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

For a constituent \(W\) with inertia subgroup \(I=I_G(W)\), choose intertwiners \(T_i:W\to {}^iW\). Their composites need not satisfy the group law strictly. Instead one obtains scalars \[ T_iT_j=\alpha (i,j)T_{ij}. \] Associativity makes \(\alpha \) a factor set. Changing the intertwiners changes \(\alpha \) by a coboundary. The resulting cohomology class measures the obstruction to replacing the projective action by an ordinary action.

Twisted Group Algebras therefore enter the Clifford-theory route. The ordinary skew Group Algebra in two routes from binary tetrahedral symmetry corresponds to trivial twisting; it should not be identified with a general \(k_\alpha [G]\). The extension and induction analysis is developed in [lux2010representations, secs. 3.6--3.7, pp. 222--240].

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

Reference. Representations of groups: A computational approach [lux2010representations]

The human-curated Group-Algebra route [fgap-001G]
📄PDF

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]

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]
✍️source

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]

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

The Group-Algebra route

The McKay route