AI 中文总结
本文研究了超Yangian $ \mathrm{Y}(\mathfrak{gl}_{m|n}) $ 评估模块张量积的不可约性条件，基于Gelfand-Tsetlin基的存在性，给出了简单模块的判据。
AI 中文摘要
从超Yangian $ \mathrm{Y}(\mathfrak{gl}_{m|n}) $ 到 $ \mathrm{U}(\mathfrak{gl}_{m|n}) $ 的评估同态诱导了任何有限维简单 $ \mathrm{U}(\mathfrak{gl}_{m|n}) $-模块 $ L(\lambda) $ 上的 $ \mathrm{Y}(\mathfrak{gl}_{m|n}) $-模块结构。在本文中，我们给出了这样的评估 $ \mathrm{Y}(\mathfrak{gl}_{m|n}) $-模块 $ L_g(\lambda) \otimes L_h(\gamma) $ 为简单模块的必要且充分条件，前提是每个 $ \lambda $ 和 $ \gamma $ 分别是协变张量或本质典型。我们的证明基于有限维简单 $ \mathrm{U}(\mathfrak{gl}_{m|n}) $-模块具有最高权属于这两种家族（协变张量和本质典型）的Gelfand-Tsetlin基的存在性。所获得的结果是Molev结果在超Yangian $ \mathrm{Y}(\mathfrak{gl}_n) $ 上的超代数类比。结合协变评估模块张量积的二元性质，我们得到了任意协变评估模块张量积的不可约性判据。
英文摘要
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 \( L(λ) \). In this paper, we give necessary and sufficient conditions for the tensor product of such evaluation \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \)-modules, \( L_g(λ) \otimes L_h(γ) \), to be simple, provided each of \( λ\) and \( γ\) 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.
