Nonvanishing of submaximal p-canonical coefficients in type C

Determine whether the coefficient \({}^p h_{w_0u,x}\) is nonzero in type \(C_n\) for characteristic \(p=2\), where \(x=w_0s_1s_2s_1\) and \(u=s_1s_2s_1[3,1]\cdots[k,1]\) for each \(3\leq k\leq n\).

Background

The paper determines the p-canonical basis elements associated with the submaximal cell in all classical types, except for the element ${}^p_{w_0s_1s_2s_1}$ in type $C_n$ when $p=2$ and $n\geq 3$. The remark records a known expansion for $n=3$ and derives, by parabolic induction, a nonzero coefficient for a particular element when $n>3$.

The authors then formulate a broader conjecture asserting nonvanishing for every element $u=s_1s_2s_1[3,1]\cdots[k,1]$ with $3\leq k\leq n$. They note that the conjecture has been verified for $n=4$, but leave the general case unresolved. Establishing it would complete the missing information about the relevant p-canonical basis expansion and support the analysis of the exceptional submaximal p-cell in type $C_n$.