rt.representation theory - What is the explanation for the special form of representations of three string braid group constructed using quantum groups information supplied

It is well-known that representations of quantised enveloping algebras give representations of braid groups. For the examples that I know explicitly the representations of the three string braid group take a specific form. Is there an explanation of this? The examples I know are the simplest examples so what can I expect in general?

More specifically: Fix a quantised enveloping algebra $U$. Let $V$ and $W$ be highest weight finite dimensional representations. Then the three string braid group acts on $Hom_U(otimes^3V,W)$.

The specific form that appears is the following. Let $P$ be the $ntimes n$ matrix with $P_{ij}=1$ if $i+j=n+1$ and $P_{ij}=0$ otherwise. Then we can write $sigma_1$ and $sigma_2$ with the following properties

• $sigma_1$ is lower triangular


• $sigma_2=Psigma_1P$


• $sigma_i^{-1}=overline{sigma}_i$ which means apply the involution $qmapsto q^{-1}$ to each entry

The simplest example is

$$sigma_1=left(begin{array}{cc} q & 0 \ 1 & -q^{-1}end{array}right)$$

I get the feeling this has something to do with canonical bases.

A specific question is: Take $V$ to be the spin representation of $Spin(2n+1)$. Then do these representations have this form and if so how do I find it?

[In fact, I have representations of this specific form which I conjecture are these representations]

Further comment Assume the eigenvalues of $sigma_i$ are distinct. This condition holds for the spin representation. Then if this basis exists it is unique. Consider a change of basis matrix $A$ which preserves this structure. Then $A$ commutes with $sigma_1$ so is lower triangular. Then $A$ also commutes with $P$ so is diagonal. Then the final condition requires $A$ to be a scalar matrix.

The problem is existence. The Tuba-Wenzl paper shows such a basis exists in small examples.

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