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