For any $k$ sums $T\_k = 1/|G|\sum\_{g\in G} g^k$ belong to the center of the group algebra, for finite group G.
For $k=2$ they "are" (up to details and interpretation) [Frobenius-Schur indicators](http://en.wikipedia.org/wiki/Frobenius-Schur_indicator). 
For $k>2$ similar things called "higher" FS indicators.


**Question 1**
Any element of the center of C[G], and $T\_k$ in particular, acts by scalars in any irreducible representation. For $k=2$
the scalar can be $-1,0,1$ and brings important group theory information on the type of irrep 
(see WP and/or [this MO question](https://mathoverflow.net/questions/46900/are-there-real-vs-quaternionic-conjugacy-classes-in-finite-groups)). 


Is it true that values of higher $T\_k$ in irreps are also integers ? Do they have some group theory interpretation ? (These MO questions [1](https://mathoverflow.net/questions/20795/frobenius-schur-indicator-and-irreducible-representations-over-r), [2](https://mathoverflow.net/questions/42653/number-of-n-th-roots-of-elements-in-a-finite-group-and-higher-frobenius-schur-ind) discuss FS and relate them to dim of invariants on tensor powers of representation, but it seems different from my question.)


**Question 1b** (later edit)


One can generalize this consturction as follows consider free associative ring Z < x\_1...x\_|G| > . Consider S\_|G| invariants in this algebra. Any such invariant can be mapped
to the center of the group algebra, just substituting "g" instead of x\_|g|.
(see [MO quest](https://mathoverflow.net/questions/107312/s-n-invariants-in-a-free-associative-algebra-noncommutative-symmetric-polynoms)).  

It will clearly give central element in C[G].


Is it true that there values will be interges in any irrep ? Moreover by analogy with T\_k their values should be divisible by |G|.


**Question 2**


How far is the linear space/ algebra generated by $T\_k$ from the whole center of group algebra?
( What can be said in general and in particular for S\_n,A\_n, GL\_n(F\_q), UT(n,q) )?


Same question not about T\_k, but also about elements discussed in questions 1b above.


PS


**Bonus Question 3** WP- article contains the following remark: 



> 
> It resembles the Casimir invariants for Lie algebra irreducible representations.
> 
> 
> 


In the sense that Casimirs are the center of U(g), while T\_k are in the center of C[G].


Question: are there some more precise analogies ? 


In some my research we studied the center of U(gl\_n) and its loop algebra,  

so such expression resembles "Gelfand" generators of U(gl\_n) (which are traces of matrices "E^k", "E" given [here](http://en.wikipedia.org/wiki/Capelli_identity)), but I do not see any way
to go beyond "resemble". While it would be nice seems now I am trying to think on some finite group questions.



sum_g g^k, Frobenius-Schur indicators, S_n-invariants in freeAss(x_i),  center of the group algebra 
For question 1, they are integers, and this is related to Adams operations. The result basically follows from Newton's identities. However, there is no general bound on the size of the integers which may occur when $k >2.$


Later edit: In general, the $T\_{k}$'s can fail very badly to generate the center of the group algebra. For example, if $p$ is an odd prime, and $G$ is a $p$-group of exponent $p,$ then $T\_{k}$ is either $1\_{G}$ (when $p$ divides $k$) or $\frac{1}{|G|} \sum\_{g \in G}g$ (when $p$ does not divide $k$), whereas the dimension of the center of the group algebra of $G$ can be made as large as desired by choosing $G$ appropriately.


