Join operation for the Bruhat order and Verma modules
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866915008880836608 |
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| author | Ko, Hankyung Mazorchuk, Volodymyr Mrđen, Rafael |
| author_facet | Ko, Hankyung Mazorchuk, Volodymyr Mrđen, Rafael |
| contents | We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type $A$. The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext spaces between a simple module and a Verma module. We give a conjectural complete description of such socles, which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan-Lusztig polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_01067 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Join operation for the Bruhat order and Verma modules Ko, Hankyung Mazorchuk, Volodymyr Mrđen, Rafael Representation Theory Combinatorics We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type $A$. The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext spaces between a simple module and a Verma module. We give a conjectural complete description of such socles, which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan-Lusztig polynomials. |
| title | Join operation for the Bruhat order and Verma modules |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2109.01067 |