  We consider the problem of finding orthogonal projections $P$ of a rank $r$
that give rise to representations of the Hecke algebra $H_N(q)$ in which the
generators of the algebra act locally on the $N$-th tensor power of the space
${\mathbb C}^n$. 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 $k=r n$ if and only if $P$
is of the Temperley-Lieb type. Besides, we propose an approach to constructing
projections $P$ and give some novel examples for $n=3$.
