What is known about S_n group invariants in a free associative (noncommutative) algebra k < x_1, ...x_n >) ? (S_n natural acts by permutations on generators).
What is Poincare series ? Is it finitely generated, is it free ? Are the generators as algebra/vector space known ?
The same question for the commutative algebra - gives algebra of symmetric polynoms, which is for-ever-young research topic. To what extent non-commutative version is the same rich ?
Is it commutative ? Probably no - however, pay attention on the following simple fact: consider group algebra of C[G], represent it as a factor of k< x_1, ...., x_G > , then S_n invariants go into the center of C[G] (since we need to check invariance with respect to conjugaction - but we have much bigger invariance with respect to S_G action). This somewhat may be considered as indication that non-commutativity is not that much big...
In particular T_k = \sum_i (x_i)^k will be mapped to higher Frobenius-Schur indicators see
This question is motivation to ask present.