Spin characters of the symmetric group which are proportional to linear characters in characteristic 2
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918159505686528 |
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| author | Fayers, Matthew McDowell, Eoghan |
| author_facet | Fayers, Matthew McDowell, Eoghan |
| contents | For a finite group, it is interesting to determine when two ordinary irreducible representations have the same $p$-modular reduction; that is, when two rows of the decomposition matrix in characteristic $p$ are equal, or equivalently when the corresponding $p$-modular Brauer characters are the same. We complete this task for the double covers of the symmetric group when $p=2$, by determining when the $2$-modular reduction of an irreducible spin representation coincides with a $2$-modular Specht module. In fact, we obtain a more general result: we determine when an irreducible spin representation has $2$-modular Brauer character proportional to that of a Specht module. In the course of the proof, we use induction and restriction functors to construct a function on generalised characters which has the effect of swapping runners in abacus displays for the labelling partitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_08243 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spin characters of the symmetric group which are proportional to linear characters in characteristic 2 Fayers, Matthew McDowell, Eoghan Representation Theory 20C30 (primary) 20C20, 05E10 (secondary) For a finite group, it is interesting to determine when two ordinary irreducible representations have the same $p$-modular reduction; that is, when two rows of the decomposition matrix in characteristic $p$ are equal, or equivalently when the corresponding $p$-modular Brauer characters are the same. We complete this task for the double covers of the symmetric group when $p=2$, by determining when the $2$-modular reduction of an irreducible spin representation coincides with a $2$-modular Specht module. In fact, we obtain a more general result: we determine when an irreducible spin representation has $2$-modular Brauer character proportional to that of a Specht module. In the course of the proof, we use induction and restriction functors to construct a function on generalised characters which has the effect of swapping runners in abacus displays for the labelling partitions. |
| title | Spin characters of the symmetric group which are proportional to linear characters in characteristic 2 |
| topic | Representation Theory 20C30 (primary) 20C20, 05E10 (secondary) |
| url | https://arxiv.org/abs/2403.08243 |