Semisimple representations are determined by their characters

Over a field of characteristic zero, two finite-dimensional semisimple representations of a monoid with the same character are equivalent (Representation.nonempty_equiv_of_character_eq_of_isSemisimpleRepresentation). The monoid need not be finite: for a finite group in characteristic zero every representation is semisimple by Maschke's theorem, but the statement here is what applies to an infinite group such as GL n ℂ, once complete reducibility is known by other means.

The character determines the trace of every element a = ∑ a_g g of the monoid algebra k[G] acting on the representation, namely ∑ a_g χ(g). So the k[G]-modules of the two representations are semisimple with equal trace functions, and IsSemisimpleModule.nonempty_linearEquiv_of_trace_eq identifies them.

Main results

• Representation.trace_asAlgebraHom_eq_of_character_eq: representations with the same character have the same trace function on k[G].
• Representation.nonempty_equiv_of_character_eq_of_isSemisimpleRepresentation: semisimple representations with the same character are equivalent.

References

• S. Lang, Algebra, revised third edition, Chapter XVII, Corollary 3.8.

theorem Representation.trace_asAlgebraHom_eq_of_character_eq {k : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [Field k] [Monoid G] [AddCommGroup V] [Module k V] [AddCommGroup W] [Module k W] {ρ : Representation k G V} {σ : Representation k G W} (h : ρ.character = σ.character) (a : MonoidAlgebra k G) : (LinearMap.trace k V) (ρ.asAlgebraHom a) = (LinearMap.trace k W) (σ.asAlgebraHom a)

The character determines the trace on the monoid algebra. If two representations have the same character, then every element of k[G] acts on them with the same trace.

theorem Representation.nonempty_equiv_of_character_eq_of_isSemisimpleRepresentation {k : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} [Field k] [Monoid G] [AddCommGroup V] [Module k V] [AddCommGroup W] [Module k W] [CharZero k] [FiniteDimensional k V] [FiniteDimensional k W] (ρ : Representation k G V) (σ : Representation k G W) [ρ.IsSemisimpleRepresentation] [σ.IsSemisimpleRepresentation] (h : ρ.character = σ.character) : Nonempty (ρ.Equiv σ)

Semisimple representations are determined by their characters. Over a field of characteristic zero, two finite-dimensional semisimple representations of a monoid with the same character are equivalent. The monoid need not be finite.