Groups with pairings, Hall modules, and Hall-Littlewood polynomials
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914249615343616 |
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| author | Shen, Jiahe Van Peski, Roger |
| author_facet | Shen, Jiahe Van Peski, Roger |
| contents | We relate the combinatorics of Hall-Littlewood polynomials to that of abelian $p$-groups with alternating or Hermitian perfect pairings. Our main result is an analogue of the classical relationship between the Hall algebra of abelian $p$-groups (without pairings) and Hall-Littlewood polynomials. Specifically, we define a module over the classical Hall algebra with basis indexed by groups with pairings, and explicitly relate its structure constants to Hall-Littlewood polynomials at different values of the parameter $t$. We also show certain expectation formulas with respect to Cohen-Lenstra type measures on groups with pairings. In the alternating case this gives a new and simpler proof of previous results of Delaunay-Jouhet. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12405 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Groups with pairings, Hall modules, and Hall-Littlewood polynomials Shen, Jiahe Van Peski, Roger Combinatorics Number Theory Probability Representation Theory We relate the combinatorics of Hall-Littlewood polynomials to that of abelian $p$-groups with alternating or Hermitian perfect pairings. Our main result is an analogue of the classical relationship between the Hall algebra of abelian $p$-groups (without pairings) and Hall-Littlewood polynomials. Specifically, we define a module over the classical Hall algebra with basis indexed by groups with pairings, and explicitly relate its structure constants to Hall-Littlewood polynomials at different values of the parameter $t$. We also show certain expectation formulas with respect to Cohen-Lenstra type measures on groups with pairings. In the alternating case this gives a new and simpler proof of previous results of Delaunay-Jouhet. |
| title | Groups with pairings, Hall modules, and Hall-Littlewood polynomials |
| topic | Combinatorics Number Theory Probability Representation Theory |
| url | https://arxiv.org/abs/2504.12405 |