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

Abstract

If $\hat{I}\pm \in S_n$ is a permutation of $\{1,2,\dots ,n\}$, the inversion set of $\hat{I}\pm$ is $\hat{I}¦(\hat{I}\pm )=\{(i,j)|1\le i<j\le n,\hat{I}\pm (i)>\hat{I}\pm (j)\}$. We describe all $r$-tuples $\hat{I}{\pm }_1,\hat{I}{\pm }_2,\dots ,\hat{I}{\pm }_r\in S_n$ such that $\hat{I}{”}_n^{+}=\{(i,j)|1\le i<j\le n\}$ is the disjoint union of $\hat{I}¦(\hat{I}{\pm }_1),\hat{I}¦(\hat{I}{\pm }_2),\dots ,\hat{I}¦(\hat{I}{\pm }_r)$. 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 discuss analogous problems for the Weyl groups of root systems of types $B$, $C$ and $D$ providing solutions for types $B$ and $C$. Finally we provide some enumerative results and introduce a useful tool for visualizing inversion sets.

43 pages