# Generalization of the formula representing power sums as sums of Schur polynomials

Let $$m_1\ge m_2\ge\ldots\ge m_k$$ are nonnegative integers. Then, we can consider the following product of power sums: $$p_m(x_1,\ldots,x_n)=\prod_{i=1}^{k}\sum_{j=1}^{n}x_j^{m_i}.$$ Since Schur polynomials $$\{s_{\lambda}\}$$ ($$\lambda$$ is a partition with $$n$$ nonnegative parts) form a basis in the space of the symmetric polynomials in $$n$$ variables $$x_1,\ldots,x_n$$, we can represent $$p_m$$ as a linear combination of $$s_{\lambda}$$.

Question 1: Does there exist a formula for the corresponding coefficients? I almost sure that there is such formula in Hall's book but I haven't found it yet.

Question 2: Even if this formula exist, I suppose that they consider only nonnegative $$m_i$$. I am also interested in the case when some of $$m_i$$ are negative. However, I don't know whether there exist a similar formula. (We can also define Schur functions for $$m=(m_1,\ldots,m_n)$$ via bialternant formula, cf. Weyl character fromula)

Some motivation: it's known that Schur functions are characters of irreducible (unitary) representations of the $$U(n)$$, so we encounter the following problem: decide whether a given function is a character of some representation of the $$U(n)$$ or not. In other words, we need to check whether this function can be represented as sum of Schur functions with nonnegative coefficients or not. That's why I am interested in exact formula for coefficients for $$p_m$$.

Remark. For simplicity we can suppose that number $$n$$ of variables is large enough (or work in the ring of the symmetric functions).

• Can you give the name of Hall's book that you mention in Q1?
– ArB
Nov 6, 2021 at 9:30
• Sorry, I meant Macdonald book on symmetric functions, of course. Nov 6, 2021 at 11:19

Partial answer. I found the answer for the first question (cf. Symmetric functions). It turns out that in the ring of symmetric functions the following identity holds: $$p_{\mu}(x)=\sum_{\lambda}\chi_{\mu}^{\lambda}\cdot s_{\lambda}(x),$$ where $$\chi_{\mu}^{\lambda}$$ is a value of the character $$\chi^{\lambda}$$ of the irreducible representation of $$S_n$$ which corresponds to partition $$\lambda\vdash n$$ (or, equivalently, Young diagram) on an element $$\sigma\in S_n$$ whose cycle structure corresponds to the partition $$\mu\vdash n$$.
Moreover, there is a formula for $$\chi_{\mu}^{\lambda}$$ (the so called Murnaghan-Nakayama rule) and the positivity question can be resolved via orthogonal relations between characters of the irreducible representations.