nLab trivial representation

Definition

A representation/action $V \times G \longrightarrow V$ is trivial if it is given by the projection out of the product onto $V$.

Properties

Example

(induced representation of the trivial representation)

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).

Related concepts

• scalar

• alternating representation

• regular representation

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theory	representation theory
pointed connected context $\mathbf{B}G$	∞-group $G$
dependent type on $\mathbf{B}G$	$G$-∞-action/∞-representation
dependent sum along $\mathbf{B}G \to \ast$	coinvariants/homotopy quotient
context extension along $\mathbf{B}G \to \ast$	trivial representation
dependent product along $\mathbf{B}G \to \ast$	homotopy invariants/∞-group cohomology
dependent product of internal hom along $\mathbf{B}G \to \ast$	equivariant cohomology
dependent sum along $\mathbf{B}G \to \mathbf{B}H$	induced representation
context extension along $\mathbf{B}G \to \mathbf{B}H$	restricted representation
dependent product along $\mathbf{B}G \to \mathbf{B}H$	coinduced representation
spectrum object in context $\mathbf{B}G$	spectrum with G-action (naive G-spectrum)