The semisimplicity criterion for cyclotomic Hecke–Clifford superalgebras

Let $\mathbb{K}$ be the ground field, let $q$ be the parameter, let $\underline{Q}=(Q_1,Q_2,\ldots,Q_m)\in(\mathbb{K}^*)^m$, and let $f=f^{(\bullet)}_{\underline{Q}}$ with $\bullet\in\{\mathtt{0},\mathtt{s},\mathtt{ss}\}$. Write $\mathcal{H}^f_{\Delta}(n)$ for the corresponding cyclotomic Hecke–Clifford superalgebra, and call a module completely splittable when the elements $X_1,X_2,\ldots,X_n$ act semisimply on it. Let $P^{\bullet}_{n}(q^2,\underline{Q})$ be the associated parameter polynomial. Semisimplicity and complete splittability conjecture. The following are equivalent:

1. The algebra $\mathcal{H}^f_{\Delta}(n)$ is semisimple.
2. Every irreducible $\mathcal{H}^f_{\Delta}(n)$-module is completely splittable.
3. $P^{\bullet}_{n}(q^2,\underline{Q})\neq 0$.

This would characterize semisimplicity through the complete splittability of irreducible modules and the nonvanishing of the parameter polynomial. The preceding results establish that the nonvanishing condition implies semisimplicity and that, under this condition, every irreducible module is completely splittable; the converse implications are stated here as a conjecture and are left for future work.