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][1]. 
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][2]). 

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][3], [2][4] 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][5]).  
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][6]), 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.
  


  [1]: http://en.wikipedia.org/wiki/Frobenius-Schur_indicator
  [2]: https://mathoverflow.net/questions/46900/are-there-real-vs-quaternionic-conjugacy-classes-in-finite-groups
  [3]: https://mathoverflow.net/questions/20795/frobenius-schur-indicator-and-irreducible-representations-over-r
  [4]: https://mathoverflow.net/questions/42653/number-of-n-th-roots-of-elements-in-a-finite-group-and-higher-frobenius-schur-ind
  [5]: https://mathoverflow.net/questions/107312/s-n-invariants-in-a-free-associative-algebra-noncommutative-symmetric-polynoms
  [6]: http://en.wikipedia.org/wiki/Capelli_identity