The orthogonal Brauer action is faithful in large dimension; B_k(δ) is generically semisimple

The Brauer algebra B_k(n) acts on the tensor power (Rⁿ)^{⊗k} through the standard symmetric form (TauCeti.brauerActionOrth), a diagram D acting by the matrix M(D) whose entry at a labelling Sum.elim u v of the boundary points is 1 when the labelling is constant on the arcs of D, and 0 otherwise. In small dimension this action is not faithful. Once k ≤ n, however, it is: the k arcs of a diagram D can be given pairwise distinct labels, and the resulting labelling is constant on the arcs of D and of no other diagram (TauCeti.BrauerDiagram.exists_forall_isConstantOnArcs_iff_eq). So the entry of ∑ D, c_D • M(D) at that labelling is the single coefficient c_D, the matrices of the diagrams are linearly independent (TauCeti.BrauerDiagram.linearIndependent_orthogonalActionMatrix), and the action is injective over every commutative semiring (TauCeti.brauerActionOrth_injective).

Over a star-ordered field K, such as ℚ, ℝ or ℂ, faithfulness makes the Brauer algebra B_k(n) semisimple for k ≤ n (TauCeti.BrauerAlgebra.isSemisimpleRing_of_le). The matrices of the diagrams have entries 0 and 1, and the transpose of M(D) is the matrix of the reflected diagram D.flip, so the image of the action is closed under conjugate transposition. If x lies in the Jacobson radical, which is a nilpotent ideal, then so does y * x for the element y = ∑ D, star c_D • D.flip acting by the conjugate transpose of the matrix A of x; hence Aᴴ * A is a nilpotent self-adjoint matrix, so it vanishes, so A = 0, and x = 0 by faithfulness. The radical is zero, and a finite-dimensional algebra with zero radical is semisimple.

The loop value here is the dimension n itself, at least the number k of strands. A single such loop value is enough to reach the generic one. The Gram determinant of the regular trace form of B_k(δ) in the diagram basis is the value at δ of one integer polynomial, the discriminant TauCeti.BrauerAlgebra.discr k, and B_k(δ) is semisimple wherever it does not vanish. Over ℚ, a field of characteristic zero, the converse holds as well, so the discriminant does not vanish at δ = k, where B_k(k) is semisimple; it is a nonzero polynomial (TauCeti.BrauerAlgebra.discr_ne_zero). Hence B_k(δ) is semisimple at every loop value δ transcendental over ℤ, in any field (TauCeti.BrauerAlgebra.isSemisimpleRing_of_transcendental), and over a field of characteristic zero it fails to be semisimple at only finitely many loop values (TauCeti.BrauerAlgebra.finite_setOfPred_not_isSemisimpleRing). Which loop values these are, for instance whether the symplectic value -2l is among them, this argument does not decide.

Main results

• TauCeti.BrauerDiagram.linearIndependent_orthogonalActionMatrix: with at least k labels, the matrices of the Brauer diagrams on k strands are linearly independent.
• TauCeti.brauerActionOrth_injective: the action of B_k(n) on (Rⁿ)^{⊗k} is faithful when k ≤ n.
• TauCeti.BrauerAlgebra.isSemisimpleRing_of_le: the Brauer algebra B_k(n) is semisimple over a star-ordered field when k ≤ n.
• TauCeti.BrauerAlgebra.discr_ne_zero: the discriminant of the Brauer algebra is a nonzero polynomial in the loop value.
• TauCeti.BrauerAlgebra.isSemisimpleRing_of_transcendental: the Brauer algebra is semisimple at a transcendental loop value.
• TauCeti.BrauerAlgebra.finite_setOfPred_not_isSemisimpleRing: over a field of characteristic zero, the Brauer algebra is semisimple at all but finitely many loop values.

References

• R. Brauer, On algebras which are connected with the semisimple continuous groups, Annals of Mathematics 38 (1937), 857-872.
• W. P. Brown, An algebra related to the orthogonal group, Michigan Mathematical Journal 3 (1955), 1-22.
• R. Goodman and N. R. Wallach, Symmetry, Representations, and Invariants, Springer GTM 255 (2009), §10.1.

theorem TauCeti.BrauerDiagram.linearIndependent_orthogonalActionMatrix (R : Type u) [CommSemiring R] (ι : Type u_1) [DecidableEq ι] [Fintype ι] {k : ℕ} (hk : k ≤ Fintype.card ι) : LinearIndependent R fun (D : BrauerDiagram k) => orthogonalActionMatrix R ι D

The matrices of the Brauer diagrams are linearly independent as soon as there are at least as many labels as strands.

theorem TauCeti.brauerActionOrth_injective (R : Type u) [CommSemiring R] (n k : ℕ) (hk : k ≤ n) : Function.Injective ⇑(brauerActionOrth R n k)

The orthogonal Brauer action is faithful in large dimension: when the number k of strands is at most the dimension n, distinct elements of B_k(n) act differently on (Rⁿ)^{⊗k}.

theorem TauCeti.BrauerAlgebra.isSemisimpleRing_of_le {n k : ℕ} {K : Type u_1} [Field K] [PartialOrder K] [StarRing K] [StarOrderedRing K] (hk : k ≤ n) : IsSemisimpleRing (BrauerAlgebra K (↑n) k)

The Brauer algebra B_k(n) is semisimple over a star-ordered field, such as ℚ, ℝ or ℂ (with ComplexOrder), when the loop value n is a natural number at least the number k of strands.

Generic loop values

theorem TauCeti.BrauerAlgebra.discr_ne_zero (k : ℕ) : discr k ≠ 0

The discriminant of the Brauer algebra is a nonzero polynomial.

theorem TauCeti.BrauerAlgebra.isSemisimpleRing_of_transcendental {k : ℕ} {K : Type u_1} [Field K] {δ : K} (hδ : Transcendental ℤ δ) : IsSemisimpleRing (BrauerAlgebra K δ k)

The Brauer algebra is semisimple at a transcendental loop value: over every field, if δ is a root of no nonzero integer polynomial then B_k(δ) is semisimple. In positive characteristic no element is transcendental over ℤ, so this is a statement about fields of characteristic zero.

theorem TauCeti.BrauerAlgebra.finite_setOfPred_not_isSemisimpleRing (k : ℕ) (K : Type u_1) [Field K] [CharZero K] : {δ : K | ¬IsSemisimpleRing (BrauerAlgebra K δ k)}.Finite

The Brauer algebra is generically semisimple: over a field of characteristic zero, B_k(δ) is semisimple for all but finitely many loop values δ, the exceptions being roots of the discriminant.