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

作者：Bell J; Launois S^*; Lutley J

来源：Advances in Mathematics, 2010, 223(2): 476-510.

DOI：10.1016/j.aim.2009.08.013

• Primitive ideals
• Quantum matrices
• Cauchon diagrams
• Perfect matchings
• Pfaffians
• Automata

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 combinatories. We show that for a fixed number in, the torus-invariant primitive ideals in m x 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 x n quantum matrices satisfies a linear recurrence in n over the rational numbers. In the 3 x 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).

• 出版日期2010-1-30