Question

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 $n\times n$ 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
• $\sigma_2=P\sigma_1P$
• $\sigma_i^{-1}=\overline{\sigma}_i$ which means apply the involution $q\mapsto q^{-1}$ 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.

Answer 1

If it would be useful for you, I can show you explicit matrices for the 3-strand braid group with V the spin representation of $Spin(5)$ and $Spin(7)$, and probably $Spin(9)$ and $Spin(11)$ as well. These won't be in any particular basis of $Hom(\otimes^3 V, W)$, however.

Answer 2

The R-matrix is always upper triangular, for any basis of the tensor product which is compatible with the weight spaces on the two factors; depending on your convention, the R-matrix only decreases weight in the first factor and increases it in the second.

The second element in your list is essentially by definition: $\sigma_1$ and $\sigma_2$ involve doing the exact same thing to the first two or last two factors, so they are conjugate by, say, the map that cyclically permutes the factors.

I believe the last condition is what's usually called "unitarity" of the R-matrix, but I should probably just wait for Noah to show up and give a better answer for that one with references and such.