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

If $\alpha \in S_n$ is a permutation of $\{1, 2, \ldots, n\}$, the inversion set of $\alpha$ is $\Phi(\alpha) = \{(i, j) \, | \, 1 \leq i < j \leq n, \alpha(i) > \alpha(j)\}$. We describe all $r$-tuples $\alpha_1, \alpha_2, \ldots, \alpha_r \in S_n$ such that $\Delta_n^+ = \{(i, j) \, | \, 1 \leq i < j \leq n\}$ is the disjoint union of $\Phi(\alpha_1), \Phi(\alpha_2), \ldots, \Phi(\alpha_r)$. 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.