Bubble sort and Howe duality for staircase matrices
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913799133462528 |
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| author | Khoroshkin, Anton Makedonskyi, Ievgen |
| author_facet | Khoroshkin, Anton Makedonskyi, Ievgen |
| contents | In this paper, we present an independent proof of the Cauchy identities for staircase matrices, originally discovered in arXiv:2411.03117, using the combinatorics of the Bruhat poset and the bubble-sort procedure. Additionally, we derive new insights into certain coefficients appearing in one of these identities. The first part of the paper focuses on combinatorial aspects. It is self-contained, of independent interest, and introduces a generalization of parabolic Bruhat graphs for monotone functions on an arborescent poset. The second part examines the intersections of Demazure modules within a given integrable representation. Finally, we propose a generalization of the classical Howe duality for staircase matrices in terms of the corresponding distributive lattice of Demazure submodules. Computing the associated character yields the desired Cauchy identities for staircase matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_21184 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bubble sort and Howe duality for staircase matrices Khoroshkin, Anton Makedonskyi, Ievgen Representation Theory Combinatorics 17B10, 33C52, 06A11 In this paper, we present an independent proof of the Cauchy identities for staircase matrices, originally discovered in arXiv:2411.03117, using the combinatorics of the Bruhat poset and the bubble-sort procedure. Additionally, we derive new insights into certain coefficients appearing in one of these identities. The first part of the paper focuses on combinatorial aspects. It is self-contained, of independent interest, and introduces a generalization of parabolic Bruhat graphs for monotone functions on an arborescent poset. The second part examines the intersections of Demazure modules within a given integrable representation. Finally, we propose a generalization of the classical Howe duality for staircase matrices in terms of the corresponding distributive lattice of Demazure submodules. Computing the associated character yields the desired Cauchy identities for staircase matrices. |
| title | Bubble sort and Howe duality for staircase matrices |
| topic | Representation Theory Combinatorics 17B10, 33C52, 06A11 |
| url | https://arxiv.org/abs/2502.21184 |