Oddification of the cohomology of type \(A\) Springer varieties (Q2925297)

The authors consider the space NEWLINE\[NEWLINE\text{OPol}_n=\mathbb Z\langle x_1,\ldots, x_n\rangle/\langle x_ix_j+x_jx_i \quad\text{for}\quad i\neq j\rangleNEWLINE\]NEWLINE of skew polynomials in variables \(x_1,\ldots, x_n\) and identify the ring of odd symmetric functions introduced in [\textit{A. P. Ellis} and \textit{M. Khovanov}, Adv. Math. 231, No. 2, 965--999 (2012; Zbl 1248.05210)] as the space of elements of \(\text{OPol}_n\) fixed by a natural action of \({\mathcal H}_{-1}(n)\) (the Hecke algebra at \(q=-1\)) on \(\text{OPol}_n\). This allows them to define the ``odd'' analogs of the cohomology of type \({\mathsf A}\) Springer fibers as some graded \({\mathcal H}_{-1}(n)\)-modules. The top degree component of this odd cohomology is identified with the corresponding Specht \({\mathcal H}_{-1}(n)\)-module.