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

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## 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.

## Keywords

Inversion set Simple permutation Littlewood–Richardson cone Catalan numbers## Mathematics Subject Classification

05E15 05A05 05E10 52B20## Notes

### Acknowledgements

The authors thank the referee who suggested a number of improvements in the exposition. This work was partially supported by NSERC. In particular, most of it was done with the support of NSERC’s Undergraduate Summer Research Awards program.

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