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

TL;DR

This paper introduces a canonical decomposition of affine permutations into cyclically decreasing elements, linking it to affine codes and providing new insights into the combinatorics of $k$-Schur functions and their Littlewood-Richardson rule.

Contribution

It presents a novel maximal decomposition method for affine permutations and connects it to affine codes, enabling new combinatorial proofs related to $k$-Schur functions.

Findings

Established a unique maximal decomposition for affine permutations.

Connected affine codes to combinatorial properties of permutations.

Proved a new special case of the Littlewood-Richardson rule for $k$-Schur functions.

Abstract

We study the unique maximal decomposition of an arbitrary affine permutation into a product of cyclically decreasing elements, providing a new perspective on 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 show 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.