Odd symmetric functions and categorification.

摘要

We introduce q- and signed analogues of several constructions in and around the theory of symmetric functions. The most basic of these is the Hopf superalgebra of odd symmetric functions. This algebra is neither (super-)commutative nor (super-)cocommutative, yet its combinatorics still exhibit many of the striking integrality and positivity properties of the usual symmetric functions. In particular, we give odd analogues of Schur functions, Kostka numbers, and Littlewood-Richardson coefficients.;Using an odd analogue of the nilHecke algebra, we give a categorification of the integral divided powers form of $U+qsl2$ inequivalent to the one due to Khovanov-Lauda. Along the way, we develop a graphical calculus for indecomposable modules for the odd nilHecke algebra.