ergodicity and the torelli subgroup⁷

in 2008, gelander proved that if G is a compact, connected, semisimple lie group, the orbit of almost any point in Hom(F_n, G) under the natural action of the automorphism group of Aut(F_n) is dense, and furthermore that this action is ergodic. in this paper, we generalize this result to the torelli subgroup, a normal subgroup of Aut(F_n). that is, we prove that the orbit of almost any point in Hom(F_n, G) under action of Tr(n) is dense and that the action is ergodic.

tokuyama's formula⁸

tokuyama’s formula connects combinatorics and representation theory by interpreting an expression involving the characters of general linear groups as a sum over combinatorial objects (e.g. gelfand-tsetlin patterns, shifted tableaux, or gamma ice models). this paper reviews existing proofs of tokuyama’s formula, presents two novel proofs avoiding machinery required by previous proofs, and describes progress in extending these results toward an analogue for symplectic groups.