On semisimple representations of universal lattices
Daniel K. Shenfeld · Groups Geometry and Dynamics · 2009
We study finite-dimensional semisimple complex representations of the universal lattices Γ_{n,k} = \mathrm{SL}_n(ℤ[x_1, \ldots, x_k]) ( n ≥ 3 ). One may obtain such a representation by specializing x_1, \ldots, x_k to some complex values and composing the induced homomorphism Γ_{n,k} → \mathrm{SL}_n(ℂ) with a rational representation of \mathrm{SL}_n(ℂ) . We show that any semisimple representation coincides, on a subgroup of finite index, with a direct sum of tensor products of representations obtained in this way.