An automaton-theoretic approach to the representation theory of quantum algebras

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).