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On the essential algebra of the shifted Burnside biset functor

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Nadia Romero

Representation Theory · Submitted 1 Dec 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this case, $\widehat{kB_T}(G)$ is isomorphic to a semidirect product of $Out(G)$ and $kB^{Z(G)}(T)$, the monomial Burnside ring of $T$ with coefficients in $Z(G)$.

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arXiv ID2212.00511 (version 1)
Pith IDO777CJER, Pith's permanent record of this paper and its review
CitationNadia Romero. "On the essential algebra of the shifted Burnside biset functor." arXiv:2212.00511 (2022). https://pith.science/paper/O777CJER
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,224 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.33 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.09 · mentions code or data availability 0.01.

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arXiv listing abstract v1. Results and the main claim are underlined; .

We describe the essential algebra, $\widehat{kB_T}(G)$, of the Burnside biset functor shifted by a group $T$, at a group $G$, in two cases. First, when $G$ and $T$ are both finite abelian groups and $k$ is a field of characteristic $0$. In this case, $\widehat{kB_T}(G)$ is isomorphic to a quotient of the shifted star algebra, which is defined in terms of the subgroups of $G\times G\times T$. The second case is when $G$ and $T$ are any finite groups satisfying $(|G|, |T|)=1$ and $k$ is a commutative unitary ring. In this case, $\widehat{kB_T}(G)$ is isomorphic to a semidirect product of $Out(G)$ and $kB^{Z(G)}(T)$, the monomial Burnside ring of $T$ with coefficients in $Z(G)$. The aim of the article is to consider the natural set of generators of $\widehat{kB_T}(G)$ coming from the transitive elements in $kB_T(G\times G)$ and explore some cases in which it is possible to give a basis for $\widehat{kB_T}(G)$ in this set.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We describe the essential algebra, $\widehat{kB_T}(G)$, of the Burnside biset functor shifted by a group $T$, at a group $G$, in two cases. contribution 0.98 0.93 0.13 asserted 1.00
s1 First, when $G$ and $T$ are both finite abelian groups and $k$ is a field of characteristic $0$. contribution 0.54 0.26 0.03 unclear 0.56
s2 In this case, $\widehat{kB_T}(G)$ is isomorphic to a quotient of the shifted star algebra, which is defined in terms of the subgroups of $G\times G\times T$. result 1.00 0.85 0.18 asserted 1.00
s3 The second case is when $G$ and $T$ are any finite groups satisfying $(|G|, |T|)=1$ and $k$ is a commutative unitary ring. contribution 0.73 0.35 0.06 asserted 0.82
s4 In this case, $\widehat{kB_T}(G)$ is isomorphic to a semidirect product of $Out(G)$ and $kB^{Z(G)}(T)$, the monomial Burnside ring of $T$ with coefficients in $Z(G)$. result 1.00 0.90 0.27 asserted 1.00
s5 The aim of the article is to consider the natural set of generators of $\widehat{kB_T}(G)$ coming from the transitive elements in $kB_T(G\times G)$ and explore some cases in which it is possible to give a basis for $\widehat{kB_T}(G)$ in this set. contribution 0.93 0.65 0.12 hedged possibility 0.64

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

We describe the essential algebra, $\widehat{kB_T}(G)$, of the Burnside biset functor shifted by a group $T$, at a group $G$, in two cases. First, when $G$ and $T$ are both finite abelian groups and $k$ is a field of characteristic $0$. In this case, $\widehat{kB_T}(G)$ is isomorphic to a quotient of the shifted star algebra, which is defined in terms of the subgroups of $G\times G\times T$. The second case is when $G$ and $T$ are any finite groups satisfying $(|G|, |T|)=1$ and $k$ is a commutative unitary ring. In this case, $\widehat{kB_T}(G)$ is isomorphic to a semidirect product of $Out(G)$ and $kB^{Z(G)}(T)$, the monomial Burnside ring of $T$ with coefficients in $Z(G)$. The aim of the article is to consider the natural set of generators of $\widehat{kB_T}(G)$ coming from the transitive elements in $kB_T(G\times G)$ and explore some cases in which it is possible to give a basis for $\widehat{kB_T}(G)$ in this set.

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