Roberto Conti: Automorphisms of the Cuntz algebras - small subgroups of outer reduced Weyl groups

Abstract: Weyl groups for the Cuntz algebras were introduced implicitly by Cuntz
himself at the end of the seventies. However, they remained largely
ignored probably because of computational difficulties until about 30
years later, when the breakthrough work by Conti and Szymanski allowed to
determine explicitly a huge number of their elements by a sophisticated
condition with a clear combinatorial flavour. In recent times, Brenti,
Conti and Nenashev pushed the boundaries of the involved combinatorial
structures, obtaining for instance the first enumerative results (at the
moment, only for cycles).
In this talk I will report on recent work by BCN (in preparation) where,
building on the combinatorial machinery developed over the last few years,
we construct certain subgroups of outer automorphisms of O_n. In
particular, we are able to describe in detail the 46 distinct finite groups of outer
automorphisms of O_4 lying in the outer reduced Weyl group and maximal at
level 2, which were first determined by Szymanski and collaborators by
clever computer-assisted methods. The notion of bicompatible subgroup of
the permutations of a square grid will play a role in the discussion.