Canonical Decompositions of Affine Permutations, Affine Codes, and Split $k$-Schur Functions

Abstract

We develop a new perspective on the unique maximal decomposition of an arbitrary affine permutation into a product of cyclically decreasing elements, implicit in work of Thomas Lam.  This decomposition is closely related to the affine code, which generalizes the $k$-bounded partition associated to Grassmannian elements.  We also prove that the affine code readily encodes a number of basic combinatorial properties of an affine permutation.  As an application, we prove a new special case of the Littlewood-Richardson Rule for $k$-Schur functions, using the canonical decomposition to control for which permutations appear in the expansion of the $k$-Schur function in noncommuting variables over the affine nil-Coxeter algebra.