Representations of The Coordinate Ring of $ GL_{q}(n) $}

It is shown that the finite dimensional irreducible representations of the quantum matrix algebra $M_q(n)$ ( the coordinate ring of $GL_q(n)$ ) exist only when q is a root of unity ( $q^p = 1$ ). The dimensions of these representations can only be one of the following values: ${p^N \over 2^k }$ where $N = {n(n-1)\over 2 }$ and $k \in \{ 0, 1, 2, . . . N \}$ For each $k$ the topology of the space of states is $(S^1)^{\times(N-k)} \times [ 0 , 1 ] ^{(\times (k)}$ (i.e. an $N$ dimensional torus for $k=0$ and an $N$ dimensional cube for $k = N$ ).