Matthias Lederer and I are studying a deformation of the Littlewood-Richardson product of Schur functions. It's a bit complicated to define (and work in progress) so I won't give the full definition here, but nonetheless my question is:

Has the following deformation appeared before?

We believe the rule for multiplying by a single box is $$ S_{\square} S_\lambda = \sum_{\nu \supset \lambda,\ \nu\ \subseteq \lambda_+} S_\nu $$ where $\lambda_+$ is defined as follows, for $\lambda$ an English partition: draw the $i=j$ line from the NW corner of the partition to where it meets $\lambda$ at $(p,p)$, push all the edges on the NE side of $(p,p)$ one step to the right, all edges on the SW side one step down, and push $(p,p)$ out to $(p+1,p+1)$.

Examples: if $\lambda = (k)$, i.e. a single row, then $\lambda_+ = (k+1,2)$. $(3,1,1) \mapsto (4,2,1,1), (3,3,1) \mapsto (4,4,3,1)$.

(There's a similar rule for the $K$-theory product, where $\lambda_+$ is $\lambda$ plus boxes at all its inner corners. That's not the $\lambda_+$ above.)