Representations of The Coordinate Ring of $GL_{q}(3)$

It is shown that the finite dimensional ireducible representations of the quantum matrix algebra $M_q(3)$ ( the coordinate ring of $GL_q(3)$ ) 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: $p^3 , { p^3 \over 2 } , { p^3 \over 4 }$ or ${ p^3 \over 8 }$ . The topology of the space of states ranges between two extremes , from a 3-dimensional torus $S^1 \times S^1 \times S^1$ ( which may be thought of as a generalization of the cyclic representation ) to a 3-dimensional cube $[ 0 , 1 ]\times [ 0 , 1 ]\times [ 0 , 1 ]$ .