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

sum_g g^k, Frobenius-Schur indicators, S_n-invariants in freeAss(x_i), center of the group algebra

This question is motivation to ask present.

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    $\begingroup$ There is a huge literature on noncommutative symmetric functions. Try Googling noncommutative symmetric functions. One paper that seems to address your question is arxiv.org/abs/0907.0814 $\endgroup$ – Benjamin Steinberg Sep 16 '12 at 11:10
  • $\begingroup$ @Benjamin Steinberg thanks for the reference. I am not sure that "noncommutative symmetric functions" is standard term everybody understand in the same sense as my question. There is quite well-known paper by Gelfand&K arxiv.org/abs/hep-th/9407124 Noncommutative symmetric functions which seems to be different from what I am asking $\endgroup$ – Alexander Chervov Sep 16 '12 at 11:29
  • $\begingroup$ PS But the paper you mention is indeed somewhat close, thanks again $\endgroup$ – Alexander Chervov Sep 16 '12 at 11:32
  • $\begingroup$ @Alexander, I think these ones are nowadays called quasisymmetric functions, but I may be wrong. There are however papers on symmetric functions in noncommuting variables, like the one I linked. You might also look at arxiv.org/pdf/math/0502082.pdf $\endgroup$ – Benjamin Steinberg Sep 16 '12 at 11:57
  • $\begingroup$ The paper I just linked embeds the noncommutative symmetric functions in the sense of Gelfand et al into the invariants. $\endgroup$ – Benjamin Steinberg Sep 16 '12 at 12:01

The paper http://arxiv.org/pdf/math/0502082.pdf shows the invariants are a free associative algebra and give an explicit basis. Hence it is not commutative. This is proved first in M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626–637 without an explicit basis.

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