# Induced representation

## Definition

### In abstract terms

Suppose $G$ is a group, $H$ is a subgroup of $G$, and $\alpha:H \to GL(V)$ is a linear representation of $H$ on a vector space $V$ over a field $K$. The induced representation of $G$ is a linear representation of $G$ on a new (bigger) vector space $W$ over the same field $K$, defined as follows.

1. $W$ is defined as the $K$-vector space of functions (under pointwise addition and scalar multiplication) $f:G \to V$ satisfying $\! f(hg) = \alpha(h)(f(g))$ for all $h \in H$ and $g \in G$. Note here that $f(g) \in V$ while $\alpha(h) \in GL(V)$ is a linear transformation of $V$, so it makes sense to apply $\alpha(h)$ to $f(g)$. The dimension of $W$ is the product of the dimension of $V$ and the index $[G:H]$ of $H$ in $G$.
2. The action of $G$ on $W$ is defined by the following map $\beta:G \to GL(W)$. For $g \in G$, $\beta(g)$ is the linear transformation that sends $f \in W$ to the following function $f_0$: $f_0(k) := f(kg)$ for $k \in G$. Note that the $g$ gets multiplied on the right in order to make this a left action, because the multiplication is happening on the inside rather than the outside.

### In matrix terms

Using the same notation as the previous definition, this more concrete description works when the index $[G:H]$ as well as the degree of $\alpha$ are both finite. Suppose $[G:H] = n$ and $\alpha$ has degree $m$, with $V$ identified with $K^m$ via a basis, so that $\alpha$ is now a map from $H$ to $GL(m,K)$. Then, we do the following:

• Choose a left transversal of $H$ in $G$, i.e., a set $S$ that intersects every left coset of $H$ in $G$ at exactly one point. Choose a bijection between this left transversal and $\{ 1,2,\dots,n\}$, and label the coset representatives $s_1, s_2, \dots, s_n$.
• If $\beta$ is the induced representation, we define $\beta(g)$, for any $g \in G$, as the following $mn \times mn$ matrix. We first begin by viewing it as a $n \times n$ block matrix with each block a $m \times m$ matrix. Define $q = s_i^{-1}gs_j$. For $i,j \in \{ 1,2,\dots,n\}$, the $ij^{th}$ block is defined as $\alpha(q)$ if $q \in H$ and $0$ otherwise. Note that the matrix is a block monomial matrix in the sense that, as a block matrix, every row has exactly one nonzero block and every column has exactly one nonzero block.

## Facts

### Iteration

• Induction of representations is transitive: If $A \le B \le C$ are groups then $\operatorname{Ind}_B^C \circ \operatorname{Ind}_A^B = \operatorname{Ind}_A^C$.