How to expand certain sums of powers in terms of Schur polyomials. I have been gaining proficiency with symmetric polynomials, today I would like to expand:
$$ a^5 + b^5 + c^5 = \sum_{\lambda_1 + \lambda_2 + \lambda_3 = 5} s_\lambda (a,b,c) \tag{$*$} $$
My first take is to write out all the partition of 5 into three parts, and I am going to write them both as sums and as "Frobenius partitions"
- $5 = 5 + 0 + 0 = (4|0)$
- $5 = 4 + 1 + 0 = (3|1)$
- $5 = 3 + 1 + 1 = (2|2)$
- $5 = 2 + 2 + 1 = (2,0|1,0)$ (not sure about this one)
So now I am going to write out all these determinants in terms of elementary symmetric functions:
$$ s_{5,0,0}(a,b,c) = \frac{1}{\Delta} \left| \begin{array}{lll} a^7 & b^7 & c^7 \\ a & b & c \\ 1 & 1 & 1 \end{array} \right| $$
$$ s_{4,1,0}(a,b,c) = \frac{1}{\Delta} \left| \begin{array}{lll} a^6 & b^6 & c^6 \\ a^2 & b^2 & c^2 \\ 1 & 1 & 1 \end{array} \right| $$
$$ s_{3,1,1}(a,b,c) = \frac{1}{\Delta} \left| \begin{array}{lll} a^5 & b^5 & c^5 \\ a^2 & b^2 & c^2 \\ a & b & c \end{array} \right| $$
$$ s_{2,2,1}(a,b,c) = \frac{1}{\Delta} \left| \begin{array}{lll} a^4 & b^4 & c^4 \\ a^3 & b^3 & c^3 \\ a & b & c \end{array} \right| $$
I am still working out what the Jacobi-Trudi identities say in these circumstances. In any case, there seem to be 4 Schur polynomials that I need (at this moment) and wish to do the expansion at $(*)$.
The power sum symmetric function $p_n$ is the alternating sum of hooks of size $n$, that is: $$p_n=\sum_{i=0}^{n-1}(-1)^{i}s_{(n-i,1^{i})}$$