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

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 [Lam06]. This decomposition is closely related to the affine code, which generalizes the kbounded 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. ∗Supported by York University and the Fields Institute. the electronic journal of combinatorics 19(4) (2012), #P19 1