Abstract
We develop a new approach to the representation theory of quantum algebras supporting a torus action via methods from the theory of finite-state automata and algebraic combinatorics. We show that for a fixed number m, the torus-invariant primitive ideals in m × n quantum matrices can be seen as a regular language in a natural way. Using this description and a semigroup approach to the set of Cauchon diagrams, a combinatorial object that parameterizes the primes that are torus-invariant, we show that for m fixed, the number P ( m , n ) of torus-invariant primitive ideals in m × n quantum matrices satisfies a linear recurrence in n over the rational numbers. In the 3 × n case we give a concrete description of the torus-invariant primitive ideals and use this description to give an explicit formula for the number P ( 3 , n ) .
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