nLab induced representation of the trivial representation

Statement

Let $G$ be a finite group and $H \overset{\iota}{\hookrightarrow} G$ a subgroup-inclusion. Then the induced representation in Rep(G) of the 1-dimensional trivial representation $\mathbf{1} \in Rep(H)$ is the permutation representation $k[G/H]$ of the coset G-set $G/H$:

$$\mathrm{ind}_H^G\big( \mathbf{1}\big) \;\simeq\; k[G/H] \,.$$

This follows directly as a special case of the general formula for induced representations of finite groups (this Example).

It follows that every virtual permutation representation (hence every element of the representation ring $R_k(G)$ in the image of the canonical morphism $A(G) \overset{\beta}{\to} R_k(G)$ from the Burnside ring) is a virtual combination of induced representations of trivial representations.

A generalization of this statement including non-permutation representations is the Brauer induction theorem.

References

See also

• Groupprops, Induced representation from trivial representation of subgroup is permutation representation for action on coset space