Generating functions for fixed points of the Mullineux map
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929724387753984 |
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| author | Hemmer, David J. |
| author_facet | Hemmer, David J. |
| contents | Mullineux defined an involution on the set of $e$-regular partitions of $n$. When $e=p$ is prime, these partitions label irreducible symmetric group modules in characteristic $p$. Mullineux's conjecture, since proven, was that this ``Mullineux map" described the effect on the labels of taking the tensor product with the one-dimensional signature representation. Counting irreducible modules fixed by this tensor product is related to counting irreducible modules for the alternating group $A_n$ in prime characteristic. In 1991, Andrews and Olsson worked out the generating function counting fixed points of Mullineux's map when $e=p$ is an odd prime (providing evidence in support of Mullineux's conjecture). In 1998, Bessenrodt and Olsson counted the fixed points in a $p$-block of weight $w$. We extend both results to arbitrary $e$, and determine the corresponding generating functions. When $e$ is odd but not prime the extension is immediate, while $e$ even requires additional work and the results, which are different, have not appeared in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03643 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generating functions for fixed points of the Mullineux map Hemmer, David J. Combinatorics Representation Theory 05E10 Mullineux defined an involution on the set of $e$-regular partitions of $n$. When $e=p$ is prime, these partitions label irreducible symmetric group modules in characteristic $p$. Mullineux's conjecture, since proven, was that this ``Mullineux map" described the effect on the labels of taking the tensor product with the one-dimensional signature representation. Counting irreducible modules fixed by this tensor product is related to counting irreducible modules for the alternating group $A_n$ in prime characteristic. In 1991, Andrews and Olsson worked out the generating function counting fixed points of Mullineux's map when $e=p$ is an odd prime (providing evidence in support of Mullineux's conjecture). In 1998, Bessenrodt and Olsson counted the fixed points in a $p$-block of weight $w$. We extend both results to arbitrary $e$, and determine the corresponding generating functions. When $e$ is odd but not prime the extension is immediate, while $e$ even requires additional work and the results, which are different, have not appeared in the literature. |
| title | Generating functions for fixed points of the Mullineux map |
| topic | Combinatorics Representation Theory 05E10 |
| url | https://arxiv.org/abs/2402.03643 |