Reference for Clifford theory (of algebras)

2013-01-23T10:19:31Z

Clifford theory relates the representation theory of a group to that of a normal subgroup. A good reference for this is Curtis and Reiner's "Methods in Representation theory II" Theorem 11.1.

Theorem: Clifford theory

Le $N $ be a normal subgroup of a finite group $G$. Let $M$ be a simple $kG$-module and $L$ a simple $kN$-submodule of ${\rm res}^G_N(M)$. Then the following statements hold:

(i) ${\rm res}^G_N(M)$ is a semisimple $kN$-module, and is isomorphic to a direct sum of conjugates of $L$

(ii) the $kN$-homogenous components of ${\rm res}^G_N(M)$ are permuted transitively by $G$.

(iii) Let $\hat{L}$ be a $kN$-homogenous component of ${\rm res}^G_N(M)$ containing $L$, and let $\hat{N}={ x\in G: x \hat{L}=\hat{L} }$. Write $G$ as a disjoint union $G=\cup_{i=1}^n g_i \hat{N}$. Then ${g_iL: 1 \leq i \leq n}$ is a complete set of non-isomorphic conjugates of $L$, and each appears with equal multiplicity in ${\rm res}^G_N(M)$.

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I am looking for a reference which generalises this theorem to other algebras. In particular the "skew-group ring" situation where a finite group, $G$, acts by automorphisms on an algebra $A$. We then get that Clifford theory relates the representation theories of $A$ and $$A \rtimes G = \textbraceleft \sum_{x \in G} a_x x : a \in A \textbraceright.$$

One possible reference for this is Ram and Rammage "Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory", however they focus on induction and restriction between $A \rtimes G $ and $A \rtimes H$ where $H$ is the inertia group of a given simple module. I would prefer a reference to a theorem of the above form, directly relating $A \rtimes G$ and $A$, so that I can just write "please see...." without going into any more detail. This is lazy of me, I know, but I think that such a reference should exist.

Answer by Julian Kuelshammer for Reference for Clifford theory (of algebras)

2013-01-23T13:23:45Z

I would suggest [Reiten, Riedtmann: Skew group algebras in the representation theory of Artin algebras, Journal of Algebra 92, 1985] if you have that $|G|$ is invertible in $A$ and $A$ is an Artin algebra.

If not, there is also a recent preprint (which I haven't read yet): [Liping Li: Representations of Modular skew group algebras, arXiv: 1211.0333] which may contain some results that are of interest to you.

Reference for Clifford theory (of algebras) - MathOverflow

Reference for Clifford theory (of algebras)

Answer by Julian Kuelshammer for Reference for Clifford theory (of algebras)