The Weyl group of the Cuntz algebra

The Weyl group of the Cuntz algebra O_n is investigated. This is (isomorphic to) the group of polynomial automorphisms lambda_u of O_n, namely those induced by unitaries u that can be written as finite sums of words in the canonical generating isometries S_i and their adjoints. A necessary and sufficient algorithmic combinatorial condition is found for deciding when a polynomial endomorphism lambda_u restricts to an automorphism of the canonical diagonal MASA. Some steps towards a general criterion for invertibility of lambda_u on the whole of O_n are also taken. A condition for verifying invertibility of a certain subclass of polynomial endomorphisms is given. First examples of polynomial automorphisms of O_n not inner related to those induced by unitaries from the core UHF subalgebra are exhibited, for every n. In particular, the image of the Weyl group in the outer automorphism group of O_n is strictly larger than the image of the restricted Weyl group analyzed in previous papers. Results about the action of the Weyl group on the spectrum of the diagonal are also included.