What is sum of degrees of the irreducible complex characters of the alternating groups?
The background of this question is to calculate the diminsion of a maximal torus of the associated Lie algebra of the group algebra $\mathbb{C} A_n$ for all $n\in \mathbb{N}$.
The corresponding sum for the symmetric group is known by the Frobenius-Schur-Indicators that is to calculate the involutions plus 1 in the symmetric group. This value is also an upper bound for the sum related to the alternating group (see also Number of involution in symmetric group)
I would like to know the difference betweeen these two values.
In addition the character theory of the alternating groups are closely related to the symmetric group by using Clifford-Theory: some irreducible are still irreducible by restricting them to the alternative group the other ones decompose into two irreducible characters.
In the article http://www.kurims.kyoto-u.ac.jp/EMIS/journals/JACO/Volume5_2/lmm440901p5p3660.fulltext.pdf these two irreducible characters of $A_n$ are investigated based on theorems by Thrall.