Canonical decompositions of affine permutations, affine codes, and split \(k\)-Schur functions. (English) Zbl 1267.05290

Denton, Tom

Electron. J. Comb. 19, No. 4, Research Paper P19, 41 p. (2012).

Summary: 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.

MSC:

05E05	Symmetric functions and generalizations
05E15	Combinatorial aspects of groups and algebras (MSC2010)

Keywords:

symmetric functions; affine permutations; maximal decomposition; affine permutations; product of cyclically decreasing elements; affine code; Littlewood-Richardson rule; Schur functions; Coxeter algebra

Software:

SageMath