Pieri-type rule for Schur P-functions

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.

• I am interested in seeing the proof of your formula (or, more specifically, the $d_1$ case of it). I'll send an e-mail with more of my motivation. Commented Jul 19, 2023 at 6:38
• For the help of other people finding this post in the future, Matt Fayers e-mailed me to let me know that this proposition appears as Prop 3.6 in his paper "Irreducible projective representations of the alternating group which remain irreducible in characteristic 2" (Advances in Math, 2020) arxiv.org/abs/2003.07713 . Commented Jul 19, 2023 at 13:30

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