Wei et al. (2023)
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Mathematical research on skew Hecke algebras and group theory; while not directly about AI safety, abstract algebra and formal mathematical structures are foundational for cryptography and formal verification methods used in AI safety research.
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Abstract
Let $G$ be a finite group, $H \le G$ a subgroup, $R$ a commutative ring, $A$ an $R$-algebra, and $α$ an action of $G$ on $A$ by $R$-algebra automorphisms. We study the associated \emph{skew Hecke algebra} $\mathcal{H}_{R}(G,H,A,α)$, which is the convolution algebra of $H$-invariant functions from $G/H$ to $A$. We prove for skew Hecke algebras a number of common generalisations of results about skew group algebras and results about Hecke algebras of finite groups. We show that skew Hecke algebras admit a certain double coset decomposition. We construct an isomorphism from $\mathcal{H}_{R}(G,H,A,α)$ to the algebra of $G$-invariants in the tensor product $A \otimes \mathrm{End}_{R} ( \mathrm{Ind}_{H}^{G} R )$. We show that if $|H|$ is a unit in $A$, then $\mathcal{H}_{R}(G,H,A,α)$ is isomorphic to a corner ring inside the skew group algebra $A \rtimes G$. Alongside our main results, we show that the construction of skew Hecke algebras is compatible with certain group-theoretic operations, restriction and extension of scalars, certain cocycle perturbations of the action, gradings and filtrations, and the formation of opposite algebras. The main results are illustrated in the case where $G = S_3$, $H = S_2$, and $α$ is the natural permutation action of $S_3$ on the polynomial algebra $R[x_1,x_2,x_3]$.
Summary
This paper studies skew Hecke algebras, which generalize both skew group algebras and classical Hecke algebras of finite groups. The authors prove several fundamental structural results, including a double coset decomposition theorem and an isomorphism relating skew Hecke algebras to G-invariants in a tensor product of endomorphism rings. They also establish that under certain conditions, skew Hecke algebras embed as corner rings in skew group algebras. The construction is shown to be compatible with various algebraic operations including restriction/extension of scalars, gradings, and filtrations, with concrete illustrations using the symmetric group S₃.
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| Page | Type | Quality |
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| Sycophancy | Risk | 65.0 |
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# Skew Hecke algebras
James Waldron
School of Mathematics and Statistics, Herschel Building, Newcastle University, Newcastle-upon-Tyne, NE1 7RU
[james.waldron@ncl.ac.uk](mailto:james.waldron@ncl.ac.uk) and Leon Deryck Loveridge
Department of Science and Industry Systems, University of South-Eastern Norway, 3616 Kongsberg, Norway
Okinawa Institute of Science and Technology Graduate University,
1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan
[Leon.D.Loveridge@usn.no](mailto:Leon.D.Loveridge@usn.no)
###### Abstract.
Let G𝐺G be a finite group, H≤G𝐻𝐺H\\leq G a subgroup, R𝑅R a commutative ring, A𝐴A an R𝑅R-algebra, and α𝛼\\alpha an action of G𝐺G on A𝐴A by R𝑅R-algebra automorphisms.
Following Baker, we associate to this data the _skew Hecke algebra_ ℋR(G,H,A,α)subscriptℋ𝑅𝐺𝐻𝐴𝛼\\mathcal{H}\_{R}(G,H,A,\\alpha), which is the convolution algebra of H𝐻H-invariant functions from G/H𝐺𝐻G/H to A𝐴A.
In this paper we study the basic structure of these algebras, proving for skew Hecke algebras a number of common generalisations of results about skew group algebras and results about Hecke algebras of finite groups.
We show that skew Hecke algebras admit a certain double coset decomposition.
We construct an isomorphism from ℋR(G,H,A,α)subscriptℋ𝑅𝐺𝐻𝐴𝛼\\mathcal{H}\_{R}(G,H,A,\\alpha) to the algebra of G𝐺G-invariants in the tensor product A⊗EndR(IndHGR)tensor-product𝐴subscriptEnd𝑅superscriptsubscriptInd𝐻𝐺𝑅A\\otimes\\mathrm{End}\_{R}(\\mathrm{Ind}\_{H}^{G}R).
We show that if \|G\|𝐺\|G\| is a unit in A𝐴A, then ℋR(G,H,A,α)subscriptℋ𝑅𝐺𝐻𝐴𝛼\\mathcal{H}\_{R}(G,H,A,\\alpha) is isomorphic to a corner ring inside the skew group algebra A⋊Gright-normal-factor-semidirect-product𝐴𝐺A\\rtimes G.
Alongside our main results, we show that the construction of skew Hecke algebras is compatible with certain group-theoretic operations, restriction and extension of scalars, certain cocycle perturbations of the action, gradings and filtrations, and the formation of opposite algebras.
The main results are illustrated in the case where G=S3𝐺subscript𝑆3G=S\_{3}, H=S2𝐻subscript𝑆2H=S\_{2}, and α𝛼\\alpha is the natural permutation action of S3subscript𝑆3S\_{3} on the polynomial algebra R\[x1,x2,x3\]𝑅subscript𝑥1subscript𝑥2subscript𝑥3R\[x\_{1},x\_{2},x\_{3}\].
###### Key words and phrases:
Skew group rings, Hecke algebras, finite groups.
## 1\. Introduction
### 1.1. Skew Hecke algebras
Let R𝑅R be a commutative ring, A𝐴A an R𝑅R-algebra, G𝐺G a finite group, H≤G𝐻𝐺H\\leq G a subgroup, and α:G→AutR-alg(A):𝛼→𝐺subscriptAut𝑅-alg𝐴\\alpha:G\\to\\mathrm{Aut}\_{R\\text{-alg}}(A) a group homomorphism from G𝐺G to the group of R𝑅R-algebra automorphisms of A𝐴A.
Our main object of study in this paper is the associated _skew Hecke algebra_ ℋR(G,H,A,α)subscriptℋ𝑅𝐺𝐻𝐴𝛼\\mathcal{H}\_{R}(G,H,A,\\alpha).
These algebras were introduced by Baker in \[ [Bak98](https://ar5iv.labs.arxiv.org/html/2311.09038#bib.bibx2 "")\], where they are called _
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