Sums of degrees of irreducible complex characters

Question

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant to determine the dimension of a maximal torus of the Lie algebra associated to the group algebra of a finite group over the complex field.

I have investigated certain group classes as extra-special groups, abelian groups, direct products, quaternion groups, (semi-)dihedral groups, A4, SL(2,q), GL(2,q), meta-cyclic groups, meta-abelian-groups, central products, frobenius groups and determine this sum. Also symmetric groups are now known counting involutions (a case of groups for which all irre. characters are real valued). Nilpotent groups are reduced to p-groups using direct products.

My question is whether there is a nice sum-formula (maybe with recursion) for groups like: alternating groups, p-groups (maybe special classes), simple groups, general linear, special linear groups, groups with united factor group, groups with normal p-Sylow-subgroup and abelian Hall-Complement, soluble groups, M-groups, ... as well. `

Maybe there is an argument or idea how to determine this sum for all groups by reduction on certain classes of groups.

In the literature there is a nice inequality saying that this sum is at least the double of the linear characters for a non-soluble groups (using the classification atlas of simple groups). Maybe there are some more upper and lower bounds to derive here.

Answer 1

I just realised that a result of mine ("On the minimal norm of a non-regular generalized character of a finite group" Bull LMS 2010) can be used to give a slightly better upper bound for the sum of the irreducible character degrees of finite non-Abelian group $G$. For such a group $G,$ with irreducible characters $\chi_{1},\chi_{2}, \ldots , \chi_{k},$ labelled so that $\chi_{k}(1)$ is maximal, we have $\sum_{i = 1}^{k} \chi_{i}(1) \leq \sqrt{k|G|+ 1 - \frac{|G|}{\chi_{k}(1)}}$ (and the $``+1"$ inside the square root sign can be omitted as long as $G$ is not a Frobenius group with Abelian Frobenius kernel). This follows from the main theorem of that paper, since $\sum_{i=1}^{k} \chi_{i}$ is not a multiple of the regular character when $G$ is non-Abelian. Notice that when $G$ is dihedral of order $2h$ with $h$ odd, we have $\sum_{i=1}^{k} \chi_{i}(1) = 1 + 1 + 2\frac{h-1}{2} = h+1,$ while the above inequality gives $\sum_{i=1}^{k} \chi_{i}(1) \leq \sqrt{ 2h(2 + \frac{h-1}{2}) - h + 1} = h+1,$ so the stated inequality becomes equality for all such dihedral groups.

By an earlier paper in that series, we have $\sum_{i = 1}^{k} \chi_{i}(1) \leq \sqrt{(k-1)|G| + \chi_{k}(1)^{2}}$ when $G$ is nilpotent, but non-Abelian, (where again we label so that $\chi_{k}(1)$ is maximal).

Answer 2

For $G=GL(n,F_q)$ and for $G=GSp(2g,F_q)$, there are exact formulas due to Gow and Klyachko and Vinroot: the sum of the degrees is equal to the number of symmetric matrices in $G$.

Good upper and lower bounds exist for many groups of Lie type, when the size of the finite field is large, using Deligne-Lusztig theory.

R. Gow: Properties of the characters of the finite general linear group related to the transpose-inverse involution, Proc. London Math. Soc. 47 (1983), 493–506.

A.A. Klyachko: Models for complex representations of the groups GL(n, q), Mat. Sb. (1983), 371–386.

C.R. Vinroot: Twisted Frobenius-Schur indicators of finite symplectic groups, J. Algebra 293 (2005), 279–311.