Braided finite automata and representation theory

Abstract

We introduce classical and non-deterministic finite automata associated with representations of the braid group. After briefly reviewing basic definitions on finite automata, Coxeter's groups and the associated word problem, we turn to the Artin presentation of the braid group and its quotients. We present various representations of the braid group as deterministic or non-deterministic finite state automata and discuss connections with $q$-Dicke states, as well as Lusztig and crystal bases. We propose the study of the eigenvalue problem of the $\mathfrak{U}_q(\mathfrak{gl}_n)$ invariant spin-chain like ``Hamiltonian'' as a systematic means for constructing canonical bases for irreducible representations of $\mathfrak{U}_q(\mathfrak{gl}_n).$ This is explicitly proven for the algebra $\mathfrak{U}_q(\mathfrak{gl}_2).$ Special braid representations associated with self-distributive structures are also studied as finite automata. These finite state automata organize clusters of eigenstates of these braid representations.