The reduced decomposition-matrix conjecture for spin symmetric groups

Let $m$ be a positive integer, let $p$ be an odd prime, and let ${\rm DP}(m)$ denote the relevant set of strict partitions of $m$. Let ${\rm DPR}_p(m)$ denote the indexing set for the reduced decomposition matrix in characteristic $p$. For each $\mu\in {\rm DPR}_p(m)$, let $\underline{\underline{G}}(\mu)$ be the column vector obtained by removing the largest common power of $2$ from the coefficients of $\underline{G}(\mu)$, where

$$\underline{G}(\mu)=\sum_{\lambda\in {\rm DP}(m)}2^{b(\lambda)-a_p(\lambda)}d_{\lambda\mu}(1)\langle\widehat{\lambda}\rangle,$$

$b(\lambda)=\left\lfloor(m-\ell(\lambda))/2\right\rfloor$, and $a_p(\lambda)$ is the quantity defined in the source. Let the reduced decomposition matrix of the spin symmetric group be obtained from the usual spin decomposition matrix by combining associate columns and expressing the resulting columns in the basis $\langle\widehat{\lambda}\rangle$. Reduced decomposition-matrix conjecture. (i) If $p^2>m$, the set of columns of the reduced decomposition matrix coincides with $\{\underline{\underline{G}}(\mu)\mid\mu\in {\rm DPR}_p(m)\}$. (ii) If $p^2\leq m$, the reduced decomposition matrix is obtained by postmultiplying the matrix with columns $\underline{\underline{G}}(\mu)$ by a unitriangular matrix with nonnegative entries. The conjecture is supported by numerical tables for $p=3,5,7,11$ reported in the source, but no resolution is given in the supplied text.