Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911780377198592 |
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| author | Fayers, Matthew Kleshchev, Alexander Morotti, Lucia |
| author_facet | Fayers, Matthew Kleshchev, Alexander Morotti, Lucia |
| contents | We calculate the (super)decomposition matrix for a RoCK block of a double cover of the symmetric group with abelian defect, verifying a conjecture of the first author. To do this, we exploit a theorem of the second author and Livesey that a RoCK block $\mathcal B^{ρ,d}$ is Morita superequivalent to a wreath superproduct of a certain quiver (super)algebra with the symmetric group $\mathfrak S_d$. We develop the representation theory of this wreath superproduct to compute its Cartan invariants. We then directly construct projective characters for $\mathcal B^{ρ,d}$ to calculate its decomposition matrix up to a triangular adjustment, and show that this adjustment is trivial by comparing Cartan invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_03304 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups Fayers, Matthew Kleshchev, Alexander Morotti, Lucia Representation Theory 20C30, 20C20, 05E10 We calculate the (super)decomposition matrix for a RoCK block of a double cover of the symmetric group with abelian defect, verifying a conjecture of the first author. To do this, we exploit a theorem of the second author and Livesey that a RoCK block $\mathcal B^{ρ,d}$ is Morita superequivalent to a wreath superproduct of a certain quiver (super)algebra with the symmetric group $\mathfrak S_d$. We develop the representation theory of this wreath superproduct to compute its Cartan invariants. We then directly construct projective characters for $\mathcal B^{ρ,d}$ to calculate its decomposition matrix up to a triangular adjustment, and show that this adjustment is trivial by comparing Cartan invariants. |
| title | Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups |
| topic | Representation Theory 20C30, 20C20, 05E10 |
| url | https://arxiv.org/abs/2303.03304 |