Decomposing Inversion Sets of Permutations and Applications to Faces of the Littlewood-Richardson Cone

Details Decomposing Inversion Sets of Permutations and Applications to Faces of the Littlewood-Richardson Cone Decomposing Inversion Sets of Permutations and Applications to Faces of the Littlewood-Richardson Cone

2011-10-26

arxiv.org/abs/1110.5880v1

Mathematics

Combinatorics

25 pages

Scientific paper

If $\sigma \in S_n$ is a permutation of $\{1, 2, \ldots, n\}$, the inversion set of $\sigma$ is $\Phi(\sigma) = \{ (i, j) \, | \, 1 \leq i < j \leq n, \sigma(i) > \sigma(j)\}$. We describe all $r$-tuples $\sigma_1, \sigma_2, \ldots, \sigma_r \in S_n$ such that $\Delta_n^+ = \{ (i, j) \, | \, 1 \leq i < j \leq n\}$ is the disjoint union of $\Phi(\sigma_1), \Phi(\sigma_2), \ldots, \Phi(\sigma_r)$. Using this description we prove that certain faces of the Littlewood-Richardson cone are simplicial and provide an algorithm for writing down their sets of generating rays. We also consider and solve the analogous problem for the Weyl groups of root systems of type $B$ and $C$ and provide some enumerative results.

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