arXiv:2608.18328 2026 Architecture 2 ideas extracted · analyzed Sep 1, 2026
The paper identifies the δ=2 Brauer planar algebra with the D∞ subfactor and provides exact diagrammatic operations: permutations, orthogonal contractions, cups, caps, and dimension-specific annihilation relations. These are precisely the tensor-network generators of O(2)-equivariant maps, while the identity A_3=0 expresses that the third exterior power of a two-dimensional representation vanishes. A practical transfer is to replace unconstrained tensor mixing with Brauer-generated equivariant layers and to use exterior-power relations to eliminate redundant channels before training. The most direct test is an O(2)-equivariant graph or attention layer on randomly rotated vector data.
Construct a neural mixing layer only from Brauer generators for the orthogonal group: identity, pairwise swaps, and pairwise contractions with the Euclidean metric. This gives an exactly O(2)-equivariant alternative to unconstrained tensor mixing, with trainable coefficients but fixed symmetry-preserving basis maps.
Use the dimension-specific relation A_3=0 to remove all intermediate channels transforming as the third exterior power of the two-dimensional vector representation. In tensor-product attention or equivariant MLPs, this is an exact algebraic pruning rule rather than approximate low-rank compression.