The discriminant of the Brauer algebra

In the diagram basis the structure constants of the Brauer algebra B_k(δ) are powers of the loop value δ: two diagrams multiply to δ ^ c times their stack, where c is the number of closed loops left in the middle. So the trace of left multiplication by a product of two diagrams is an integer polynomial in δ, namely

tr(L_{D D'}) = δ ^ c(D, D') · ∑_{E, (D ∘ D') ∘ E = E} δ ^ c(D ∘ D', E),

and the Gram determinant of the regular trace form in the diagram basis, the discriminant TauCeti.BrauerAlgebra.discr k of the Brauer algebra on k strands, is a single integer polynomial whose value at δ is the discriminant of B_k(δ), over every commutative ring (TauCeti.BrauerAlgebra.aeval_discr).

By the trace-form criterion for semisimplicity, B_k(δ) is semisimple over a field wherever the discriminant does not vanish at δ, and over a field of characteristic zero exactly there (TauCeti.BrauerAlgebra.isSemisimpleRing_iff_aeval_discr_ne_zero). This is how the semisimplicity of the Brauer algebra at a generic loop value is reduced to the nonvanishing of one polynomial, that is, to its semisimplicity at a single loop value.

Main definitions

• TauCeti.BrauerAlgebra.discr: the discriminant of the Brauer algebra on k strands, an integer polynomial in the loop value.

Main results

• TauCeti.BrauerAlgebra.trace_mulLeft_of_mul_of: the trace of left multiplication by the product of two diagrams.
• TauCeti.BrauerAlgebra.aeval_discr: the value of the discriminant at δ is the Gram determinant of the regular trace form of B_k(δ) in the diagram basis.
• TauCeti.BrauerAlgebra.isSemisimpleRing_of_aeval_discr_ne_zero: over a field, B_k(δ) is semisimple when the discriminant does not vanish at δ.
• TauCeti.BrauerAlgebra.isSemisimpleRing_iff_aeval_discr_ne_zero: over a field of characteristic zero, B_k(δ) is semisimple exactly when the discriminant does not vanish at δ.

References

• R. Brauer, On algebras which are connected with the semisimple continuous groups, Annals of Mathematics 38 (1937), 857-872.
• H. Wenzl, On the structure of Brauer's centralizer algebras, Annals of Mathematics 128 (1988), 173-193.

noncomputable def TauCeti.BrauerAlgebra.discr (k : ℕ) : Polynomial ℤ

The discriminant of the Brauer algebra on k strands: the Gram determinant of the regular trace form (x, y) ↦ tr(L_{xy}) in the diagram basis, as an integer polynomial in the loop value. Its value at δ is the discriminant of B_k(δ) (TauCeti.BrauerAlgebra.aeval_discr).

theorem TauCeti.BrauerAlgebra.trace_mulLeft_of_mul_of {k : ℕ} {R : Type u_1} [CommSemiring R] {δ : R} (D D' : BrauerDiagram k) : (LinearMap.trace R (BrauerAlgebra R δ k)) (LinearMap.mulLeft R (of D * of D')) = δ ^ middleLoopCount D D' * ∑ E : BrauerDiagram k, if composeDiagram (composeDiagram D D') E = E then δ ^ middleLoopCount (composeDiagram D D') E else 0

The trace of left multiplication by a product of two diagrams in the Brauer algebra: it is δ ^ c(D, D') times the sum of δ ^ c(D ∘ D', E) over the diagrams E fixed by stacking D ∘ D' on top.

theorem TauCeti.BrauerAlgebra.aeval_discr {k : ℕ} {R : Type u_1} [CommRing R] (δ : R) : (Polynomial.aeval δ) (discr k) = ((LinearMap.BilinForm.toMatrix (basis R δ k)) (Algebra.regularTraceForm R (BrauerAlgebra R δ k))).det

The value of the discriminant at δ is the Gram determinant of the regular trace form of the Brauer algebra B_k(δ) in the diagram basis.

theorem TauCeti.BrauerAlgebra.isSemisimpleRing_of_aeval_discr_ne_zero {k : ℕ} {K : Type u_1} [Field K] {δ : K} (h : (Polynomial.aeval δ) (discr k) ≠ 0) : IsSemisimpleRing (BrauerAlgebra K δ k)

The Brauer algebra is semisimple where the discriminant does not vanish. Over every field, if the discriminant of the Brauer algebra on k strands does not vanish at δ, then B_k(δ) is semisimple.

theorem TauCeti.BrauerAlgebra.isSemisimpleRing_iff_aeval_discr_ne_zero {k : ℕ} {K : Type u_1} [Field K] [CharZero K] {δ : K} : IsSemisimpleRing (BrauerAlgebra K δ k) ↔ (Polynomial.aeval δ) (discr k) ≠ 0

The semisimplicity locus of the Brauer algebra: over a field of characteristic zero, the Brauer algebra B_k(δ) is semisimple exactly when the discriminant does not vanish at δ.