The type A–C correspondence conjecture for homomorphisms between hook Specht modules

Let $R_n(\Gamma,p)$ be the quiver Hecke algebra corresponding to a quiver $\Gamma$ over a field of characteristic $p$, and let $\mu,\lambda\vdash n$ be arbitrary partitions. Type A–C correspondence conjecture.

1. For $e\in\mathbb{Z}_{\geq 2}$, whenever

$$\operatorname{Hom}_{R_n(A_{2e-1}^{(1)},p)}(S^\mu,S^\lambda)\neq\{0\},$$

we also have

$$\operatorname{Hom}_{R_n(C_e^{(1)},p)}(S^\mu,S^\lambda)\neq\{0\}$$

for any charge $\kappa$.

2. Whenever

$$\operatorname{Hom}_{R_n(C_2^{(1)},2)}(S^\mu,S^\lambda)\neq\{0\}$$

for any charge $\kappa$, we also have

$$\operatorname{Hom}_{R_n(A_1^{(1)},2)}(S^\mu,S^\lambda)\neq\{0\}.$$

The conjecture proposes inclusions between families of nonzero homomorphism spaces for Specht modules over quiver Hecke algebras of types A and C. The preceding examples and the paper's main results motivate these patterns, but the source provides no resolution of the general assertions.