General block-size braid-group action conjecture

Let $m$ be the size of the matrix blocks, let ${\mathbb A}_{I,I}$ be the corresponding block of the operator matrix, and let $a$ and $b$ be the exponents appearing in the quantum braid-group action formulas and. The inverse ${\mathbb A}_{I,I}^{-1}$ is determined by

$${\mathbb A}_{I,I}{\mathbb A}_{I,I}^{-1}={\mathbb E},$$

and the conjugation rules are those in.

General block-size braid-group action conjecture. The braid-group action for general $m\times m$ matrix blocks should have the same form as and, with

$$a=b+1,\qquad b=m-1.$$

This extends the explicitly checked cases $m=1$ and $m=2$ and predicts the parameters and inverse-block construction needed for the quantum action in arbitrary block size; the general case is presented as plausible rather than established.