Clifford Theory

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 20 May 2010

Clifford Theory

Let $R$ be an algebra over $ℂ$, $K$ a finite group and fix $\beta :K\times K\to R$ and automorphisms $c_k:R\to R,k\in K,$ such that the semidirect product algebra $R⋊{\left(ℂK\right)}_{\beta }$ is associative, where $$R⋊{\left(ℂK\right)}_{\beta }=\left\{\sum _{k\in K}{r}_k{t}_k|r_k\in R\right\},$$ with multiplication determined by the multiplication in $R$ and the relations $$t_{k_1}{t}_{k_2}=\beta \left(k_1,k_2\right)t_{k_1}{t}_{k_2},t_{k}r=c_k\left(r\right)t_k,\text{and}{t}_1=1,$$ for $k,k_1,k_2\in K$ and $r\in R.$

Let $N$ be an $R$-module. Define a new $R$-module $k:N$ to have the same underlying vector space $N$ but with $R$-action given by $$c_k\left(r\right)\circ n=rn,\text{for all }r\in R,n\in N.$$

If $P\subseteq N$ is an $R$-submodule of $N$ then $k:P$ is an $R$-submodule of $k:N,$ and so, $N$ is a simple $R$-module iff $k:N$ is a simple $R$-module. In this way $K$ acts on the (isomorphism classes of) simple $R$-modules.

The inertia group of a simple $R$-module $R^{\lambda }$ is $$H=\left\{h\in K|h:R^{\lambda }\cong R^{\lambda }\right\}.$$ For each $h\in H$ fix an $R$-module homomorphism ${\phi }_h:h:R^{\lambda }\to R^{\lambda }.$ The condition that ${\phi }_h$ is an $R$-module homomorphism is the same as saying that, as liner transformations on the vector space $R^{\lambda },$ $$c_h\left(r\right){\phi }_h={\phi }_{h}r,\text{for all }r\in R,h\in H.$$(Proof: $\left(c_h\left(r\right){\phi }_{h}m\right)={\phi }_h\left(c_h\left(r\right)\circ m\right)={\phi }_{h}rm.$) Thus $$\beta {\left(h_1,h_2\right)}^{-1}{\phi }_{h_1}{\phi }_{h_2}r=\beta {\left(h_1,h_2\right)}^{-1}{c}_{h_1}\left(c_{h_2}\left(r\right)\right){\phi }_{h_1}{\phi }_{h_2}=c_{h_1{h}_2}\left(r\right)\beta {\left(h_1,h_2\right)}^{-1}{\phi }_{h_1}{\phi }_{h_2}.$$ By Schur's lemma ${\phi }_h$ is unique up to constant multiples and so $$\beta {\left(h_1,h_2\right)}^{-1}{\phi }_{h_1}{\phi }_{h_2}=\alpha \left(h_1,h_2\right){\phi }_{h_1{h}_2},\text{for some }\alpha \left(h_1,h_2\right)\in ℂ.$$ Let ${\left(ℂ,H\right)}_{\frac{1}{\alpha }}$ be the algebra over $ℂ$ with basis $\left\{b_h|h\in H\right\}$ and mutliplication given by

$$b_{h_1}{b}_{h_2}=\alpha {\left(h_1,h_2\right)}^{-1}{b}_{h_1{h}_2},h_1,h_2\in H.$$

If $H^{\mu }$ is a ${\left(ℂH\right)}_{\frac{1}{\alpha }}$-module then $R^{\lambda }\otimes H^{\mu }$ is an $R⋊{\left(ℂH\right)}_{\beta }$ module with action given by $$r{t}_h\left(m\otimes n\right)=r{\phi }_{h}m\otimes b_{h}n,$$ for all $r\in R,h\in H,m\in R^{\lambda }$ and $n\in H^{\mu }.$ This is an $R⋊{\left(ℂH\right)}_{\beta }$ action since $$\begin{array}{rcl}{t}_{h}r\left(m\otimes n\right)& =& {\phi }_{h}rm\otimes b_{h}n=c_h\left(r\right){\phi }_{h}m\otimes b_{h}n,\\ t_{h_1}{t}_{h_2}\left(m\otimes n\right)& =& {\phi }_{h_1}{\phi }_{h_2}m\otimes b_{h_1}{b}_{h_2}n=\beta \left(h_1,h_2\right)\alpha \left(h_1,h_2\right)\alpha {\left(h_1,h_2\right)}^{-1}{\phi }_{h_1{h}_2}m\otimes b_{h_1{h}_2}n.\end{array}$$

(Clifford theory)

1. Let $M$ be a finite dimensional simple $R⋊{\left(ℂK\right)}_{\beta }$ module. Then $$M\cong {\mathrm{Ind}}_{R⋊{\left(ℂK\right)}_{\beta }}^{R⋊{\left(ℂH\right)}_{\beta }}\left(R^{\lambda }\otimes H^{\mu }\right),$$ where
   1. $R^{\lambda }$ is a simple $R$ submodule of $M,$
   2. $H=\left\{h\in K|h:R^{\lambda }\cong R^{\lambda }\text{ as }R\text{-modules}\right\},$
   3. $\alpha :H\times H\to ℂ$ is determined by choices of $R$ module isomorphisms ${\phi }_h:t_h{R}^{\lambda }\to R^{\lambda },$ and
   4. $H^{\mu }$ is the simple ${\left(ℂH\right)}_{\frac{1}{\alpha }}$-module given by $H^{\mu }={\mathrm{Hom}}_R\left(R^{\lambda },M\right)$ with

      $$\left(b_h\psi \right)\left(m\right)=\alpha {\left(h,h^{-1}\right)}^{-1}{t}_h\psi \left({\phi }_{h^{-1}}\beta {\left(h,h^{-1}\right)}^{-1}m\right),h\in H,\psi \in H^{\mu },m\in R^{\lambda },$$

where $$\left\{b_h|h\in H\right\}$$ is the basis of ${\left(ℂH\right)}_{1/\alpha }$ in (???).

		Proof.
	Let $M$ be a simple $R⋊{\left(ℂK\right)}_{\beta }$ module. Let $R^{\lambda }$ be a simple $R$-submodule of $M.$ Then $t_k{R}^{\lambda }$ is an $R$-submodule of $M$ isomorphic to $k:R^{\lambda }.$ The sum ${\sum }_{k\in K}{t}_k{R}^{\lambda }$ is an $R⋊{\left(ℂK\right)}_{\beta }$ submodule of $M,$ and, since $M$ is simple, $$M=\sum _{k\in K}{t}_k{R}^{\lambda }=\sum _{k_i\in K/H}{t}_{k_i}N,\text{where }N=\sum _{h\in H}{t}_h{R}^{\lambda }$$ and the second sum is over a set of coset representatives of the cosets in $K/H.$ So $$M\cong {\mathrm{Ind}}_{R⋊{\left(ℂH\right)}_{\beta }}^{R⋊{\left(ℂK\right)}_{\beta }}\left(N\right),$$ as $R⋊{\left(ℂK\right)}_{\beta }$-modules. We shall define a $R⋊{\left(ℂH\right)}_{1/\alpha }$ action on $H^{\mu }={\mathrm{Hom}}_R\left(R^{\lambda },N\right)={\mathrm{Hom}}_R\left(R^{\lambda },M\right)$ so that $$\begin{array}{rcll}\Theta :& R^{\lambda }\otimes {\mathrm{Hom}}_R\left(R^{\lambda },N\right)& \to & N\\ & m\otimes \psi & \mapsto & \psi \left(m\right)\end{array}$$ is an $R⋊{\left(ℂH\right)}_{\beta }$ module isomorphism. The condition that $\Theta$ is an $R⋊{\left(ℂH\right)}_{\beta }$ module homomorphism is that $$t_h\left(\Theta \left(m\otimes \psi \right)\right)=\Theta \left(t_h\left(m\otimes \psi \right)\right),\text{for all }m\in M,\psi \in H^{\mu },h\in H,$$ and so $$t_h\psi \left(m\right)=\Theta \left({\phi }_{h}m\otimes b_h\psi \right)=\left(b_h\psi \right)\left({\phi }_{h}m\right),\text{for all }m\in M,h\in H,\psi \in H^{\mu }.$$ Thus the appropriate formula for the action of ${\left(ℂH\right)}_{1/\alpha }$ on $H^{\mu }$ must be $$\left(b_h\psi \right)\left(m\right)=t_h\psi \left({\phi }_h^{-1}m\right)=\alpha {\left(h,h^{-1}\right)}^{-1}{t}_h\psi \left({\phi }_{h^{-1}}\beta {\left(h,h^{-1}\right)}^{-1}m\right).$$ Given these formulas, it is straightforward to check that $H^{\mu }$ is a well defined ${\left(ℂH\right)}_{1/\alpha }$ module and that $\Theta$ is an $R⋊{\left(ℂH\right)}_{\beta }$ isomorphism. The $R⋊{\left(ℂH\right)}_{\beta }$ module $N$ is simple, since if $P$ is an $R⋊{\left(ℂH\right)}_{\beta }$ submodule of $N$ then ${\mathrm{Ind}}_{R⋊{\left(ℂH\right)}_{\beta }}^{R⋊{\left(ℂK\right)}_{\beta }}\left(P\right)$ is an $R⋊{\left(ℂK\right)}_{\beta }$ module of $M.$ Since $M$ is simple $P$ must be equal to $N.$ Since $N$ is simple the map $\Phi$ is surjective. If $\Phi :R^{\lambda }\to N$ is a nonzero $R$ module homomorphism then it must be injective, since $R^{\lambda }$ is simple. So $\Phi \left(m\right)\ne 0$ for all nonzero $m\in R^{\lambda }.$ So the map $\Phi$ is injective. Thus $$R^{\lambda }\otimes {\mathrm{Hom}}_R\left(R^{\lambda },N\right)\cong N.$$ It follows that ${\mathrm{Hom}}_R\left(R^{\lambda },N\right)$ is a simple ${\left(ℂH\right)}_{1/\alpha }$ module since any submodule $P$ would yield a submodule $\Phi \left(R^{\lambda }\otimes P\right)$ of $N.\square$

Remark. Note that the map $$\begin{array}{rcl}k:R^{\lambda }& \to & t_k{R}^{\lambda }\\ n& \mapsto & t_{k}n\end{array}$$ is an isomorphism.

1. The simple $R⋊{\left(ℂK\right)}_{\beta }$ modules $R{K}^{\left(\lambda ,\mu \right)}$ are indexed by pairs $\left(\lambda ,\mu \right)$ with $\lambda \in \hat{R}/K,\mu \in R⋊{\widehat{\left(ℂH\right)}}_{1/\alpha },$ where
   1. $\hat{R}/K$ is a set of representatives of the $K$ orbits on $\hat{R},$ and
   2. ${\widehat{\left(ℂH\right)}}_{1/\alpha }$ is an index set for the simple ${\left(ℂH\right)}_{1/\alpha }$ modules.
2. $\dim \left(R{K}^{\left(\lambda ,\mu \right)}\right)=\dim \left(R^{\lambda }\right)\dim \left(H^{\mu }\right)\mathrm{Card}\left(K/H\right).$
3. The irreducible $R⋊{\left(ℂK\right)}_{\beta }$ module $R{K}^{\left(\lambda ,\mu \right)}$ is given by $$R{K}^{\left(\lambda ,\mu \right)}\cong {\mathrm{Ind}}_{R⋊{\left(ℂH\right)}_{\beta }}^{R⋊{\left(ℂK\right)}_{\beta }}\left(R^{\lambda }\otimes H^{\mu }\right).$$

		Proof.
	Let $R^{\lambda }$ be a simple $R$ module. If the inertia group of $R^{\lambda }$ is $H$ then the inertia group of $k:R^{\lambda }$ is $kH{k}^{-1}\cong H.$ If $h\in H$ and $${\phi }_h:h:R^{\lambda }\to R^{\lambda }$$ is an isomorphism, then $$\begin{array}{rcll}{\psi }_{gh{g}^{-1}}:& gh:R^{\lambda }& \to & g:R^{\lambda }\\ & m& \mapsto & {\phi }_h\left(m\right)\end{array}\text{is an }R\text{ module isomorphism.}$$ This is because $$\begin{array}{rcl}r\circ {\psi }_{gh{g}^{-1}}\left(m\right)& =& r\circ {\phi }_h\left(m\right)=g^{-1}\left(r\right){\phi }_h\left(m\right)\\ & =& {\phi }_h\left(g^{-1}\left(r\right)\circ m\right)={\phi }_h\left(h^{-1}{g}^{-1}\left(r\right)m\right)\\ & =& {\psi }_{gh{g}^{-1}}\left(r\circ m\right).\end{array}$$ So, in fact we may choose ${\phi }_{gh{g}^{-1}}.$ Then the factor set for $gH{g}^{-1}$ is $\alpha :H\times H\to ℂ$ and clearly ${\left({ℂ}^{g}H\right)}_{1/\alpha }\cong {\left(ℂH\right)}_{1/\alpha }$ and $$\begin{array}{rcl}{R}^{\lambda }\otimes H^{\mu }& \to & g:R^{\lambda }\otimes g:H^{\mu }\\ r\otimes m& \mapsto & r\otimes m\end{array}$$ is an $R$-module isomorphism. A different choice $${\tilde{\phi }}_h:h:R^{\lambda }\to R^{\lambda }$$ may yield a different factor set $\tilde{\alpha }:H\times H\to ℂ,$ $${\tilde{\phi }}_{h_1}{\tilde{\phi }}_{h_2}=\tilde{\alpha }\left(h_1,h_2\right){\tilde{\phi }}_{h_1{h}_2}.$$ By Schur's lemma $${\tilde{\phi }}_h=\gamma \left(h\right){\phi }_h,\text{for some constant }\gamma \left(h\right)\in {ℂ}^{*}.$$ So $${\tilde{\phi }}_{h_1}{\tilde{\phi }}_{h_2}=\gamma \left(h_1\right)\gamma \left(h_2\right){\phi }_{h_1}{\phi }_{h_2}=\gamma \left(h_1\right)\gamma \left(h_2\right)\alpha \left(h_1,h_2\right){\phi }_{h_1{h}_2}$$ implies $$\tilde{\alpha }\left(h_1,h_2\right)=\frac{\gamma \left(h_1\right)\gamma \left(h_2\right)}{\gamma \left(h_1{h}_2\right)}\alpha \left(h_1,h_2\right).$$ Then the algebra ${\left(ℂH\right)}_{1/\tilde{\alpha }}$ given by $${\left(ℂH\right)}_{1/\tilde{\alpha }}=\mathrm{span}\left\{d_h|h\in H\right\},\text{with}{d}_{h_1}{d}_{h_2}=\tilde{\alpha }{\left(h_1,h_2\right)}^{-1}{d}_{h_1{h}_2},$$is isomorphic to ${\left(ℂH\right)}_{1/\alpha }$ via the isomorphism $$\begin{array}{rcl}{\left(ℂH\right)}_{1/\tilde{\alpha }}& \to & {\left(ℂH\right)}_{1/\alpha }\\ d_h& \mapsto & c_h\gamma {\left(h\right)}^{-1}.\end{array}$$ Just to check, $$\Phi \left(d_{h_1},d_{h_2}\right)=c_{h_1}\gamma {\left(h_1\right)}^{-1}{c}_{h_2}\gamma {\left(h_2\right)}^{-1}=\gamma \left({h_1}^{-1}\right)\gamma \left({h_2}^{-1}\right)c_{h_1{h}_2}\alpha {\left(h_1,h_2\right)}^{-1}=\tilde{\alpha }{\left(h_1,h_2\right)}^{-1}\Phi \left(d_{h_1{h}_2}\right)$$$\square$

Example 1. Let $G$ be a group and let $N$ be a normal subgroup of $G.$ Let $K$ be the quotient group $N\backslash G$ and choose a set $\left\{t_k|k\in K\right\}$ of the representatives of the cosets in $N\backslash G.$ If the map $\beta :K\times K\to N$ and automorphisms $c_k:N\to N$ are $$t_{k_1}{t}_{k_2}=\beta \left(k_1,k_2\right)t_{k_1{k}_2}\text{and}{c}_k\left(n\right)=t_{k}n{t_k}^{-1},\text{then}ℂG\cong ℂN⋊{\left(ℂK\right)}_{\beta }.$$

Example 2. $G\left(r,1,n\right)={\left(\mathbb{Z}/r\mathbb{Z}\right)}^n⋉S_n.$
Recall that $S_n$ acts on ${\left(\mathbb{Z}/r\mathbb{Z}\right)}^n$ by permuting the factors. The simple $\mathbb{Z}/r\mathbb{Z}$ representations are indexed by $\lambda =0,1,2,\dots ,r-1$ and are given by $$\begin{array}{rcll}{x}^{\lambda }:& \mathbb{Z}/r\mathbb{Z}& \to & ℂ\\ & e^{2\pi ik/r}& \mapsto & e^{2\pi i\lambda k/r}\end{array}$$ The simple ${\left(\mathbb{Z}/r\mathbb{Z}\right)}^n$ modules are indexed by $n$-tuples $\lambda =\left({\lambda }_1,\dots ,{\lambda }_n\right)$ with ${\lambda }_i\in \left\{0,1,2,\dots ,r-1\right\}.$ The group $S_n$ acts on the simple ${\left(\mathbb{Z}/r\mathbb{Z}\right)}^n$-modules and on the indexes $\lambda =\left({\lambda }_1,\dots ,{\lambda }_n\right)$ by permuting the factors. Each orbit under this action has a unique representation of the form $$\lambda =\left(0,\dots ,0,1,\dots ,1,\dots ,r-1,\dots ,r-1\right),$$ with $m_0$ consecutive zeroes, $m_1$ consecutive ones, etc. The inertia group of this representation is $$S_{\overrightarrow{m}}=S_{m_0}\times S_{m_1}\times \dots \times S_{m_{r-1}}$$ and ${\left(\mathbb{Z}/r\mathbb{Z}\right)}^n$ module isomorphisms ${\phi }_h:h:R^{\lambda }\to R^{\lambda }$ can be fixed to be ${\phi }_h={\mathrm{Id}}_{R^{\lambda }}$ for all $h\in H.$ So the factor set is trivial in this case.

The simple $S_{m_0}tgimes{S}_{m_1}\times \dots S_{m_{r-1}}$ modues are indexed by $r$-tuples of partitions $$\mu =\left({\mu }^{\left(0\right)},\dots ,{\mu }^{\left(r-1\right)}\right)\text{with}{\mu }^{\left(i\right)}⊢m_i.$$ So the simple $G\left(r,1,n\right)$-modules are indexed by pairs $$\left(\lambda ,\mu \right),\text{with }\lambda \text{ as in ??? and }\mu \text{ as in ????.}$$ Given $\mu =\left({\mu }^{\left(0\right)},\dots ,{\mu }^{\left(r-1\right)}\right)$ the information in $\lambda$ is redundant and so we may index the simple $G\left(r,1,n\right)$ modules by $r$-tuples of partitions $\mu =\left({\mu }^{\left(0\right)},\dots ,{\mu }^{\left(r-1\right)}\right)$ with $n$ boxes in total.

The same type of anlysis can be carried through for the affine symmetric group ${\tilde{S}}_n={\mathbb{Z}}^n⋉S_n.$ The simple $\mathbb{Z}$-modules are given by $$\begin{array}{rcll}{x}^{\lambda }:& \mathbb{Z}& \to & ℂ\\ & 1& \mapsto & \lambda \end{array},\lambda \in {ℂ}^{*}.$$ Thus the representations of ${\tilde{S}}_n$ are indexed by ${ℂ}^{*}$-tuples of partitions $\mu ={\left({\mu }^{\left(\lambda \right)}\right)}_{\lambda \in {ℂ}^{*}}$ with $n$ boxes in totl. Alternatively the simple ${\tilde{S}}_n$ modules are indexed by functions from ${ℂ}^{*}$ to the set $𝒫$ of partitions $$\begin{array}{rcll}\mu :& {ℂ}^{*}& \to & 𝒫\\ & \lambda & \mapsto & {\mu }^{\left(\lambda \right)}\end{array}\text{such that }n=\sum _{\lambda \in {ℂ}^{*}}\left|{\mu }^{\left(\lambda \right)}\right|.$$

The same analysis works for $G^n⋉S_n=G\between S_n$ where $G$ is any finite group. The general statement is that the simple $G\between S_n$ modules are indexed by the functions $$\begin{array}{rcll}\mu :& \hat{G}& \to & 𝒫\\ & \lambda & \mapsto & {\mu }^{\left(\lambda \right)}\end{array}\text{such that }n=\sum _{\lambda \in \hat{G}}\left|{\mu }^{\left(\lambda \right)}\right|,$$ where $\hat{G}$ is an index set for the simple $G$-modules.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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