Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

What are the places where character table values of $S_n$ occurs naturally? one such an example is when we write power sum symmetric function of order n in terms of Schur function of order n the coefficient of each monomial will be the character value of the respective conjugacy class and respective irreducible representation in the character table of $S_n$.

share|improve this question
3  
One is tempted to say: In the representation theory of the symmetric groups! ;-) –  Johannes Hahn Apr 12 '13 at 17:54
3  
It is very weird that you accepted an answer so soon: you asked for what is essentially a list... –  Mariano Suárez-Alvarez Apr 12 '13 at 18:21
1  
Should be CW it would seem. –  Benjamin Steinberg Apr 12 '13 at 18:32
    
I am new to mathoverflow. cant I accept more than one answer? I found that answer interesting. –  GA316 Apr 12 '13 at 18:35
add comment

2 Answers

up vote 2 down vote accepted

My favourite place where the character table of $S_n$ occurs naturally is in the enumeration of branched covers of Riemann surfaces (Hurwitz numbers).

In more detail, let $\Sigma$ be a closed Riemann surface of $g$ with $k$ marked points $x_1, \ldots, x_k$, and fix an integer $n\geq 1$. For each marked point $x_i$ pick a partition $\mu^i$ of $n$.

We define $\mathcal H_n(g,\mu_1 , \ldots ,\mu_k)$ to be the sum of $1/|Aut(f)|$ for each isomorphism class of branched covers $f: \widetilde{\Sigma} \to \Sigma$ which are ramified precisely at $x_i$ with ramification type $\mu^i$.

Then there is a formula due to Hurwitz (or perhaps Frobenius, or Burnside, depending on which version...?):

$\mathcal H_n(g,k) = \sum_{\nu \in P(n)} \left(\prod_{i=1} ^k|ccl(\mu^i)| \frac{\chi_\nu(\mu^i)}{\chi_\nu(1)}\right) \left( \frac{\chi_\nu(1)}{n!} \right)^{2-2g}$.

Here $\nu \in P(n)$ indexes the irreducible representations $\chi_\nu$ of $S_n$, and the $\mu^i$ are identifed with conjugacy classes in $S_n$.

share|improve this answer
    
Of course, this isn't really specific to the symmetric group... we have the same formula for every finite group $G$ (enumerating $G$-Galois covers). But the symmetric group case has historical significance. –  Sam Gunningham Apr 12 '13 at 18:09
    
thanks. this is interesting. –  GA316 Apr 12 '13 at 18:12
    
what is ccl in the expression for $Hn$? –  GA316 Apr 12 '13 at 18:14
    
That means the size of the conjugacy class in $S_n$ corresponding to the partition $\mu ^i$. There is a formula for this, I just didn't want to write it down... –  Sam Gunningham Apr 12 '13 at 18:20
add comment

As a somewhat trivial example, the sign of a permutation is a character of $S_n$, and the sign function appears in many contexts.

For instance, the sign function is used when calculating the determinant of a matrix by summing over permutations in $S_n$. Replacing the sign character in this formula with another character of $S_n$ gives the definition of the immanant of a matrix.

share|improve this answer
    
thanks. but I want little more natural answer. –  GA316 Apr 12 '13 at 18:51
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.