A conjecture on descents, inversions and the weak order
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915855322841088 |
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| author | Hohlweg, Christophe Pons, Viviane |
| author_facet | Hohlweg, Christophe Pons, Viviane |
| contents | In this article, we discuss the notion of partition of elements in an arbitrary Coxeter system $(W,S)$: a partition of an element $w$ is a subset $\mathcal P\subseteq W$ such that the left inversion set of $w$ is the disjoint union of the left inversion set of the elements in $\mathcal P$. Partitions of elements of $W$ arises in the study of the Belkale-Kumar product on the cohomology $H^*(X,\mathbb Z)$, where $X$ is the complete flag variety of any complex semi-simple algebraic group. Partitions of elements in the symmetric group $\mathcal S_n$ are also related to the {\em Babington-Smith model} in algebraic statistics or to the simplicial faces of the Littlewood-Richardson cone.
We state the conjecture that the number of right descents of $w$ is the sum of the number of right descents of the elements of $\mathcal P$ and prove that this conjecture holds in the cases of symmetric groups (type $A$) and hyperoctahedral groups (type $B$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09227 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A conjecture on descents, inversions and the weak order Hohlweg, Christophe Pons, Viviane Combinatorics Group Theory Primary 20F55, secondary 05A05, 17B22, 05E16 In this article, we discuss the notion of partition of elements in an arbitrary Coxeter system $(W,S)$: a partition of an element $w$ is a subset $\mathcal P\subseteq W$ such that the left inversion set of $w$ is the disjoint union of the left inversion set of the elements in $\mathcal P$. Partitions of elements of $W$ arises in the study of the Belkale-Kumar product on the cohomology $H^*(X,\mathbb Z)$, where $X$ is the complete flag variety of any complex semi-simple algebraic group. Partitions of elements in the symmetric group $\mathcal S_n$ are also related to the {\em Babington-Smith model} in algebraic statistics or to the simplicial faces of the Littlewood-Richardson cone. We state the conjecture that the number of right descents of $w$ is the sum of the number of right descents of the elements of $\mathcal P$ and prove that this conjecture holds in the cases of symmetric groups (type $A$) and hyperoctahedral groups (type $B$). |
| title | A conjecture on descents, inversions and the weak order |
| topic | Combinatorics Group Theory Primary 20F55, secondary 05A05, 17B22, 05E16 |
| url | https://arxiv.org/abs/2412.09227 |