Representation stability of the cohomology of Springer varieties and some combinatorial consequences

TL;DR

This paper proves that the cohomology representations of Springer varieties exhibit uniform stability as the size increases, extending known results from flag varieties and deriving combinatorial implications.

Contribution

It generalizes the uniform representation stability from flag varieties to all Springer fibers, introducing a new graded co-FI-module framework.

Findings

Cohomology of Springer varieties is uniformly representation stable.

The stability applies to all increasing subsequences of Young diagrams.

Derived combinatorial consequences from the stability results.

Abstract

A sequence of $S_n$-representations $\{V_n\}$ is said to be uniformly representation stable if the decomposition of $V_n = \bigoplus_{\mu} c_{\mu,n} V(\mu)_n$ into irreducible representations is independent of $n$ for each $\mu$---that is, the multiplicities $c_{\mu,n}$ are eventually independent of $n$ for each $\mu$. Church-Ellenberg-Farb proved that the cohomology of flag varieties (the so-called diagonal coinvariant algebra) is uniformly representation stable. We generalize their result from flag varieties to all Springer fibers. More precisely, we show that for any increasing subsequence of Young diagrams, the corresponding sequence of Springer representations form a graded co-FI-module of finite type (in the sense of Church-Ellenberg-Farb). We also explore some combinatorial consequences of this stability.