Irreducibility of the tensor product of Yangian \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \) evaluation modules

Abstract

The evaluation homomorphism from the super Yangian \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \) to \( \mathrm{U}(\mathfrak{gl}_{m|n}) \) induces a \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \)-module structure on any finite dimensional simple \( \mathrm{U}(\mathfrak{gl}_{m|n}) \)-module \( L(\lambda) \). In this paper, we give necessary and sufficient conditions for the tensor product of such evaluation \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \)-modules, \( L_g(\lambda) \otimes L_h(\gamma) \), to be simple, provided each of \( \lambda \) and \( \gamma \) is either covariant tensor or essentially typical. Our proof is based on the existence of a Gelfand--Tsetlin basis for finite dimensional simple \( \mathrm{U}(\mathfrak{gl}_{m|n}) \)-modules with highest weights that belong to these two families: covariant tensor and essentially typical. The obtained result is a super analogue of the Molev's result for the classical Yangian \( \mathrm{Y}(\mathfrak{gl}_n) \). Combining this with the binary property of tensor products of covariant evaluation modules, we obtain an irreducibility criterion for arbitrary tensor products of covariant evaluation modules.