The graded Cartan elementary-divisor conjecture for Hecke algebras

Let $p$ be a prime. For a partition $\boldsymbol{\mu}=(\mu^{(1)},\dots,\mu^{(p-1)})\in M_{p-1}(d)$, let $w_H(\mu^{(p-1)})$ be the corresponding weight, and let the block of $p$-weight $d$ be the associated block of the graded Cartan matrix. Let $C_n(q)$ be the graded Cartan matrix, and let $P_{(p)}(n)$ denote the set of $p$-regular partitions of $n$, with weights $w_E$ and $w_G$ as defined in the paper. The graded Cartan elementary-divisor conjecture. (i) The elementary divisors of the block of $p$-weight $d$ of the graded Cartan matrix over $\mathbb{Q}[q,q^{-1}]$ coincide with those of the diagonal matrix with entries

$$\left\{w_H(\mu^{(p-1)})\mid \boldsymbol{\mu}=(\mu^{(1)},\dots,\mu^{(p-1)})\in M_{p-1}(d)\right\}.$$

(ii) The elementary divisors of $C_n(q)$ over $\mathbb{Q}[q,q^{-1}]$ coincide with those of the diagonal matrices with entries

$$\left\{w_E(\lambda)\mid \lambda\in P_{(p)}(n)\right\}$$

and

$$\left\{w_G(\lambda)\mid \lambda\in P_{(p)}(n)\right\}.$$

This conjecture proposes explicit combinatorial descriptions of the elementary divisors of graded Cartan matrices, refining the known determinant formulas and relating blockwise and global descriptions.