sum of the character of the symmetric group

Question

Suppose $\mu$ is a fixed partition of $n$ of length $l(\mu)$, and I was encountered with the following sum, namely $\sum_{\nu} \chi_{\nu}(\mu)$.

I did some calculation using the character table that I can find (mainly Fulton & Harris's book, they have the character table up to $S_5$), and found that the sum does not vanish only if $\mu$ has an even number of even parts(someone call such $\mu$ an orthogonal partition).

This is actually very simple to prove, only use the fact that $\chi_{\nu^t}(\mu)=(-1)^{n-l(\mu)} \chi_{\nu}(\mu)$, if $\mu$ has an odd number of even parts, then $n-l(\mu)$ is odd.

But my calculation indicates more: the sum is nonzero only if every even part of $\mu$ occurs even times.(someone also call such partition an orthogonal partition, and I donot know which is the correct definition...can anyone help?)

I checked this for $n \leq 6$ and also for $n=11$ (I found the charater table of $S_{11}$ in some paper...)

I donot know whether this is just an coincidence or this is always true.

Since my knowledge of symmetric group is very limited, I donot hesitate to ask for help on MO. Hopefully, someone will give me an answer. Thank you all!

p.s. (1) My second question, which is quite related to the above one. We know that $\sum_{\nu} s_{\nu}(x) s_{\nu}(y)=\prod_{i,j} \frac{1}{1-x_i y_j}$, where $s_{\nu}$ is the Schur function. Is there a similar expression for $\sum_{\nu} s_{\nu}$?

(2) My third question: Is there a similar expression for $\sum_{\nu} (\frac{|\nu|!}{dim R_{\nu}})^k s_{\nu}$? Here $R_{\nu}$ is the irreducible representation indexed by $\nu$, and $k$ is a positive integer.

Answer 1

Your conjecture is correct; as a matter of fact, it is possible to compute these sums $C(\mu):=\sum_\nu\chi_\nu(\mu)$ exactly for any $\mu$:

Suppose $\mu$ has $m_i$ parts of length $i$, for $i=1,2,\ldots$. Then $C(\mu)=\prod_{i>0} c_{i,m_i}$, where $c_{i,m_i}$ is the coefficient of $t^{m_i}/({m_i}!)$ in $\exp(t+\frac{1}{2}it^2)$ if $i$ is odd, or in $\exp(\frac{1}{2}it^2)$ if $i$ is even.

A first proof can be found in Macdonald's Symmetric functions and Hall polynomials, ex.11 p.122; it relies on symmetric function techniques. A second one is given in Stanley's Enumerative Combinatorics Vol. 2, ex. 7.69, and is based on the fact that, from a general character theory result, $C(\mu)$ is the number of square roots of a given permutation of cycle-type $\mu$.