Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
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 –  Benjamin Steinberg Sep 16 '12 at 11:10
@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 –  Alexander Chervov Sep 16 '12 at 11:29
PS But the paper you mention is indeed somewhat close, thanks again –  Alexander Chervov Sep 16 '12 at 11:32
@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 –  Benjamin Steinberg Sep 16 '12 at 11:57
The paper I just linked embeds the noncommutative symmetric functions in the sense of Gelfand et al into the invariants. –  Benjamin Steinberg Sep 16 '12 at 12:01

1 Answer 1

up vote 2 down vote accepted

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.

share|improve this answer
thank you !..... –  Alexander Chervov Sep 16 '12 at 12:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.