Denton, Tom Canonical decompositions of affine permutations, affine codes, and split \(k\)-Schur functions. (English) Zbl 1267.05290 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. Cited in 5 Documents 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 PDF BibTeX XML Cite \textit{T. Denton}, Electron. J. Comb. 19, No. 4, Research Paper P19, 41 p. (2012; Zbl 1267.05290) Full Text: Link arXiv