Mirror symmetry, tropical geometry and representation theory

TL;DR

This paper introduces a new interpretation of ideal fillings and related polytopes in representation theory, connecting tropical geometry, superpotentials, and Toeplitz matrix factorizations to deepen understanding of canonical bases.

Contribution

It provides a novel coordinate system and a parabolic generalization of ideal fillings, linking them to superpotentials and matrix factorizations, expanding the geometric and combinatorial framework.

Findings

New coordinate system called 'ideal' coordinates for superpotentials.

Explicit transformations between ideal and string coordinates.

Relation established between ideal fillings and Toeplitz matrix factorizations.

Abstract

An ideal filling is a combinatorial object introduced by Judd that amounts to expressing a dominant weight $\lambda$ of $SL_n$ as a rational sum of the positive roots in a canonical way, such that the coefficients satisfy a $\max$ relation. He proved that whenever an ideal filling has integral coefficients it corresponds to a lattice point in the interior of the string polytope which parametrises the canonical basis of the representation with highest weight $\lambda$. The work of Judd makes use of a construction of string polytopes via the theory of geometric crystals, and involves tropicalising the superpotential of the flag variety $SL_n/B$ in certain `string' coordinates. He shows that each ideal filling relates to a positive critical point of the superpotential over the field of Puiseux series, through a careful analysis of the critical point conditions. In this thesis we give a new interpretation of ideal fillings, together with a parabolic generalisation. For every dominant weight $\lambda$ of $GL_n$, we also define a new family of polytopes in $\mathbb{R}^{R_+}$, where $R_+$ denotes the positive roots of $GL_n$, with one polytope for each reduced expression of the longest element of the Weyl group. These polytopes are related by piecewise-linear transformations which fix the ideal filling associated to $\lambda$ as a point in the interior of each of these polytopes. Our main technical tool is a new coordinate system in which to express the superpotential, which we call the `ideal' coordinates. We describe explicit transformations between these coordinates and string coordinates in the $GL_n/B$ case. Finally, we demonstrate a close relation between our new interpretation of ideal fillings and factorisations of Toeplitz matrices into simple root subgroups.