Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type An−1A_{n-1}

It is known that the Lawrence-Krammer representation of the Artin group of type $A_{n-1}$ based on the two parameters $t$ and $q$ that was used by Krammer and independently by Bigelow to show the linearity of the braid group on $n$ strands is generically irreducible. Here, we recover this result and show further that for some complex specializations of the parameters the representation is reducible. We give all the values of the parameters for which the representation is reducible as well as the dimensions of the invariant subspaces. We deduce some results of semisimplicity of the Birman-Murakami-Wenzl algebra of type $A_{n-1}$.Comment: 8 page

A quantum combinatorial approach for computing a tetrahedral network of Jones-Wenzl projectors

Trivalent plane graphs are used in various areas of mathematics which relate for instance to the colored Jones polynomial, invariants of 3-manifolds and quantum computation. Their evaluation is based on computations in the Temperley-Lieb algebra and more specifically the Jones-Wenzl projectors. We use the work by Kauffman-Lins to present a quantum combinatorial approach for evaluating a tetrahedral net. On the way we recover two equivalent definitions for the unsigned Stirling numbers of the first kind and we provide an equality for the quantized factorial using these numbers.Comment: 25 pages, 17 figure