Reference for Clifford theory (of algebras) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T02:32:32Z http://mathoverflow.net/feeds/question/119640 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119640/reference-for-clifford-theory-of-algebras Reference for Clifford theory (of algebras) Chris Bowman 2013-01-23T10:19:31Z 2013-01-23T13:23:45Z <p>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.</p> <p>Theorem: Clifford theory</p> <p>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:</p> <p>(i) ${\rm res}^G_N(M)$ is a semisimple $kN$-module, and is isomorphic to a direct sum of conjugates of $L$</p> <p>(ii) the $kN$-homogenous components of ${\rm res}^G_N(M)$ are permuted transitively by $G$.</p> <p>(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 ${\rm res}^G_N(M)$.</p> <p>$\$</p> <p>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 $$A \rtimes G = \textbraceleft \sum_{x \in G} a_x x : a \in A \textbraceright.$$ </p> <p>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.</p> http://mathoverflow.net/questions/119640/reference-for-clifford-theory-of-algebras/119656#119656 Answer by Julian Kuelshammer for Reference for Clifford theory (of algebras) Julian Kuelshammer 2013-01-23T13:23:45Z 2013-01-23T13:23:45Z <p>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. </p> <p>If not, there is also a recent preprint (which I haven't read yet): [Liping Li: Representations of Modular skew group algebras, arXiv: <a href="http://arxiv.org/abs/1211.0333" rel="nofollow">1211.0333</a>] which may contain some results that are of interest to you. </p>