Clifford theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 28 February 2013

Clifford theory

Let $R$ be an algebra over $ℂ$ and let $G$ be a finite group acting by automorphisms on $R\text{.}$ The skew group ring is

$$R⋊G=\{\sum _{g\in G}{r}_{g}g\hspace{0.17em}\mid \hspace{0.17em}{r}_g\in R\}$$

with multiplication given by the distributive law and the relation

$$gr=g\left(r\right)g,\text{for}\hspace{0.17em}g\in G\hspace{0.17em}\text{and}\hspace{0.17em}r\in R\text{.}$$

Let $N$ be a (finite dimensional) left $R\text{-module.}$ For each $g\in G$ define an $R\text{-module}$ ${}^{g}N,$ which has the same underlying vector space $N$ but such that

$$\begin{array}{cc}{}^{g}N\hspace{0.17em}\text{has}\hspace{0.17em}R\text{-action given by}r\circ n=g^{-1}\left(r\right)n,& \text{(A.1)}\end{array}$$

for $r\in R,n\in N\text{.}$ If $W$ is an $R\text{-submodule}$ of $N$ then ${}^{g}W$ is an $R\text{-submodule}$ of ${}^{g}N$ and so ${}^{g}N$ is simple if and only if $N$ is simple. Thus there is an action of $G$ on the set of simple $R\text{-modules.}$

Let $R^{\lambda }$ be a simple $R\text{-module.}$ The inertia group of $R^{\lambda }$ is

$$\begin{array}{cc}H=\{h\in G\hspace{0.17em}\mid \hspace{0.17em}{R}^{\lambda }\cong {}^h{R}^{\lambda }\}\text{.}& \text{(A.2)}\end{array}$$

If $h\in H$ then Schur’s lemma implies that the isomorphism $R^{\lambda }\cong {}^h{R}^{\lambda }$ is unique up to constant multiples (since both $R^{\lambda }$ and ${}^h{R}^{\lambda }$ are simple). For each $h\in H$ fix an isomorphism ${\varphi }_h:R^{\lambda }\to {}^{h^{-1}}{R}^{\lambda }\text{.}$ Then, as operators on $R^{\lambda },$

$$\begin{array}{cc}{\varphi }_{h}r=h\left(r\right){\varphi }_h,\text{and}{\varphi }_g{\varphi }_h=\alpha (g,h){\varphi }_{gh},& \text{(A.3)}\end{array}$$

where $\alpha (g,h)\in {ℂ}^{*}$ are determined by the choice of the isomorphisms ${\varphi }_h\text{.}$ The resulting function $\alpha :H\times H\to {ℂ}^{*}$ is called a factor set [CRe1981, 8.32].

Let ${\left(ℂH\right)}_{{\alpha }^{-1}}$ be the algebra with basis $\{c_h\hspace{0.17em}\mid \hspace{0.17em}h\in H\}$ and multiplication given by

$$\begin{array}{cc}{c}_g{c}_h=\alpha {(g,h)}^{-1}{c}_{gh},\text{for}\hspace{0.17em}g,h\in H\text{.}& \text{(A.4)}\end{array}$$

Let $H^{\mu }$ be a simple ${\left(ℂH\right)}_{{\alpha }^{-1}}\text{-module.}$ The putting

$$\begin{array}{cc}rh(m\otimes n)=r{\varphi }_{h}m\otimes c_{h}n,\text{for}\hspace{0.17em}r\in R,h\in H,m\in R^{\lambda },n\in H^{\mu },& \text{(A.5)}\end{array}$$

defines an action of $R⋊H$ on $R^{\lambda }\otimes H^{\mu }\text{.}$

Theorem A.6 (Clifford theory) Let $R^{\lambda }$ be a simple $R\text{-module}$ and let $H$ be the inertia group of $R^{\lambda }\text{.}$ Let $H^{\mu }$ be a simple ${\left(ℂH\right)}_{{\alpha }^{-1}}\text{-module}$ where $\alpha :H\times H\to {ℂ}^{*}$ is the factor set determined by a choice of isomorphisms ${\varphi }_h:R^{\lambda }\to {}^h{R}^{\lambda }\text{.}$ Define an action of $R⋊H$ on $R^{\lambda }\otimes H^{\mu }$ as in (A.5) and define

$$R{G}^{\lambda ,\mu }={\text{Ind}}_{R⋊H}^{R⋊G}(R^{\lambda }\otimes H^{\mu })=(R⋊G){\otimes }_{R⋊H}(R^{\lambda }\otimes H^{\mu })\text{.}$$

Then

1. $R{G}^{\lambda ,\mu }$ is a simple $R⋊G\text{-module.}$
2. Every simple $R⋊G\text{-module}$ is obtained by this construction.
3. If $R{G}^{\lambda ,\mu }\cong R{G}^{\nu ,\gamma }$ then $R^{\lambda }$ and $R^{\nu }$ are in the same $G\text{-orbit}$ of simple $R\text{-modules}$ and $H^{\mu }\cong H^{\gamma }$ as ${\left(ℂH\right)}_{{\alpha }^{-1}}\text{-modules.}$

		Proof.
	The proof of this theorem is as in [Mac1980] except that the consideration of the factor set $\alpha :H\times H\to C^{*}$ is necessary to correct an error there. We thank P. Deligne for pointing this out to us. A sketch of the proof is as follows. Let $M$ be a simple $R⋊G\text{-module}$ and let $R^{\lambda }$ be a simple $R\text{-submodule}$ of $M\text{.}$ Then $g{R}^{\lambda }\cong {}^g{R}^{\lambda }$ as $R\text{-modules}$ and $M={\sum }_{g\in G}g{R}^{\lambda }$ since the right hand side is an $R⋊G\text{-submodule}$ of $M\text{.}$ Then $$M=\sum _{g_i\in G/H}{g}_{i}N={\text{Ind}}_{R⋊H}^{R⋊G}\left(N\right),\text{where}N=\sum _{h\in H}h{R}^{\lambda },$$ and the first sum is over a set $\left\{g_i\right\}$ of coset representatives of the cosets $G/H\text{.}$ The $R\text{-module}$ $N$ is semisimple and by [Bou1958] $$\begin{array}{cc}N\cong R^{\lambda }\otimes H^{\mu },& \text{(A.7)}\end{array}$$ where $H^{\mu }={\text{Hom}}_R(R^{\lambda },N)\text{.}$ It can be checked that the vector space $H^{\mu }$ has a ${\left(ℂH\right)}_{{\alpha }^{-1}}\text{-action}$ given by $$\left(c_h\psi \right)\left(m\right)=\alpha (h,h^{-1})h\psi \left({\varphi }_{h^{-1}}\left(m\right)\right),\text{for}\hspace{0.17em}h\in H,\psi \in {\text{Hom}}_R(R^{\lambda },N),$$ where $c_h$ is as in (A.4). Then, with $R⋊H\text{-action}$ on $R^{\lambda }\otimes H^{\mu }$ given by (A.5), the isomorphism in (A.7) is an isomorphism of $R⋊H\text{-modules}$ (see [CRe1981, Thm. (11.17) (ii)]). If $P$ is an ${\left(ℂH\right)}_{{\alpha }^{-1}}\text{-submodule}$ of $H^{\mu }$ then $R^{\lambda }\otimes P$ is an $R⋊H\text{-submodule}$ of $R^{\lambda }\otimes H^{\mu }$ and ${\text{Ind}}_{R⋊H}^{R⋊G}(R^{\lambda }\otimes P)$ is an $R⋊G\text{-submodule}$ of $M\text{.}$ Thus $H^{\mu }$ must be a simple ${\left(ℂH\right)}_{{\alpha }^{-1}}\text{-module.}$ This argument shows that every simple $R⋊G\text{-module}$ is of the form $R{G}^{\lambda ,\mu }\text{.}$ The uniqueness follows as in [Mac1980, App.]. $\square$

Remark A.8. A different choice ${\psi }_h:R^{\lambda }\to {}^h{R}^{\lambda }$ of the isomorphisms in (A.3) may yield a factor set $\beta :H\times H\to {ℂ}^{*}$ which is different from the factor set $\alpha \text{.}$ However, the algebras ${\left(ℂH\right)}_{{\beta }^{-1}}$ and ${\left(ℂH\right)}_{{\alpha }^{-1}}$ are always isomorphic (a diagonal change of basis suffices).

Lemma A.9. Define $R^G=\{r\in R\hspace{0.17em}\mid \hspace{0.17em}g\left(r\right)=r\hspace{0.17em}\text{for all}\hspace{0.17em}g\in G\}$ and let

$$e=(1/\mid G\mid )\sum _{g\in G}g\in R⋊G\text{.}$$

1. The map $$\begin{array}{cccc}\theta :& R^G& ⟶& e(R⋊G)e\\ & s& ⟼& se\end{array}$$ is a ring isomorphism.
2. Left multiplication by elements of $R$ and the action of $G$ by automorphisms make $R$ into a left $R⋊G\text{-module.}$ Right multiplication makes $R$ a right $R^G\text{-module.}$ The rings $R⋊G$ and $e(R⋊G)e$ act on $(R\times G)e$ by left and right multiplication, respectively. The map $$\begin{array}{cccc}\psi :& R& \cong & (R⋊G)e\\ & r& ⟼& re\end{array}$$ is an isomorphism of $(R⋊G,R^G)\text{-bimodules.}$

		Proof.
	(a) If $r\in R^G$ then $$ere=\frac{1}{\mid G\mid }\sum _{g\in G}g\left(r\right)ge=\frac{1}{\mid G\mid }\sum _{g\in G}re=re\text{.}$$ Thus the map $\theta$ is well defined and if $r,s\in R^G$ then $rese=rse,$ so $\theta$ is a homomorphism. If $re=se$ then $r=s$ since $R⋊G$ is a free $R\text{-module}$ with basis $G\text{.}$ Thus $\theta$ is injective. If ${\sum }_{g\in G}{r}_{g}g$ is a general element of $R⋊G$ then $$e\left(\sum _{g\in G}{r}_{g}g\right)e=\sum _{g,h\in G}h\left(r_g\right)hge=\left(\sum _{g,h\in G}\right)h\left(r_g\right)e,$$ and, for each $g\in G,$ ${\sum }_{h\in G}h\left(r_g\right)\in R^G\text{.}$ So $\theta$ is surjective. The proof of (b) is straightforward. $\square$

Let ${\left(ℂH\right)}_{\alpha }$ be the algebra with basis $\{b_h\hspace{0.17em}\mid \hspace{0.17em}h\in H\}$ and multiplication given by

$$b_g{b}_h=\alpha (g,h)b_{gh},\text{for}\hspace{0.17em}g,h\in H,$$

and let ${\left(ℂH\right)}_{{\alpha }^{-1}}$ be as in (A.4). Let $M$ be a ${\left(ℂH\right)}_{\alpha }\text{-module.}$ The dual of $M$ is the ${\left(ℂH\right)}_{{\alpha }^{-1}}\text{-module}$ given by the vector space $M^{*}=\text{Hom}(M,ℂ)$ with action

$$\left(c_h\psi \right)\left(m\right)=\alpha {(h,h^{-1})}^{-1}\psi \left(b_{h^{-1}}m\right),\text{for}\hspace{0.17em}h\in H,\psi \in M^{*}\text{.}$$

This is a ${\left(ℂH\right)}_{{\alpha }^{-1}}$ action since, for all $g,h\in H,$ $\psi \in M^{*},$

$$\begin{array}{ccc}\left(c_g{c}_h\psi \right)\left(m\right)& =& \alpha {(h,h^{-1})}^{-1}\alpha {(g,g^{-1})}^{-1}\psi \left(b_{h^{-1}}{b}_{g^{-1}}m\right)\\ & =& \alpha {(h,h^{-1})}^{-1}\alpha {(g,g^{-1})}^{-1}\alpha (h^{-1},g^{-1})\psi \left(b_{{\left(gh\right)}^{-1}}m\right)\\ & =& \alpha {(h,h^{-1})}^{-1}\alpha {(g,g^{-1})}^{-1}\alpha (h^{-1},g^{-1})\alpha (gh,h^{-1}{g}^{-1})\left(c_{gh}\psi \right)\left(m\right)\\ & =& \alpha {(g,h)}^{-1}\left(c_{gh}\psi \right)\left(m\right),\end{array}$$

where the last equality follows from the associativity of the product $b_g{b}_h{b}_{h^{-1}}{b}_{g^{-1}}$ in ${\left(ℂH\right)}_{\alpha }\text{.}$ If $\rho :{\left(ℂH\right)}_{\alpha }\to \text{End}\left(M\right)$ is the representation corresponding to $M$ then the representation ${\rho }^{*}:{\left(ℂH\right)}_{{\alpha }^{-1}}\to \text{End}\left(M^{*}\right)$ corresponding to $M^{*}$ is

$$\begin{array}{cc}{\rho }^{*}\left(c_h\right)=\alpha {(h,h^{-1})}^{-1}\rho {\left(b_{h^{-1}}\right)}^t={\left(\rho {\left(b_h\right)}^{-1}\right)}^t\text{.}& \text{(A.11)}\end{array}$$

If $M$ is a ${\left(ℂH\right)}_{\alpha }\text{-module}$ and $N$ is a ${\left(ℂH\right)}_{{\alpha }^{-1}}\text{-module}$ then $M\otimes N$ is an $ℂH\text{-module}$ with action defined by

$$\begin{array}{cc}h(m\otimes n)=b_{h}m\otimes c_{h}n,\text{for}\hspace{0.17em}h\in H,m\in M\hspace{0.17em}\text{and}\hspace{0.17em}n\in N\text{.}& \text{(A.12)}\end{array}$$

The following lemma is a version of Schur’s lemma which will be used in the proof of Theorem A.13.

Lemma A.12. Suppose that $M$ and $N$ are simple ${\left(ℂH\right)}_{\alpha }\text{-modules}$ and let $N^{*}$ be the ${\left(ℂH\right)}_{{\alpha }^{-1}}\text{-module}$ which is the dual of $N\text{.}$ Let $e_H=(1/\mid H\mid ){\sum }_{h\in H}h\text{.}$ Then

$$\text{dim}\left(e_H(M\otimes N^{*})\right)=\{\begin{array}{cc}1,& \text{if}\hspace{0.17em}M\cong N,\\ 0,& \text{otherwise.}\end{array}$$

		Proof.
	Identify $M\otimes N^{*}$ with $\text{Hom}(N,M)\text{.}$ Then, by (A.10) and (A.11), the action of $ℂH$ on $\text{Hom}(N,M)$ is given by, $$hA=\rho \left(b_h\right)A\rho {\left(b_h\right)}^{-1},\text{for}\hspace{0.17em}h\in H\hspace{0.17em}\text{and}\hspace{0.17em}A\in \text{Hom}(N,M),$$ where $\rho :{\left(ℂH\right)}_{\alpha }\to \text{End}\left(M\right)$ is the representation corresponding to $M\text{.}$ If $A\in \text{Hom}(N,M)$ and $g\in H$ then $$e_{H}A=g{e}_{H}A=g\left(e_{H}A\right)=\rho \left(b_g\right)\left(e_{H}A\right)\rho {\left(b_g\right)}^{-1},$$ and so $\rho \left(b_g\right)\left(e_{H}A\right)=\left(e_{H}A\right)\rho \left(b_g\right)$ for all $g\in H\text{.}$ Then, by Schur's lemma, $e_{H}A=0$ if $M\ncong N$ and $e_{H}A$ is a constant if $M=N\text{.}$ $\square$

Theorem A.13. Let $R^{\lambda }$ be a simple $R\text{-module}$ and let $H$ be the inertia group of $R^{\lambda }\text{.}$ The ring $R^G$ acts on $R^{\lambda }$ (by restriction) and ${\left(ℂH\right)}_{\alpha }$ acts on $R^{\lambda }$ (by the $R\text{-module}$ isomorphisms ${\varphi }_h:R^{\lambda }\cong {}^h{R}^{\lambda }$ of (A.3)) and these two actions commute. Thus there is a decomposition

$$R^{\lambda }\cong \underset{\nu \in {\hat{H}}_{\alpha }}{⨁}{R}^{\lambda ,\nu }\otimes {\left(H^{\nu }\right)}^{*},$$

where ${\hat{H}}_{\alpha }$ is an index set for the simple ${\left(ℂH\right)}_{\alpha }\text{-modules,}$ ${\left(H^{\nu }\right)}^{*}$ is the dual of the simple ${\left(ℂH\right)}_{{\alpha }^{-1}}\text{-module}$ $H^{\nu },$ and $R^{\lambda ,\nu }$ is an $R^G\text{-module.}$

1. If $R^{\lambda ,\mu }\ne 0$ then it is a simple $R^G\text{-module.}$
2. Every simple $R^G\text{-module}$ is isomorphic to some $R^{\lambda ,\mu }\text{.}$
3. The nonzero $R^{\lambda ,\mu }$ are pairwise nonisomorphic.

		Proof.
	The setup of Lemma A.9(b) puts us in the situation of [Gre1980, §6.2]. If $e$ is the idempotent used in Lemma A.9 then the functor $$\begin{array}{ccc}R⋊G\text{-modules}& ⟶& R^G\text{-modules}\\ M& ⟼& eM\end{array}$$ is an exact functor such that if $M$ is a simple $R⋊G\text{-module}$ then $eM$ is either 0 or a simple $R^G\text{-module.}$ Furthermore, every simple $R^G\text{-module}$ arises as $eM$ for some simple $R⋊G\text{-module}$ $M\text{.}$ Let $R{G}^{\lambda ,\mu }$ be a simple $R⋊G\text{-module}$ as given by Theorem A.6. From the definition of $R{G}^{\lambda ,\mu }$ we obtain $$\begin{array}{ccc}eE{G}^{\lambda ,\mu }& =& e(R⋊G){\otimes }_{R⋊H}(R^{\lambda }\otimes H^{\mu })\\ & =& e\otimes (R^{\lambda }\otimes H^{\mu })\\ & =& e{e}_H\otimes (R^{\lambda }\otimes H^{\mu })\\ & =& e\otimes e_H(R^{\lambda }\otimes H^{\mu }),\end{array}$$ where $e_H=(1/\mid H\mid ){\sum }_{h\in H}h\text{.}$ Using the decomposition in the statement of the Theorem, we conclude that, as $R^G\text{-modules,}$ $$\begin{array}{ccc}eR{G}^{\lambda ,\mu }& =& e\otimes e_H(\underset{\nu \in \hat{H}}{⨁}{R}^{\lambda ,\nu }\otimes {\left(H^{\nu }\right)}^{*}\otimes H^{\mu })\\ & =& e\otimes e_H(\underset{\nu \in \hat{H}}{⨁}{R}^{\lambda ,\nu }\otimes e_H({\left(H^{\nu }\right)}^{*}\otimes H^{\mu }))\\ & \cong & R^{\lambda ,\mu }\text{.}\end{array}$$ The last isomorphism is a consequence of Lemma A.12. The statement of the Theorem now follows from the results of J.A. Green quoted above. $\square$

Remark A.14. It follows from Theorem A.13 that $R^{\lambda }$ is semisimple as an $R^G\text{-module}$ and the action of ${\left(ℂH\right)}_{\alpha }$ on $R^{\lambda }$ generates ${\text{End}}_{R^G}\left(R^{\lambda }\right)\text{.}$

Notes and References

This is an excerpt of the paper entitled Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.

Research partially supported by the Natioanl Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).

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