Kessar–Schaps crossing conjecture in the opposite-sign case

Establish the Kessar–Schaps crossing conjecture for spin $p$-blocks labeled by $p$-bar cores $kappa$ and $kappa'$ satisfying $\operatorname{sgn}(\kappa)=1$ and $\operatorname{sgn}(\kappa')=-1$, including the corresponding perfect-isometry assertion beyond the weaker form previously proved.

Background

The paper studies perfect isometries and Galois–Navarro equivariance for spin $p$-blocks of double covering groups of symmetric and alternating groups. In the crossing case with $\operatorname{sgn}(\kappa)=-1$ and $\operatorname{sgn}(\kappa')=1$, the authors prove an $\mathcal H$-equivariant correspondence under a compatibility condition on the Galois signs of the two cores.

The opposite crossing orientation, with $\operatorname{sgn}(\kappa)=1$ and $\operatorname{sgn}(\kappa')=-1$, is not covered by the equivariance theorem: the perfect isometry constructed in the cited prior work fails to commute with the action of $\mathcal H$. The paper identifies an additional factor $(-1)^{(p-1)/2}$ in this case, while the underlying Kessar–Schaps crossing conjecture is described as including these cases; thus the full conjectural statement remains beyond the results established here.