# On two types of shifted symmetric power sums

In the ring of shifted symmetric functions $\Lambda^*$ there are many ways to generalize the symmetric power sums. First of all, we have the functions $$p^*_k=\sum_{i=1} \left((x_i-i+1/2)^k-(-i+1/2)^k\right)$$ (the factor 1/2 is convenient when the $x_i$ are the parts of a partition, but can replaced by any constant). For a partition $\kappa$ we let $p^*_\kappa=p^*_{\kappa_1}p^*_{\kappa_2}\cdots.$ Another shifted analogue to the power sums are the functions $p_\kappa^\#$, defined by $$p^\#_\kappa = \sum_{\lambda \vdash k} \chi_\kappa^\lambda s_\lambda^*,$$ where k=$|\kappa|$ is the size of the partition $\kappa$. Both $p_\kappa^*$ and $p_\kappa^\#$ provide a basis for $\Lambda^*$. (For more details: see the original paper 'Shifted Schur Functions' by Okounkov and Olshanski or the post On shifted symmetric power sums).

In the literature there is a vast amount of results relating different bases of $\Lambda^*$ to each other. However, the only relation between the bases by $p_\kappa^*$ and $p_\kappa^\#$ I can find is that $$p_\kappa^\#=p_\kappa^*+\textrm{lower degree terms},$$ (because the highest degree terms of $p_\kappa^\#$ and $p_\kappa^*$ equal the ordinary symmetric power sum $p_\kappa$). How can these lower degree terms be expressed in terms of $p_\kappa^*$ (or $p_\kappa^\#$)?

Question: Are there any expressions known relating $p_\kappa^*$ and $p_\kappa^\#$?

• You might want to look at mathoverflow.net/questions/214551/… and see if there is something related – Per Alexandersson Mar 21 '18 at 16:32
• @PerAlexandersson, yes, I've seen this post (I even referred to it in my question). The posts indeed introduce $p_\kappa^*$ and $p_\kappa^\#$ but they are not related there. – Jan-Willem van Ittersum Mar 21 '18 at 17:52

The answer to this question is precisely given by the Gromov-Witten/Hurwitz correspondence by Okounkov and Pandharipande (see https://arxiv.org/abs/math/0204305). Up to the constant $(1-2^{-k})\zeta(-k)$ the functions $p_k^*$ equal the renormalized shifted symmetric power sums $\mathbf{p}_k$ which occur in expressions of integrals of descendants classes in Gromov-Witten theory (equation 0.25). The functions $p_\kappa^\#$ equal (up to a multiplicative constant depending on $\kappa$) the central characters $\mathbf{f}_\kappa$ of the symmetric group as explained in the related post On shifted symmetric power sums. These functions occur naturally in the character formula for extended Hurwitz numbers (equation 0.10). The completion coefficients relating $\mathbf{p}_k$ and $\mathbf{f}_\kappa$ are given by equation 0.22 in the before-mentioned article.