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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:
 500000 Symmetric functions and generalizations 5e+15 Combinatorial aspects of groups and algebras (MSC2010)
SageMath
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