I was reading the paper Braids, Link Polynomials and A New Algebra by J. S. Birman and H. Wenzl, and I was wondering is there a combinatorial way to compute the dimension of the algebras $\mathscr{C}_n(l, m)$ other than comparing the dimension with that of Brauer's centralizer algebra?
I know that using the Bratteli diagram associated to $\mathscr{C}_n(l, m)$ inductively one can solve the problem, but it seems even more difficult to figure out the dimension of each of the simple module a vertex stands for on the $n$-th row than to find the dimension of the algebra...
