The ring of perfect $p$-permutation bimodules for blocks with cyclic defect groups
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| Format: | Preprint |
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2024
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| _version_ | 1866909281494761472 |
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| author | Boltje, Robert Monteiro, Nariel |
| author_facet | Boltje, Robert Monteiro, Nariel |
| contents | Let $B$ be a block algebra of a group algebra $FG$ of a finite group $G$ over a field $F$ of characteristic $p>0$. This paper studies ring theoretic properties of the representation ring $T^Δ(B,B)$ of perfect $p$-permutation $(B,B)$-bimodules and properties of the $k$-algebra $k\otimes_\mathbb{Z} T^Δ(B,B)$, for a field $k$. We show that if the Cartan matrix of $B$ has $1$ as an elementary divisor then $[B]$ is not primitive in $T^Δ(B,B)$. If $B$ has cyclic defect groups we determine a primitive decomposition of $[B]$ in $T^Δ(B,B)$. Moreover, if $k$ is a field of characteristic different from $p$ and $B$ has cyclic defect groups of order $p^n$ we describe $k\otimes_\mathbb{Z} T^Δ(B,B)$ explicitly as a direct product of a matrix algebra and $n$ group algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04134 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The ring of perfect $p$-permutation bimodules for blocks with cyclic defect groups Boltje, Robert Monteiro, Nariel Representation Theory 20C20, 19A22 Let $B$ be a block algebra of a group algebra $FG$ of a finite group $G$ over a field $F$ of characteristic $p>0$. This paper studies ring theoretic properties of the representation ring $T^Δ(B,B)$ of perfect $p$-permutation $(B,B)$-bimodules and properties of the $k$-algebra $k\otimes_\mathbb{Z} T^Δ(B,B)$, for a field $k$. We show that if the Cartan matrix of $B$ has $1$ as an elementary divisor then $[B]$ is not primitive in $T^Δ(B,B)$. If $B$ has cyclic defect groups we determine a primitive decomposition of $[B]$ in $T^Δ(B,B)$. Moreover, if $k$ is a field of characteristic different from $p$ and $B$ has cyclic defect groups of order $p^n$ we describe $k\otimes_\mathbb{Z} T^Δ(B,B)$ explicitly as a direct product of a matrix algebra and $n$ group algebras. |
| title | The ring of perfect $p$-permutation bimodules for blocks with cyclic defect groups |
| topic | Representation Theory 20C20, 19A22 |
| url | https://arxiv.org/abs/2408.04134 |