Oddification of the Cohomology of Type A Springer Varieties

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.

Odd Khovanov's arc algebra

“…In Section 4, we prove an odd version of Khovanv's results from [21]. Namely, we prove that the odd center of OH n C is isomorphic to the odd cohomology of the (n, n)-Springer variety as given by Lauda and Russell in [24]. In this paper they constructed an oddification of the cohomology of the Springer variety associated to any partition, by replacing polynomial rings and symmetric functions by their odd counterparts.…”

“…Write (n, n) for the partition λ = (n, n) of 2n. Lauda and Russell constructed in [24] an oddification of the cohomology of the Springer varieties, denoted OH(B λ , Z). Like the usual cohomology is obtained as a quotient of the polynomials by the partially symmetric functions, they constructed OH(B λ , Z) as a quotient of the ring OP ol m of odd polynomials…”

The odd Littlewood–Richardson rule

“…On OPol n , the kernel and image of ∂ i coincide; by contrast with the even case, however, these do not equal the space of invariants or anti-invariants of the action of s i . (See, however, [13].) The odd divided difference operators were introduced independently in [8], and they are also closely related to the spin Hecke algebras of [11,20].…”

Spin nilHecke algebras of classical type

“…Lauda and Russell [LR14] developed an intriguing odd Springer theory, building on the spin type A nilHecke algebra and Ellis-Khovanov's theory of odd symmetric polynomials. It will be interesting to see if there is a spin/odd Springer theory of type B and D.…”