A $t$-generalization for Schubert Representatives of the Affine Grassmannian

Abstract

We introduce two families of symmetric functions with an extra parameter $t$ that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when $t=1$. The families are defined by a statistic on combinatorial objects associated to the type-$A$ affine Weyl group and their transition matrix with Hall-Littlewood polynomials is $t$-positive. We conjecture that one family is the set of $k$-atoms.

Keywords

• $k$-schur functions
• pieri rule
• bruhat order
• hall-littlewood polynomials
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]