 Induced representation

For a subgroup $H\le G$ and a left group algebra $k[H]$-module $V$, its induced representation is ${\mathrm{Ind}}_H^{G}V=k[G]{\otimes }_{k[H]}V$, with $G$ acting by left multiplication on the first tensor factor. If $H$ has finite index of a subgroup and $V$ is a finite-dimensional vector space, the dimension of the induced representation is $[G:H]\dim V$.

Inducing the trivial representation gives the permutation representation on cosets $G/H$. Frobenius reciprocity characterizes the construction: intertwining operators from the induced representation to $W$ correspond to $H$-intertwining operators from $V$ to the restriction of a representation $W$. Thus induction of representations extends a smaller group action while imposing only the identifications forced by that subgroup.

Induced representation is a concept from representation theory in mathematics, particularly in the study of group theory. It allows one to construct a representation of a larger group from a representation of a subgroup. To understand induced representations, consider the following key ideas: 1. **Groups and Representations**: A group is a mathematical structure consisting of a set of elements equipped with an operation that satisfies certain axioms (closure, associativity, identity, and invertibility).