The down operator and expansions of near rectangular k-Schur functions

Abstract

We prove that the Lam-Shimozono ``down operator'' on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of ``near rectangles'' in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood–Richardson coefficients.

Keywords

• affine schubert calculus
• dual graded graphs
• symmetric functions
• k-schur functions
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]