Relation between two geometrically defined bases in representations of $GL_n$

TL;DR

This paper proves that two different geometrically defined bases in irreducible representations of $GL_n$—one from Springer fibers and one from Mirković-Vilonen cycles—are actually the same, establishing a key connection between these constructions.

Contribution

It demonstrates the equivalence of bases obtained from Springer fibers and Mirković-Vilonen cycles in representations of $GL_n$, linking two geometric approaches.

Findings

The two bases coincide in $GL_n$ representations.

The result bridges Springer fiber and MV cycle constructions.

Provides a unified geometric understanding of bases in representation theory.

Abstract

Let $V$ be an irreducible representation of group $GL_n({\mathbb C})$, which appears as a submodule in $({\mathbb C}^n)^{\otimes d}$, where ${\mathbb C}^n$ is the tautological $n$-dimensional representation of $GL_n$, and $d$ is a non-negative integer. On the one hand, following refs [Gi] and [BG] one can produce a basis in $V$ using irreducible components of Sringer fibers over a nilpotent matrix in ${\mathfrak {gl}}_d$, whose Jordan blocks correspond to the highest weight of $V$. On the other hand, one can produce a basis in $V$ by Mirkovi\'c-Vilonen cycles, a construction that works for an arbitrary reductive group $G$. In this note we prove that the resulting to bases coincide.