The induced character ${\mathrm{Ind}}_H^G\chi$ of a complex character

Depends on

• The character $\chi_V(g)=\operatorname{tr}(\rho_V(g))$ of a finite-dimensional complex representation
• The induced $R$-linear $G$-module $\operatorname{Ind}_H^G W$ as $H$-covariant functions on $G$

Used by

• Frobenius reciprocity for complex characters Corollary
• M-groups are solvable (Taketa) Corollary
• Induced class functions and restricted class functions Definition
• Induction ideal of a subgroup family Definition
• Monomial representations, monomial characters, and M-groups Definition
• Supersolvable groups and monomial characters Definition
• The cyclic induction subgroup of the character ring Definition
• The outer induction product of symmetric-group characters Definition
• Inducing a nontrivial character of a three-cycle subgroup of S₃ gives an irreducible degree-two character Example
• Induction commutes with an external tensor factor Lemma
• Induction is invariant under conjugation of the subgroup and the representation Lemma
• Littlewood's identity - the internal product against an outer product Lemma
• Monomiality lifts along a quotient Lemma
• Restriction to the two-block Young subgroup has LR multiplicities Lemma
• The value of an irreducible character on a full cycle is supported on hooks Lemma
• A faithful irreducible is induced from a proper inertia subgroup Proposition
• The r-cycle virtual character filter Proposition
• Frobenius' formula for the character of an induced representation Theorem
• Mackey's double-coset formula for restricting an induced character Theorem
• Outer induction makes the graded symmetric-group representation group a commutative graded ring Theorem
• The outer Littlewood–Richardson rule Theorem
• Virtual characters are integrally generated by monomial characters from elementary subgroups Theorem

Dependency tree · two levels

8 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, Proposition 4.3.5 (standard reference, not scraped)
• Pavel Etingof et al., Introduction to Representation Theory, Theorem 4.32 (standard reference, not scraped)