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 $q=s_i^{-1}g{s}_j$. For $i,j\in \{1,2,\dots ,n\}$, the $i{j}^{th}$ 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 ${\mathrm{I}}_{\mathrm{n}}^{\mathrm{d}}\circ {\mathrm{I}}_{\mathrm{n}}^{\mathrm{d}}={\mathrm{I}}_{\mathrm{n}}^{\mathrm{d}}$.

Relation with induced class functions

• Character of induced representation is induced class function of character

Relation with restriction of representations

• Frobenius reciprocity relates the representation-theoretic operations of induction and restriction

Particular cases of induction of representations

• Induced representation from regular representation of subgroup is regular representation of group
• Induced representation from trivial representation on normal subgroup factors through regular representation of quotient group
• Induced representation from trivial representation of subgroup is permutation representation for action on coset space