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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 https://math.stackexchange.com/questions/590402/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.

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    $\begingroup$ This is equivalent to finding the sum of the degrees of all self-conjugate characters of the symmetric group $S_n$. It appears in oeis.org/A067136, but I don't think that any reasonable formula is known. $\endgroup$
    – Richard Stanley
    Commented Jun 8, 2015 at 16:37
  • $\begingroup$ Thank you, maybe its nice to know difference to the corresponding sum for $S_n$. Is this also in this tool? $\endgroup$
    – Sven Wirsing
    Commented Jun 10, 2015 at 13:05
  • $\begingroup$ Following up on my previous comment, an interesting related result is that $\sum_\lambda f^\lambda (-1)^{\frac 12(n-\mathrm{rank}(\lambda))}$ equals the coefficient of $x^n/n!$ in $e^{x-\frac{x^2}{2}}$. Here $\lambda$ ranges over all self-conjugate partitions of $n$, $f^\lambda$ is the dimension of the corresponding symmetric group character, and $\mathrm{rank}(\lambda)$ is size of the Durfee square of $\lambda$ (the largest $i$ for which $\lambda_i\geq i$). $\endgroup$
    – Richard Stanley
    Commented Jun 12, 2015 at 17:35

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For small $n$, you can compute the sums of irreducible complex character degrees of ${\rm A}_n$ with GAP:

gap> sums := List([3..15],n->Sum(List(CharacterDegrees(AlternatingGroup(n)),Product)));
[ 3, 6, 16, 46, 126, 448, 1366, 5356, 18568, 76296, 297012, 1264264, 5412928 ]

However I doubt there is a nice formula for these values.

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