The middle loops of a stack of three Brauer diagrams

The Brauer algebra B_k(δ) multiplies its diagram basis by loop-weighted stacking, D₁ * D₂ = δ ^ middleLoopCount D₁ D₂ • composeDiagram D₁ D₂. Stacking itself is associative (TauCeti.composeDiagram_assoc), so this multiplication is associative exactly when the powers of δ agree as well:

middleLoopCount D₁ D₂ + middleLoopCount (D₁ ∘ D₂) D₃
  = middleLoopCount D₂ D₃ + middleLoopCount D₁ (D₂ ∘ D₃),

writing D₁ ∘ D₂ for composeDiagram D₁ D₂. This file proves that identity, TauCeti.middleLoopCount_add_middleLoopCount_composeDiagram. Both sides count the closed loops of the stack of D₁ above D₂ above D₃.

Main results

• TauCeti.middleLoopCount_add_middleLoopCount_composeDiagram: the loop counts of the two bracketings of a stack of three Brauer diagrams agree, the loop-count half of the associativity of the Brauer algebra.

Implementation notes

The stack of three diagrams has two middle boundaries, an upper one between D₁ and D₂ and a lower one between D₂ and D₃, and its arcs that stay off the outer boundary form a graph on their points: the caps of D₁ and the cups of D₂ on the upper boundary, the caps of D₂ and the cups of D₃ on the lower one, and the through strands of D₂ between them. A closed loop is a connected component of that graph all of whose points keep both of their arcs inside it; the components are counted as classes of the quotient by the graph. In the bracketing (D₁ ∘ D₂) ∘ D₃, the loops lying on the upper boundary alone are the middle loops of D₁ above D₂, and the loops meeting the lower boundary are the middle loops of D₁ ∘ D₂ above D₃: a cap of D₁ ∘ D₂ is a cap of D₂, or else a strand running up through D₂, along the upper boundary, and down again.

Reflecting the stack in a horizontal line reverses it and exchanges its two middle boundaries (TauCeti.flip_composeDiagram, TauCeti.middleLoopCount_flip), so the count for the other bracketing D₁ ∘ (D₂ ∘ D₃) is the same count read upside down, and only one bracketing needs an argument. The stack of three and its loop count are private to this file.

References

• R. Brauer, On algebras which are connected with the semisimple continuous groups, Annals of Mathematics 38 (1937), 857-872.

The middle graph of a stack of three diagrams

Reflecting the stack of three

The strands of the stack of D₁ above D₂ in the stack of three

The loops of the stack of three meeting the lower boundary

The loops of the stack of three on the upper boundary alone

Counting the loops of the stack of three

theorem TauCeti.middleLoopCount_add_middleLoopCount_composeDiagram {k : ℕ} (D₁ D₂ D₃ : BrauerDiagram k) : middleLoopCount D₁ D₂ + middleLoopCount (composeDiagram D₁ D₂) D₃ = middleLoopCount D₂ D₃ + middleLoopCount D₁ (composeDiagram D₂ D₃)

The loop counts of the two bracketings of a stack of three Brauer diagrams agree. Stacking D₁ above D₂ and the composite above D₃ closes up as many loops in the middle as stacking D₂ above D₃ and D₁ above the composite. With TauCeti.composeDiagram_assoc this makes the loop-weighted stacking D₁ * D₂ = δ ^ middleLoopCount D₁ D₂ • composeDiagram D₁ D₂ associative, which is the associativity of the multiplication of the Brauer algebra on its diagram basis.