5 problems

Yvonne's conjecture. The matrix $abla^{+}(q)$ coincides with the $q$-analogue of the decomposition matrices of these cyclotomic Schur algebras.

Uglov's conjecture. The transition matrix of the homogeneous component of degree $m$ of the canonical basis of $\boldsymbol{\operatorname{F}}_q[\boldsymbol s_l]$ is equal to the de…

Let $4D(q)4$ be the $q$-analogue of the decomposition matrix of a cyclotomic $v$-Schur algebra, and let $4\Delta(q)4$ be the transition matrix between the standard basis and the ca…

For a level $\ell$ and multipartition data $\mu,\nu$, let $\mathrm{ParMat}_{\nu,\mu}^{\flat}$ denote the corresponding set of enriched block matrices, let…

Let $d_{\lambda\mu}(q)$ be the $q$-decomposition number of the cyclotomic $v$-Schur algebra, and let $\varDelta^+_{\mu^{\dag},\lambda^{\dag}}(q)$ be the corresponding canonical-bas…