{P}^{\mathbf{i}}_\lambda$. We prove that this tropical critical point is independent of the reduced expression $\mathbf{i}$, and that it is given by a pattern called the ideal filling for $\lambda$ that was introduced by Judd. Finally, combining these results with work of Rietsch (arXiv:math/0511124) relating critical points of the superpotential with Toeplitz matrices, we show that for a totally positive lower-triangular Toeplitz matrix over the field of Puiseux series factorized into simple root subgroups, the valuations of the factors give an ideal filling.