Pieri-type rule for Schur P-functions

Question

This question is about symmetric functions. As usual, let $s_\lambda$ denote the Schur function corresponding to a partition $\lambda$. For $r\geqslant1$, let $d_r$ denote the map defined on symmetric functions by mapping $s_\lambda\mapsto s_{\lambda/(1^r)}$ and extending linearly. In other words, $s_\lambda\mapsto\sum_\mu s_\mu$, summed over all partitions $\mu$ obtained from $\lambda$ by removing $r$ nodes in distinct rows.

Let $P_\lambda$ denote the Schur P-function corresponding to the strict partition $\lambda$. I would like to write $d_rP_\lambda$ in terms of Schur P-functions, and it appears that

$ d_rP_\lambda = \sum_\mu2^{j(\lambda,\mu)}P_\mu, $

summed over all strict partitions $\mu$ obtained from $\lambda$ by removing $r$ nodes in distinct columns; $j(\lambda,\mu)$ denotes the number of $i\geqslant2$ such that there is a node of $\lambda/\mu$ in column $i$ but not in column $i-1$.

I expect there is a generalisation to Hall-Littlewood functions, though for now I only need this special case. This generalisation appears to be in some sense dual to the Pieri-type rule for Hall-Littlewood functions (III.5.7 in Macdonald), but I don't know enough about duality to do this kind of thing.

Is my conjectured formula in the literature, or can it be easily deduced from known results?

Update: I can now prove my formula, using marked tableaux and the fact that the Schur P-functions are invariant under the involution $\omega$. But it would be good to know if this is already known or easy to prove.

Another update (27th February 2021) I can now prove a version for Hall-Littlewood functions using duality. Please e-mail me if you would like to see the details.

Answer 1

It seems you are aiming to skew a Schur-P function $P_\lambda$ by a single row of length $r$, i.e. computing $P_{\lambda/(r)}$. There's a natural analogue of the Hall inner product for the ring generated by odd power sums. Here, skewing Schur-P functions is equivalent to multiplying Schur-Q functions:

$$P_{\lambda/(r)} = \sum_{\mu} c_{mu(r)}^\lambda P_\mu \quad \mbox{and}\quad Q_{\mu} Q_{(r)} = \sum\lambda c^\lambda_{\mu(r)} Q_\lambda.$$

This duality should be somewhere in Macdonald and in papers by Stembridge amongst others, but likely dates back to Schur. There are many proofs of Littlewood-Richardson rules for Schur-Q functions one may now apply, e.g. here.

I'm not sure when the first proof of a Pieri rule appears. It likely occurs much earlier than Stembridge's work both in representation theory and geometry, since Schur-Q functions correspond to projective representations and are representatives for cohomology classes in the Lagrangian Grassmannian.

Answer 2

I believe this exact result is stated as Corollary 6.8 in this preprint of Soichi Okada: https://arxiv.org/abs/1904.03386v1. Okada attributes the result to A.O. Morris with citation: Morris, A. O., A note on the multiplication of Hall functions, J. Lond. Math. Soc. 39, 481-488 (1964). ZBL0125.01702.

EDIT: Ah, sorry, this is a reference for the "dual" result, which you mentioned you already know about.