# Frobenius coordinate expansion of character

Let $\lambda$ be the partition of integer $d$. The Frobenius coordinate of $\lambda$ is given $$(a_1 ,\ldots,a_{d(\lambda)}|b_1,\ldots,b_{d(\lambda)}),$$ where $d(\lambda)$ denote the diagonal of $\lambda$. Let $a_i'= a_i +\frac12$ and $b_i'= b_i +\frac12$ are modified Frobenius coordinates. By a classical theorem of Frobenius, it states that

$f(\lambda)=\frac{|C_{2,1,\ldots,1}|}{dim \lambda}\chi_{2,1,\ldots,1}^{\lambda}=\frac12\sum_{i=1}^{d(\lambda)}(a_i')^2-(b_i')^2$

Notice that the above example we treat the L.H.S as a function of $\lambda$ as we fix the partition of the form $2,1^{d-2}$

If there is any generalisation of the above theorem? That is if we replace $(2,1,\ldots,1)$ with any other partition of $d$ then say $\mu$ then what will be

$$\frac{|C_{2^{d/2}}|}{dim \lambda}\chi_{2^{d/2}}^{\lambda}$$ For example say $\mu$ is of the form $2^{d/2}$

in Frobenius coordinate? If it is already studied can anyone please site the reference.

There is a generalisation of the character formula, although it is usually stated in terms of contents rather than Frobenius coordinates. Also, it applies to conjugacy classes labelled by partitions $$(1^{m_1} 2^{m_2} \cdots)$$, where $$m_2, m_3, \ldots$$ are fixed and $$m_1$$ varies with $$n$$ (as in the example you gave, where there is one part of size $$2$$, and $$n-2$$ parts of size 1). So in particular, it does not apply directly to the partitions $$(2^{n/2})$$.

For the benefit of the uninitiated, let me state right away that $$\frac{|C_\mu| \chi_{\mu}^\lambda}{dim(\lambda)}$$ is a central character. The sum of all elements of $$S_n$$ of cycle type $$\mu$$ is a central element of the group algebra of $$S_n$$, and the central character is the scalar by which the central element acts on the irreducible representation labelled by $$\lambda$$. Central characters are important in modular representation theory, which might provide a little motivation for why the quantity in question is a natural thing to consider.

Suppose that we have a partition $$\lambda = (\lambda_1, \ldots, \lambda_l)$$. Suppose that $$d$$ is the largest integer such that $$\lambda_d \geq d$$ (the Durfee size of $$\lambda$$). Then the Frobenius coordinates of $$\lambda$$ are $$(\alpha_1, \ldots, \alpha_d | \beta_1, \ldots, \beta_d)$$, where $$\alpha_i = \lambda_i - i$$ and $$\beta_i = \lambda_i^\prime - i$$ (here $$\lambda^\prime$$ is the transpose partition of $$\lambda$$). So the $$\alpha_i$$ and $$\beta_i$$ count the number of boxes in past the diagonal in the $$i$$-th row and $$i$$-th column respectively (and the number of each is $$d$$). On the other hand, the contents of a box in row $$j$$ and column $$i$$ is $$i-j$$. It is not difficult to see that the multiset of contents of $$\lambda$$ consists of $$0$$ with multiplicity $$d$$, and $$\{1,2,\ldots, \alpha_i\}$$ for each $$i$$ together with $$\{-1, -2, \ldots, -\beta_i\}$$ for each $$i$$. So in particular, the sum of $$r$$-th powers of the contents of $$\lambda$$ can be expressed as polynomials in the $$\alpha_i$$ and $$\beta_i$$ by Faulhaber's Formula. For example, the sum of the contents ($$r=1$$) is

$$\sum_i (1 + 2 + \cdots + \alpha_i) - (1 + 2 + \cdots + \beta_i) = \sum_i {\alpha_i + 1 \choose 2} - {\beta_i + 1 \choose 2},$$

which is precisely the quantity appearing in the example you gave. We write $$cont(\lambda)$$ for the multiset of contents of $$\lambda$$.

Now, for a partition $$\mu = (\mu_1, \mu_2, \cdots, \mu_k)$$, let $$\bar{\mu} = (\mu_1 - 1, \mu_2 - 1, \ldots, \mu_k - 1)$$ be the partition obtained from $$\mu$$ by subtracting 1 from each nonzero part of $$\mu$$ (and ignoring trailing zeros). For example, if $$\mu = (2,1, \ldots, 1)$$ as in your example, then $$\bar{\mu} = (1)$$.

Theorem There exists a family $$f_{\rho}$$ of elements of $$\mathbb{Q}[n] \otimes \Lambda$$ (where $$n$$ is a polynomial variable) and $$\Lambda$$ is the ring of symmetric functions with the property that

$$\frac{|C_\mu| \chi_{\mu}^\lambda}{dim(\lambda)} = f_{\bar{\mu}}(|\lambda|, cont(\lambda))$$

By this notation I mean that we evaluate $$f_{\bar{\mu}}$$ at $$n=|\lambda|$$ and its symmetric function variables at $$cont(\lambda)$$.

It turns out that $$f_{(1)} = p_1$$ (the first power-sum symmetric function) which doesn't actually depend on $$n$$. Then putting this together, we see

$$\frac{|C_{(2,1,\ldots,1)}| \chi_{(2,1,\ldots,1)}^\lambda}{dim(\lambda)} = p_1(cont(\lambda)) = \sum_i {\alpha_i + 1 \choose 2} - {\beta_i + 1 \choose 2},$$

which recovers the result you stated. In general, you can take $$f_{\bar{\mu}}$$, express it as a polynomial in $$n$$ and the power-sums $$p_1, p_2, \ldots \in \Lambda$$, and upon evaluating at the contents of $$\lambda$$ you will obtain a polynomial in $$n$$ and the Frobenius coordinates of $$\lambda$$. For example,

$$f_{(2)} = p_2 - {n \choose 2},$$

which tells us that the central character corresponding to the conjugacy class $$(3,1,\ldots, 1)$$ may be computed using the formula

$$\left( \sum_{i} \frac{\alpha_i(\alpha_i + 1)(2\alpha_i + 1)}{6} + \frac{\beta_i (\beta_i + 1)(2\beta_i + 1)}{6} \right) - {|\lambda| \choose 2}.$$

There is a lot more to be said about the $$f_\rho$$. For example, they form a $$\mathcal{R}$$-basis of $$\mathcal{R} \otimes \Lambda$$, wher $$\mathcal{R}$$ is the subring of $$\mathbb{Q}[n]$$ consisting of integer-valued polynomials. But in this post I worked over $$\mathbb{Q}$$ because I wanted to use power-sum symmetric functions to turn contents-evaluaion into polynomials in Frobenius coordinates.

References you might find helpful (each of which contains an explanation of the above theorem):

Corteel, Sylvie; Goupil, Alain; Schaeffer, Gilles, Content evaluation and class symmetric functions, Adv. Math. 188, No. 2, 315-336 (2004). ZBL1059.05104.

Section 5.4 of

Ceccherini-Silberstein, Tullio; Scarabotti, Fabio; Tolli, Filippo, Representation theory of the symmetric groups. The Okounkov-Vershik approach, character formulas, and partition algebras., Cambridge Studies in Advanced Mathematics 121. Cambridge: Cambridge University Press (ISBN 978-0-521-11817-0/hbk). xv, 412 p. (2010). ZBL1230.20002.

(although this is based on the material of the above paper)

Section 3 of my own paper (which doesn't appear in the citation engine, so I'm providing an arXiv link instead) Stable Centres I: Wreath Products

Frobenius coordinates is very similar to multirectangular coordinates. There has been some studies on how (normalized) characters behave in these coordinates, see e.g. these slides by V. Feray.