# Explicit formulas for the action of the Hall algebra of the cyclic quiver on q-Fock space?

In their paper on the decomposition numbers of Schur algebra, Vasserot and Varagnolo introduce an action of the (twisted) Hall algebra of a cyclic quiver $\Gamma$ on q-Fock space.

Without q-shifts, it is easy to describe: the Hall algebra is generated by $\mathbf{f}_{d}$, the characteristic function of the zero representation of dimension vector $d$ (remember, this is a function on the nodes of the cyclic quiver of some fixed size). The product $\mathbf{f}_{d}\cdot |\lambda\rangle$ is the sum over all partitions $|\mu\rangle$ where $\mu$ is obtained from $\lambda$ by adding $d_i$ boxes of residue $i$ for all $i\in \Gamma$, no two in the same column (so, essentially the Pieri rule, but broken up according to the residues of the added boxes).

In order to have a $q$ in the picture, though, one has to multiply these by some power of $q$. If one is adding a single box of residue $i$, then this is easy (which means I'll probably mess up the formula); one simply multiplies by q to the number of addable minus the number of removable boxes of the right residue which are below the node (English style; I actually prefer Russian style, in which case this should be to the left of the node). But I cannot extract what the combinatorial statistic is for adding more than one box from Vasserot and Varagnolo's paper. Is it written down more explicitly somewhere else?

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