The odd nilHecke algebra and its diagrammatics

TL;DR

This paper introduces an odd version of the nilHecke algebra, develops its diagrammatic calculus, and explores its Morita equivalences with rings of odd symmetric functions and cohomology rings of Grassmannians, extending categorification of quantum sl(2).

Contribution

It presents the first construction of an odd nilHecke algebra and its diagrammatics, establishing Morita equivalences with odd symmetric functions and Grassmannian cohomology rings.

Findings

Odd nilHecke algebra categorifies positive half of quantum sl(2).

Morita equivalence with rings of odd symmetric functions.

Cyclotomic quotients relate to odd Grassmannian cohomology rings.

Abstract

We introduce an odd version of the nilHecke algebra and develop an odd analogue of the thick diagrammatic calculus for nilHecke algebras. We graphically describe idempotents which give a Morita equivalence between odd nilHecke algebras and the rings of odd symmetric functions in finitely many variables. Cyclotomic quotients of odd nilHecke algebras are Morita equivalent to rings which are odd analogues of the cohomology rings of Grassmannians. Like their even counterparts, odd nilHecke algebras categorify the positive half of quantum sl(2).