On orthogonal projections related to representations of the Hecke algebra on a tensor space

Abstract

We consider the problem of finding orthogonal projections P of a rank r that gives rise to representations of the Hecke algebra HN(q) in which the generators of the algebra act locally on the Nth tensor power of the space Cn. It is shown that such projections are global minima of a certain functional. It is also shown that a characteristic property of such projections is that a certain positive definite matrix A has only two eigenvalues or only one eigenvalue if P gives rise to a representation of the Temperley–Lieb algebra. Apart from the parameters n, r, and Q = q + q−1, an additional parameter k proves to be a useful characteristic of a projection P. In particular, we use it to provide a lower bound for Q when the values of n and r are fixed and we show that k = rn if and only if P is of the Temperley–Lieb type. In addition, we propose an approach to constructing projections P and give some novel examples for n = 3.