The Rouquier-block multiplicity formula for cyclotomic q-Schur algebras

Let $\mathcal{S}_{r,n}$ be a cyclotomic $q$-Schur algebra defined over a field of characteristic $p ot<0$. Let $boldsymbol{lambda}$ and $boldsymbol{mu}$ be $r$-multipartitions of $n$ lying in a Rouquier block, with the same multicore, and suppose that no component of $boldsymbol{mu}$ has $p$ or more removable $e$-rim hooks. Write $Delta(\boldsymbol{lambda})$ for the Weyl module indexed by $boldsymbol{lambda}$ and $L(\boldsymbol{mu})$ for the corresponding simple module.

The Rouquier-block multiplicity conjecture. The multiplicity of $L(\boldsymbol{mu})$ as a composition factor of $\Delta(\boldsymbol{\lambda})$ is

$$[\Delta(\boldsymbol{\lambda}):L(\boldsymbol{mu})] = \sum_{\boldsymbol{\alpha} \in \Gamma^{r}_{e+1}} \sum_{\boldsymbol{\beta} \in \Gamma^{r}_{e}} \sum_{\boldsymbol{\gamma} \in \Gamma^{r+1}_{e}} \sum_{\boldsymbol{\delta} \in \Gamma^{r}_{e}} \left( \prod_{k=0}^{r-1} \prod_{i=0}^{e-1} c^{\delta^k_i}_{\mu^k_i \gamma^k_i} c^{\delta^k_i}_{\alpha^k_i \beta^k_i \gamma^{k+1}_i} c^{\lambda^k_i}_{\beta^k_i (\alpha^k_{i+1})'}\right) c^{\varnothing}_{\alpha^0_0 \alpha^1_0 \ldots \alpha^{r-1}_{0}} c^{\varnothing}_{\gamma^0_0 \gamma^0_1 \ldots \gamma^{0}_{e-1}} c^{\varnothing}_{\gamma^r_0 \gamma^r_1 \ldots \gamma^r_{e-1}}.$$

The formula is intended to extend the known multiplicity result for $e$-regular $boldsymbol{mu}$ to the stated characteristic and multipartition hypotheses, and its general validity is the conjectural part of the paper.