Hilbert series of fixed free algebras and noncommutative classical invariant theory (Q1092984)

Let G be a group acting linearly on a finite-dimensional vector space V over a field of characteristic 0. The authors study the invariants \(T(V)^ G\) of G on the tensor algebra T(V). They prove a Molien-Weyl- type formula for the Poincaré series \(H(T(V)^ G)\) for a compact G (generalizing the earlier result of the last two authors for finite groups [see Linear Multilinear Algebra 12, 21-30 (1982; Zbl 0493.15020)]. They describe \(H(T(V)^ G)\) for a polynomial representation of \(G=SL_ r\) on V restricted to the group of unipotent \(r\times r\) upper triangular matrices. They show that \(H(T(V))^{SL_ 2}\), for a polynomial representation of \(SL_ 2\) on V, is algebraic over Z[t]. Finally, they develop a theory of invariants of a binary form with noncommuting coefficients analogous to the classical invariant theory of Sylvester and Franklin, and compute an asymptotic formula for the number of invariants of a given degree.