DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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Conjugate representations and conjugate characters on conjugate subgroups

Definition

Let $H \leq G$ , let $g \in G$ , and let $W$ be a finite-dimensional representation of $H$ over a field $k$ (A finite-dimensional representation $ρ : G \to GL (V)$ over a field, and its degree).

The conjugate representation $^{g} W$ is the same vector space, now regarded as a representation of the conjugate subgroup $g H g^{- 1}$ by the rule

$(g h g^{- 1}) \cdot w : = h \cdot w (h \in H, w \in W) .$

If $χ$ is the complex character of $W$ , the conjugate character $^{g} χ$ is the character of $^{g} W$ :

$^{g} χ (g h g^{- 1}) = χ (h) .$

Remarks

Every element of $g H g^{- 1}$ has a unique form $g h g^{- 1}$ with $h \in H$ , so the displayed formulas are well defined.
Restricting $^{g} W$ or $^{g} χ$ to a subgroup of $g H g^{- 1}$ will be used in Mackey's formula.

Depends on

Used by

Mackey's double-coset formula for restricting an induced character Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter Webb, A Course in Finite Group Representation Theory, Section 5.2 (standard reference, not scraped)
Anupam Singh, Representation Theory of Finite Groups, Section 20.1 (standard reference, not scraped)