Theorem. Clifford homogeneity and the semidirect route [fgap-001Q]
✍️source

Let \(N\trianglelefteq G\) and let \(V\) be an irreducible complex representation of \(G\). If \(W_1,\ldots ,W_r\) represent the distinct conjugates of an irreducible constituent of \(V|_N\), then for 1 positive integer \(e\), \[ V|_N\cong e(W_1\oplus \cdots \oplus W_r). \] Thus restriction is homogeneous along a single orbit. The common multiplicity is an additional assertion beyond the single orbit in Clifford restriction. This is Clifford's theorem, [james2001representations, thm. 20.8, pp. 216--217].

The inertia subgroup is the intermediate group where one studies whether \(W\) extends, possibly projectively, before inducing to \(G\). For semidirect products, this reduces the problem to the action of the complement. For \(Q_8\rtimes C_3\cong 2T\), it connects the normal quaternion subgroup to the skew-Group-Algebra model in two routes from binary tetrahedral symmetry. This theorem does not supply the full character table of \(2T\).

Theorem. Clifford homogeneity and the semidirect route [fgap-001Q]
✍️source

Reference. Representations and characters of groups [james2001representations]

The human-curated Group-Algebra route [fgap-001G]
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Definition. the Group Algebra and its universal extension [fgap-001H]

MonoidAlgebra.lift MonoidAlgebra.lift_single
L∃∀N

Theorem. representations are Group-Algebra modules [fgap-001I]

Representation.asModule Representation.ofModule Rep.equivalenceModuleMonoidAlgebra
L∃∀N

Proof.

Example. the regular representation reads the basis [fgap-001J]

Theorem. Maschke averaging [fgap-001K]

MonoidAlgebra.Submodule.exists_isCompl
L∃∀N

Proof.

Theorem. Wedderburn blocks over an algebraically closed field [fgap-001L]

exists_algEquiv_pi_matrix_conjClasses
L∃∀N

Definition. characters, class functions, and the character table [fgap-001M]

basisOfIrreducibleCharacters characterTable card_inv_mul_sum_characterTable_mul_characterTable_inv card_conjClass_mul_sum_characterTable_mul_characterTable_inv
L∃∀N

Theorem. primitive central idempotents as block projectors [fgap-001N]

primitiveCentralIdempotent asAlgebraHom_primitiveCentralIdempotent primitiveCentralIdempotent_mul_primitiveCentralIdempotent
L∃∀N

Theorem. the Frobenius--Schur trichotomy [fgap-0021]

frobeniusSchurIndicator_eq_one_or_eq_zero_or_eq_neg_one frobeniusSchurIndicator_eq_one_iff frobeniusSchurIndicator_eq_neg_one_iff frobeniusSchurIndicator_eq_zero_iff
L∃∀N

Remark. real types are not Schur indices [fgap-001Y]

Definition. conjugate constituents and the inertia subgroup [fgap-001O]

Theorem. the single orbit in Clifford restriction [fgap-001P]

isSemisimpleRepresentation_comp_subtype exists_isAtom_forall_nonempty_linearEquiv_conjSubrep
L∃∀N

Theorem. Clifford homogeneity and the semidirect route [fgap-001Q]
✍️source

Example. semidirect products make Clifford theory concrete [fgap-0020]

Remark. from a presentation to checked representation data [fgap-001R]

Definition. factor sets and twisted Group Algebras [fgap-001S]

of_mul_of lift isProjectiveRepEquivAlgHom
L∃∀N

Remark. the qualified Clifford bridge [fgap-001Z]

Remark. the inertia obstruction is projective [fgap-001T]

proposition. the canonical grading of a Group Algebra [fgap-001U]

AddMonoidAlgebra.grade.isInternal AddMonoidAlgebra.grade.gradedAlgebra AddMonoidAlgebra.single_mem_grade
L∃∀N

proposition. the standard Hopf structure on a Group Algebra [fgap-001V]

MonoidAlgebra.instHopfAlgebra MonoidAlgebra.counit_single MonoidAlgebra.comul_single MonoidAlgebra.antipode_single
L∃∀N

Proof.

Remark. what the real quaternion example settles [fgap-001W]

Remark. the Hessian-group outlook [fgap-001X]

Example. semidirect products make Clifford theory concrete [fgap-0020]

Remark. the qualified Clifford bridge [fgap-001Z]

Theorem. the single orbit in Clifford restriction [fgap-001P]

isSemisimpleRepresentation_comp_subtype exists_isAtom_forall_nonempty_linearEquiv_conjSubrep
L∃∀N

Remark. two routes from binary tetrahedral symmetry [fgap-001E]

The Group-Algebra route

The McKay route