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It is shown that finite-dimensional irreducible 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: $p^3, p^3/2, p^3/4$, or $p^3/8$. The topology
of the space of states ranges between two extremes, from a three-dimensional
torus $S^1
\times S^1 \times S^1$ (which may be thought of as a generalization
of the cyclic representation) to a
three-dimensional cube $[0,1]\times [0,1] \times [0,1]$.
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