Geodesic
Canonical decompositions of affine permutations, affine codes, and split \(k\)-Schur functions
Tom Denton1
1York University, Toronto
The electronic journal of combinatorics, Tome 19 (2012) no. 4
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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.
DOI : 10.37236/2248
Classification : 05E05
Mots-clés : symmetric functions, affine permutations, maximal decomposition, affine permutations, product of cyclically decreasing elements, affine code, Littlewood-Richardson rule, Schur functions, Coxeter algebra

Tom Denton  1

1 York University, Toronto 
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     author = {Tom Denton},
     title = {Canonical decompositions of affine permutations, affine codes, and split {\(k\)-Schur} functions},
     journal = {The electronic journal of combinatorics},
     year = {2012},
     volume = {19},
     number = {4},
     doi = {10.37236/2248},
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     url = {http://geodesic.mathdoc.fr/articles/10.37236/2248/}
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Tom Denton. Canonical decompositions of affine permutations, affine codes, and split \(k\)-Schur functions. The electronic journal of combinatorics, Tome 19 (2012) no. 4. doi: 10.37236/2248

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