Oddification of the cohomology of type A Springer varieties

TL;DR

This paper introduces 'odd' analogs of the cohomology of type A Springer varieties by identifying a ring of odd symmetric functions with fixed points of skew polynomials under a Hecke algebra action at q=-1, leading to new graded modules and connections with Specht modules.

Contribution

It defines graded modules over the Hecke algebra at q=-1 as odd analogs of Springer variety cohomology, linking odd symmetric functions to Springer theory.

Findings

Identifies the ring of odd symmetric functions with fixed points of skew polynomials.

Constructs graded modules as odd analogs of Springer cohomology.

Connects top degree odd cohomology to Specht modules.

Abstract

We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q=-1. This allows us to define graded modules over the Hecke algebra at q=-1 that are `odd' analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient of the skew polynomial ring by the left ideal of nonconstant odd symmetric functions. The top degree component of the odd cohomology of Springer varieties is identified with the corresponding Specht module of the Hecke algebra at q=-1.