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## The induced representation

Charles Siegel over at Rigorous Trivialities suggested a NaNoWriMo for math bloggers: instead of writing a 50,000-word novel, just write a blog post every day. I have to admit I rather like the idea, so we’ll see if I can keep it up.

Continuing the previous post, what we want to do now is to think of restriction $\text{Res}_H^G : \text{Rep}(G) \to \text{Rep}(H)$ as a forgetful functor, since restricting a representation just corresponds to forgetting some of the data that defines it. Its left adjoint, if it exists, should be a construction of the “free $G$-representation” associated to an $H$-representation. Given a representation $\rho : H \to \text{Aut}(V)$ we therefore want to find a representation $\rho' : G \to \text{Aut}(V')$ with the following universal property: any $H$-intertwining operator $\phi : V \to W$ for $\tau$ a $G$-representation on $W$ naturally determines a unique $G$-intertwining operator $\phi' : V' \to W$. In other words, we want to construct a functor $\text{Ind}_G^H : \text{Rep}(H) \to \text{Rep}(G)$ such that

$\text{Hom}_{\text{Rep}(G)}(\text{Ind}_H^G \rho, \tau) \simeq \text{Hom}_{\text{Rep}(H)}(\rho, \text{Res}_H^G \tau)$.

Cosets

To gain some insight into what’s going on, consider the special case that $\rho$ is the trivial representation on $\mathbb{C}$, hence $\rho(h) = I$ for all $h$. Given another $G$-representation $\tau$ on a vector space $W$, an $H$-intertwining operator $\phi : \mathbb{C} \to W$ must satisfy $\phi \rho(h) = \phi = \tau(h) \phi$, hence $\phi$ is identically zero or $\tau$ is constant on $H$. What representation is universal for representations constant on $H$?

The answer is immediate: it’s the coset representation! The formal way to “pretend that elements of $H$ are the identity” is to consider two group elements $g_1, g_2$ to be equivalent if $g_1 = g_2 h$ for some $h \in H$, and the equivalence classes of this relation are precisely the cosets $G/H$, on which $G$ acts by left multiplication. Extend this action to a representation $\rho'$ on $V' = \mathbb{C}^{|G/H|}$ in the obvious way. I now claim that this representation has the desired universal property, as follows:

If $\tau$ is not constant on $H$, the only $H$-intertwining operator $\mathbb{C} \to W$ is identically zero. Similarly, the only $G$-intertwining operator $\phi' : V' \to W$ is identically zero, since $\phi' \rho'(h) = \phi' = \tau(h) \phi$ for all $h \in H$ if and only if either $\phi'$ is identically zero or $\tau$ is constant on $H$. If $\tau$ is constant on $H$, then the $H$-intertwining operators $\mathbb{C} \to W$ are determined by the image $w$ of $1$, and any choice of $w$ uniquely determines a $G$-intertwining operator $V' \to W$ by sending a coset $g_i H$ to $\tau(g_i H) w$ and extending by linearity, and every intertwining operator arises in this way.

This coset construction generalizes as follows, following the Wikipedia article: given a representation $\rho$ on $V$, define

$\displaystyle V' = \bigoplus_{i=1}^{|G/H|} x_i V$

where $\displaystyle G = \bigsqcup_{i=1}^{|G/H|} x_i H$ is a decomposition of $G$ into cosets of $H$ and $x_i V$ is just a copy of $V$. The representation $\rho'$ is defined as follows: for $g \in G$, $gx_i = x_j h$ for unique $j$ and unique $h \in H$, and $\rho'(g)$ sends a vector $x_i v \in x_i V$ to $x_j \rho(h) v \in x_j V$. This action is extended by linearity. Again, we verify the universal property, as follows:

Any $H$-intertwining operator $\phi : V \to W$ satisfies $\phi \rho(h) = \tau(h) \phi$ for all $h$. It defines an operator $\phi' : V' \to W$ by sending $x_i v$ to $\tau(x_i) \phi(v)$, again extended by linearity, and we claim that this operator is $G$-intertwining. This is equivalent to the claim that, if $gx_i = x_j h$ as above, then $\phi' \rho'(g) = \tau(g) \phi'$, or

$(\phi' \rho'(g)) x_i v = \tau(x_j) \phi(\rho(h) v) = (\tau(g) \phi') x_i v = \tau(x_j) \tau(h) \phi(v)$

and this property is satisfied if and only if $\phi \rho(h) = \tau(h) \phi$, which is precisely the $H$-intertwining condition. As before, any $G$-intertwining operator $V' \to W$ arises in this way, although this is somewhat messy.

The module perspective

The generalization of the above argument in module theory gives what is called the extension of scalars functor. Given a ring $R$ and a subring $S$, any left $R$-module $M$ naturally acquires the structure of a left $S$-module by restriction of scalars, which is the generalization of the restriction functor $\text{Res}_H^G$ above if we set $R = \mathbb{C}[G], S = \mathbb{C}[H]$. Restriction defines a functor $\text{Res}_S^R : \text{R-Mod} \to \text{S-Mod}$, and as above restriction has a left adjoint called extension. This adjunction follows from a more general adjunction, which is as follows:

Suppose $M$ is an $S$-module and $N, P$ are $R$-modules, and consider the set of all maps $f : M \times N \to P$ which are $S$-linear in the first variable and $R$-linear in the second variable. For fixed $m \in M$, the map $n \mapsto f(m, n)$ is $R$-linear, and this assignment of maps is itself $S$-linear, so $f$ defines an $S$-linear map $M \to \text{Hom}_{\text{R-Mod}}(N, P)$. Moreover, all such maps arise this way. On the other hand, any map $f$ must also be $S$-bilinear, hence it factors through the tensor product $M \otimes_S N$. This tensor product comes equipped with a natural action of $R$ on the right factor turning it into an $R$-module, and then the extra condition that $f$ be $R$-linear in $N$ is precisely the condition that the assignment $M \otimes_S N \to P$ be $R$-linear. Again, all such maps arise this way. It follows that we have a natural equivalence

$\text{Hom}_{\text{R-Mod}}(M \otimes_S N, P) \simeq \text{Hom}_{\text{S-Mod}}(M, \text{Hom}_{\text{R-Mod}}(N, P))$.

In other words, for any $N \in \text{R-Mod}$ the functor $- \otimes_S N : \text{S-Mod} \to \text{R-Mod}$ (a free construction) is left adjoint to the functor $\text{Hom}_{\text{R-Mod}}(N, -) : \text{R-Mod} \to \text{S-Mod}$ (a forgetful functor).

Here’s why this is relevant: let $R = N = \mathbb{C}[G], S = \mathbb{C}[H], M \in \text{Rep}(H), P \in \text{Rep}(G)$. Our goal is to multiply elements of $M$ by elements of $\mathbb{C}[G]$ in a universal fashion (this defines an action of $G$ on $M$), hence to find a universal object for functions $M \times \mathbb{C}[G] \to P$ which are $\mathbb{C}[H]$-linear in the first variable and $\mathbb{C}[G]$-linear in the second; this is precisely the setup as above. Since $\text{Hom}_{\text{Rep}(G)}(N, P) \simeq \text{Res}_H^G P$, we are then led to the following conclusion.

Proposition: $\text{Ind}_H^G M \simeq M \otimes_{\mathbb{C}[H]} \mathbb{C}[G]$.

A little thought will reveal that this is essentially the same construction as above.

Covering

A combinatorial way to think about restriction and induction is as follows: given a subgroup $H$ of a subgroup $G$, construct a category whose objects are irreducible representations of either $G$ or $H$ where there are $k$ arrows from a representation $\beta$ of $H$ to a representation $\alpha$ of $G$ if $\left< \text{Ind}_H^G \beta, \alpha \right> = \left< \beta, \text{Res}_H^G \alpha \right>$. What Frobenius reciprocity tells us is that these arrows can be interpreted in two dual ways: either as the number of times $\alpha$ appears in the induced representation of $\beta$, or as the number of times $\beta$ appears in the restricted representation of $\alpha$. This turns the category into a generalization of a graded poset with two ranks where two objects may be related in more than one way.

This construction generalizes to any chain $G_1 \le G_2 \le ...$ of groups. Next time we’ll see what this means for the symmetric groups. (There’s one more thing I need the induced representation for, too.)