The blob simple-module dimension and intersection-form conjecture

Let $P_1^l(n)$ be the set of $l$-tuples of non-negative integers summing to $n$, and let $P(\boldsymbol{\lambda})$ denote the orbit of $\boldsymbol{\lambda}$. For a regular $\boldsymbol{\lambda}\in P_1^l(n)$ and $\boldsymbol{\mu}\in P(\boldsymbol{\lambda})$, let $L^p_{\boldsymbol{\lambda}}(\boldsymbol{\mu})$ be the corresponding graded simple module, and let $I^p_{\underline{w}_{\boldsymbol{\lambda}}}(w_{\boldsymbol{\mu}})$ be the associated intersection form. Blob simple-module dimension and intersection-form conjecture.

$$\operatorname{dim}_v L^p_{\boldsymbol{\lambda}}(\boldsymbol{\mu})=\operatorname{rk}_v I^p_{\underline{w}_{\boldsymbol{\lambda}}}(w_{\boldsymbol{\mu}}).$$

This conjecture supplies the graded simple-module dimensions needed to compute decomposition numbers in positive characteristic, in direct analogy with the role of intersection forms in $p$-Kazhdan–Lusztig theory.