Symmetric functions and Springer representations

TL;DR

This paper provides an algebraic proof that Springer representations of GL(n) are Ext-orthogonal, linking them to symmetric functions and categorification, thus deepening the understanding of their algebraic structure.

Contribution

It offers a purely algebraic proof of Ext-orthogonality of Springer representations and connects this property with the categorification of symmetric functions.

Findings

Springer representations are Ext-orthogonal to each other.

The proof aligns with the categorification of symmetric functions.

Links Springer representations to Green functions and Hall-Littlewood Q-functions.

Abstract

The characters of the (total) Springer representations are identified with the Green functions by Kazhdan [Israel J. Math. {\bf 28} (1977)], and the latter are identified with Hall-Littlewood's $Q$-functions by Green [Trans. Amer. Math. Soc. (1955)]. In this paper, we present a purely algebraic proof that the (total) Springer representations of $\mathop{GL} ( n )$ are $\mathrm{Ext}$-orthogonal to each other, and show that it is compatible with the natural categorification of the ring of symmetric functions.