The lattice of submonoids of the uniform block permutations containing the symmetric group
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908273814274048 |
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| author | Orellana, Rosa Saliola, Franco Schilling, Anne Zabrocki, Mike |
| author_facet | Orellana, Rosa Saliola, Franco Schilling, Anne Zabrocki, Mike |
| contents | We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the $\mathscr{J}$-classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09710 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The lattice of submonoids of the uniform block permutations containing the symmetric group Orellana, Rosa Saliola, Franco Schilling, Anne Zabrocki, Mike Combinatorics Group Theory 06D10, 05E16, 20M18 We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the $\mathscr{J}$-classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra. |
| title | The lattice of submonoids of the uniform block permutations containing the symmetric group |
| topic | Combinatorics Group Theory 06D10, 05E16, 20M18 |
| url | https://arxiv.org/abs/2405.09710 |