Decomposing inversion sets of permutations and applications to faces of the Littlewood–Richardson cone

Abstract

If \(\alpha \in S_n\) is a permutation of \(\{1, 2,\ldots ,n\}\), the inversion set of \(\alpha \) is \(\Phi (\alpha ) = \{ (i, j) \,| \, 1 \leqslant i < j \leqslant n, \alpha (i) > \alpha (j)\}\). We describe all r-tuples \(\alpha _1, \alpha _2, \ldots , \alpha _r \in S_n\) such that \(\Delta _n^+ = \{ (i, j) \, | \, 1 \leqslant i < j \leqslant n\}\) 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.