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 H N( q) in which the generators of the algebra act locally on the Nth tensor power of the space [Formula: see text]. 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.