Brauer duality for the orthogonal group on two strands

Let V = Rⁿ carry its standard symmetric form. The Brauer algebra B₂(n) on two strands acts on V ⊗ V (TauCeti.brauerActionOrth) through its three diagrams: the identity, the crossing, which flips the two tensor factors, and the cap-cup diagram, which acts by the contraction-expansion x ⊗ y ↦ (x ⬝ᵥ y) • ∑ᵢ eᵢ ⊗ eᵢ. These operators commute with the diagonal action of the orthogonal group (TauCeti.range_brauerActionOrth_le_centralizer). This file proves the converse on two strands: every operator on V ⊗ V commuting with the orthogonal group is a combination of them, so the image of B₂(n) is the whole commutant (TauCeti.range_brauerActionOrth_two_eq_centralizer_of_noZeroDivisors). This is the two-strand case of Brauer's theorem that the image of B_k(n) is the commutant of the orthogonal group on V^{⊗k}.

The theorem holds over a commutative ring without zero divisors in which 2 ≠ 0 and which contains a point (c, s) of the unit circle with c * s ≠ 0, for instance over any field of characteristic zero, with (3 / 5, 4 / 5) (TauCeti.range_brauerActionOrth_two_eq_centralizer in TauCeti.RepresentationTheory.ClassicalGroups.BrauerGenerators.Orthogonal.Commutant). Some such point is needed: over ZMod 3 the orthogonal group of the plane consists of the signed permutation matrices, and these commute with the projection onto the span of e₀ ⊗ e₀ and e₁ ⊗ e₁, which is not in the image.

The argument

In the monomial basis an operator on V ⊗ V is a matrix M whose rows and columns are indexed by pairs of labels, and commuting with g ⊗ g is commuting with the Kronecker square of g.

• Signed permutations. The diagonal sign matrix negating one label is orthogonal, so an entry of M vanishes as soon as some label occurs an odd number of times among its four indices (Matrix.apply_eq_zero_of_commute_kroneckerPower). Permutation matrices are orthogonal, so an entry depends only on the pattern of equalities among its indices (Matrix.apply_comp_comp_of_commute_kroneckerPower). The surviving patterns are those of the identity, the crossing and the cap-cup diagram, together with the pattern in which all four indices agree. So M is a combination of the three diagram matrices and of the projection D onto the span of the tensors e_a ⊗ e_a.
• A rotation. The diagram matrices commute with every orthogonal matrix, but D does not commute with the plane rotation A through (c, s) (TauCeti.exists_orthogonal_apply_mul_apply_ne_zero): the two products with its Kronecker square differ at the entry (e_a ⊗ e_a, e_a ⊗ e_b) by A a a * A a b = -c * s ≠ 0. So the coefficient of D vanishes.

In dimension at most one the tensor square has rank at most one, and every operator is a scalar. In dimension at least two the action of B₂(n) is faithful (TauCeti.brauerActionOrth_injective), so the commutant has rank three (TauCeti.finrank_centralizer_stdOrthogonalRep_tensorPower_two).

Main results

• TauCeti.range_brauerActionOrth_two_eq_centralizer_of_noZeroDivisors: Brauer duality on two strands, the image of B₂(n) is the commutant of the orthogonal group on Rⁿ ⊗ Rⁿ.
• TauCeti.finrank_centralizer_stdOrthogonalRep_tensorPower_two: the commutant has rank three once n ≥ 2.

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), §10.1.

The matrices commuting with the orthogonal group on two strands

Brauer duality on two strands

theorem TauCeti.range_brauerActionOrth_two_eq_centralizer_of_noZeroDivisors (R : Type u) [CommRing R] (n : ℕ) [NoZeroDivisors R] (h2 : 2 ≠ 0) (hrot : ∃ (c : R) (s : R), c ^ 2 + s ^ 2 = 1 ∧ c * s ≠ 0) : (brauerActionOrth R n 2).range = Subalgebra.centralizer R (Set.range ⇑((stdOrthogonalRep R n).tensorPower 2))

Brauer duality for the orthogonal group on two strands. Over a commutative ring without zero divisors in which 2 ≠ 0 and which contains a point (c, s) of the unit circle with c * s ≠ 0, the operators on Rⁿ ⊗ Rⁿ commuting with the diagonal action of the orthogonal group are exactly the image of the Brauer algebra B₂(n): the combinations of the identity, the flip of the two factors, and the contraction-expansion x ⊗ y ↦ (x ⬝ᵥ y) • ∑ᵢ eᵢ ⊗ eᵢ.

Some point of the circle with c * s ≠ 0 is needed when n = 2: over ZMod 3 the orthogonal group of the plane consists of the signed permutation matrices, which commute with the projection onto the span of e₀ ⊗ e₀ and e₁ ⊗ e₁, and that projection is not in the image.

theorem TauCeti.finrank_centralizer_stdOrthogonalRep_tensorPower_two (R : Type u) [CommRing R] [IsDomain R] {n : ℕ} (hn : 2 ≤ n) (h2 : 2 ≠ 0) (hrot : ∃ (c : R) (s : R), c ^ 2 + s ^ 2 = 1 ∧ c * s ≠ 0) : Module.finrank R ↥(Subalgebra.centralizer R (Set.range ⇑((stdOrthogonalRep R n).tensorPower 2))) = 3

The commutant of the orthogonal group on Rⁿ ⊗ Rⁿ has rank three when n ≥ 2, under the hypotheses of TauCeti.range_brauerActionOrth_two_eq_centralizer_of_noZeroDivisors: it is the image of the Brauer algebra B₂(n), on which the action is faithful, and B₂(n) is free on its three diagrams.