The Unapologetic Mathematician

Mathematics for the interested outsider

Restricting and Inducing Representations

Two of the most interesting constructions involving group representations are restriction and induction. For our discussion of both of them, we let H\subseteq G be a subgroup; it doesn’t have to be normal.

Now, given a representation \rho:G\to\mathrm{End}(V), it’s easy to “restrict” it to just apply to elements of H. In other words, we can compose the representing homomorphism \rho with the inclusion \iota:H\to G: \rho\circ\iota:H\to\mathrm{End}(V). We write this restricted representation as \rho\!\!\downarrow^G_H; if we are focused on the representing space V, we can write V\!\!\downarrow^G_H; if we pick a basis for V to get a matrix representation X we can write X\!\!\downarrow^G_H. Sometimes, if the original group G is clear from the context we omit it. For instance, we may write V\!\!\downarrow_H.

It should be clear that restriction is transitive. That is, if K\subseteq H\subseteq G is a chain of subgroups, then the inclusion mapping \iota_{K,G}K\hookrightarrow G is the exactly composition of the inclusion arrows \iota_{K,H}K\hookrightarrow H and \iota_{H,G}H\hookrightarrow G. And so we conclude that

\displaystyle\begin{aligned}\rho\!\!\downarrow^G_K&=\rho\circ\iota_{K,G}\\&=\rho\circ\iota_{K,H}\circ\iota_{H,G}\\&=\left(\rho\circ\iota_{K,H}\right)\!\!\downarrow^G_H\\&=\left(\rho\!\!\downarrow^H_K\right)\!\!\downarrow^G_H\end{aligned}

So whether we restrict from G directly to K, or we stop restrict from G to H and from there to K, we get the same representation in the end.

Induction is a somewhat more mysterious process. If V is a left H-module, we want to use it to construct a left G-module, which we will write V\!\!\uparrow_H^G, or simply V\!\!\uparrow^G if the first group H is clear from the context. To get this representation, we will take the tensor product over H with the group algebra of G.

To be more explicit, remember that the group algebra \mathbb{C}[G] carries an action of G on both the left and the right. We leave the left action alone, but we restrict the right action down to H. So we have a G\times H-module {}_G\mathbb{C}[G]_H, and we take the tensor product over H with {}_HV. We get the space V\!\!\uparrow_H^G=\mathbb{C}[G]\otimes_HV; in the process the tensor product over H “eats up” the right action of H on the \mathbb{C}[G] and the left action of H on V. The extra left action of G on \mathbb{C}[G] leaves a residual left action on the tensor product, and this is the left action we seek.

Again, induction is transitive. If K\subseteq H\subseteq G is a chain of subgroups, and if V is a left K-module, then

\displaystyle\begin{aligned}\left(V\!\!\uparrow_K^H\right)\!\!\uparrow_H^G&=\mathbb{C}[G]\otimes_H\left(V\!\!\uparrow_K^H\right)\\&=\mathbb{C}[G]\otimes_H\mathbb{C}[H]\otimes_KV\\&\cong\mathbb{C}[G]\otimes_KV\\&=V\!\!\uparrow_K^G\end{aligned}

The key step here is that \mathbb{C}[G]\otimes_H\mathbb{C}[H]\cong\mathbb{C}[G]. But if we have any simple tensor g\otimes h\in\mathbb{C}[G]\otimes_H\mathbb{C}[H], we can use the relation that lets us pull elements of H across the tensor product. We get gh\otimes1\in\mathbb{C}[G]\otimes_H\mathbb{C}[H]. That is, we can specify any tensor by an element in \mathbb{C}[G] alone.

November 23, 2010 Posted by | Algebra, Group theory, Representation Theory | 9 Comments

   

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