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

Details Canonical Decompositions of Affine Permutations, $k$-Castles, and Split $k$-Schur Functions Canonical Decompositions of Affine Permutations, $k$-Castles, and Split $k$-Schur Functions

2012-04-11

arxiv.org/abs/1204.2591v2

Mathematics

Combinatorics

46 pages, 14 figures

Scientific paper

We develop a unique decomposition of an arbitrary affine permutation into a product of cyclically decreasing elements. To this decomposition, we associate a combinatorial object called a $k$-castle, which generalizes the $k$-bounded partition associated to Grassmannian elements. The $k$-castle readily encodes a number of basic combinatorial properties of a given affine permutation. As an application, we prove a special case of the Littlewood-Richardson Rule for $k$-Schur functions, using our decomposition to control for which permutations appear in the expansion of the $k$-Schur function in noncommuting variables over the affine nil-Coxeter algebra.

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