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trivial representation

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Definition

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

Properties

Example

(induced representation of the trivial representation)

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

ind H G(1)k[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).

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

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type on BG\mathbf{B}GGG-∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}Hrestricted representation
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum with G-action (naive G-spectrum)

Last revised on January 28, 2019 at 05:02:59. See the history of this page for a list of all contributions to it.