Non-triviality of the Poisson boundary of random walks on the group of Monod

We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by Monod [Groups of piecewise projective homeomorphisms. Proc. Natl Acad. Sci. USA110(12) (2013), 4524–4527]. For a finitely generated subgroup $H$ of $H(\mathbb{Z})$ , we prove that either $H$ is solvable or every measure on $H$ with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson’s group $F$ that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich [Thompson’s group $F$ is not Liouville. Groups, Graphs and Random Walks (London Mathematical Society Lecture Note Series). Eds. T. Ceccherini-Silberstein, M. Salvatori and E. Sava-Huss. Cambridge University Press, Cambridge, 2017, pp. 300–342, 7.A].

Affine quantum Schur algebras at roots of unity

Finite dimensional irreducible modules for the affine quantum Schur algebra [Formula: see text] were classified in [B. Deng, J. Du and Q. Fu, A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory, London Mathematical Society Lecture Note Series, Vol. 401 (Cambridge University Press, Cambridge, 2012), Chapt. 4] when [Formula: see text] is not a root of unity. We will classify finite-dimensional irreducible modules for affine quantum Schur algebras at roots of unity and generalize [J. A. Green, Polynomial Representations of [Formula: see text] , 2nd edn., with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J. A. Green and M. Schocker, Lecture Notes in Mathematics, Vol. 830 (Springer-Verlag, Berlin, 2007), (6.5f) and (6.5g)] to the affine case in this paper.