The commutant of the orthogonal group on the tensor square

Let K be a field and V = Kⁿ with its standard symmetric bilinear form. The Brauer algebra B₂(n) on two strands has three diagrams: the identity, the crossing σ and the cap-cup e. Its action TauCeti.brauerActionOrth on V ⊗ V sends them to the identity, the flip of the two tensor factors, and the map x ⊗ y ↦ ⟨x, y⟩ ∑ⱼ eⱼ ⊗ eⱼ. These operators commute with the diagonal action g ↦ g ⊗ g of the orthogonal group O(n, K) (TauCeti.range_brauerActionOrth_le_centralizer). This file proves the converse on two strands when 2 ≠ 0 in K and the unit circle of K has a point (c, s) with c s ≠ 0, in particular in characteristic zero: every operator on V ⊗ V commuting with every g ⊗ g is a combination of the identity, σ and e, so the image of B₂(n) is exactly the commutant of O(n, K) (TauCeti.range_brauerActionOrth_two_eq_centralizer_of_sq_add_sq_eq_one, and TauCeti.range_brauerActionOrth_two_eq_centralizer in characteristic zero). This is the two-strand case of Brauer's description of the commutant of the orthogonal group on V^{⊗k}, the first fundamental theorem of invariant theory for O(n).

The proof is a computation with the matrix M of such an operator in the monomial basis eₐ ⊗ e_b, using three kinds of orthogonal matrices.

• The diagonal reflections diag(±1, …, ±1) force M to vanish at every entry whose four indices do not pair off. An entry M_{(a,b),(c,d)} can only be nonzero when a = b and c = d, when (a, b) = (c, d), or when (a, b) = (d, c).
• The permutation matrices make M invariant under renaming the indices, so on each of these three patterns with two distinct indices M is constant. So M = q • 1 + r • σ + p • e + y • Δ, where Δ is the diagonal projection onto the span of the eₐ ⊗ eₐ.
• A reflection g with g₀₀ g₀₁ ≠ 0 does not commute with Δ, so y = 0.

The last step needs such a reflection. Its entries come from a point (c, s) of the unit circle with c s ≠ 0, which exists over every field of characteristic zero, for instance (3/5, 4/5); the diagonal reflections need 2 ≠ 0. The unit-circle condition cannot be dropped: over 𝔽₃ with n = 2 the orthogonal group consists of the signed permutation matrices, and Δ commutes with all of them without lying in the image of B₂(2).

Main results

• TauCeti.eq_and_eq_or_eq_or_eq_and_eq_of_forall_update_mul_eq: when 2 ≠ 0, if every sign vector Function.update 1 a (-1) gives the same product of signs on two pairs of indices, the four indices pair off.
• TauCeti.range_brauerActionOrth_two_eq_centralizer_of_sq_add_sq_eq_one: over a field with 2 ≠ 0 and a point (c, s) of the unit circle with c ≠ 0 and s ≠ 0, the image of the Brauer algebra B₂(n) is the commutant of the orthogonal group on (Kⁿ)^{⊗2}.
• TauCeti.range_brauerActionOrth_two_eq_centralizer: the same over every field of characteristic zero.

References

• R. Brauer, On algebras which are connected with the semisimple continuous groups, Annals of Mathematics 38 (1937), 857-872.
• R. Goodman and N. R. Wallach, Symmetry, Representations, and Invariants, Springer GTM 255 (2009), Chapter 5 (the first fundamental theorem for the orthogonal group) and §10.1.

theorem TauCeti.eq_and_eq_or_eq_or_eq_and_eq_of_forall_update_mul_eq {R : Type u_1} {ι : Type u_2} [CommRing R] [DecidableEq ι] (h2 : 2 ≠ 0) {v u : Fin 2 → ι} (h : ∀ (a : ι), Function.update 1 a (-1) (v 0) * Function.update 1 a (-1) (v 1) = Function.update 1 a (-1) (u 0) * Function.update 1 a (-1) (u 1)) : v 0 = v 1 ∧ u 0 = u 1 ∨ v = u ∨ u 0 = v 1 ∧ u 1 = v 0

Let 2 ≠ 0 in R. If, for every a, the sign vector Function.update 1 a (-1) (the diagonal of TauCeti.orthogonalDiagonalReflection R a) gives the same product of signs on the two pairs of indices (v 0, v 1) and (u 0, u 1), then the four indices pair off: the two pairs are both constant, or equal, or swapped.

theorem TauCeti.range_brauerActionOrth_two_eq_centralizer_of_sq_add_sq_eq_one (K : Type u_1) [Field K] (n : ℕ) (h2 : 2 ≠ 0) {c s : K} (hcs : c ^ 2 + s ^ 2 = 1) (hc : c ≠ 0) (hs : s ≠ 0) : (brauerActionOrth K n 2).range = Subalgebra.centralizer K (Set.range ⇑((stdOrthogonalRep K n).tensorPower 2))

Brauer duality on the tensor square. Over a field K with 2 ≠ 0 that has a point (c, s) of the unit circle with c ≠ 0 and s ≠ 0, the image of the Brauer algebra B₂(n) in the endomorphisms of (Kⁿ)^{⊗2} is the commutant of the diagonal action of the orthogonal group: an operator commuting with every g ⊗ g is a combination of the identity, the flip of the two tensor factors and the cup-cap generator.

theorem TauCeti.range_brauerActionOrth_two_eq_centralizer (K : Type u_1) [Field K] [CharZero K] (n : ℕ) : (brauerActionOrth K n 2).range = Subalgebra.centralizer K (Set.range ⇑((stdOrthogonalRep K n).tensorPower 2))

Brauer duality on the tensor square in characteristic zero. Over a field of characteristic zero, the image of the Brauer algebra B₂(n) in the endomorphisms of (Kⁿ)^{⊗2} is the commutant of the diagonal action of the orthogonal group.