Let \(G=GL(m|n)\) be the general linear supergroup over an algebraically closed field K of characteristic zero, and let \(G_{ev}=GL(m)\times GL(n)\) be its even subsupergroup. The induced supermodule \(H^0_G(\lambda )\), corresponding to a dominant weight \(\lambda \) of G, can be represented as \(H^0_{G_{ev}}(\lambda )\otimes \Lambda (Y)\), where \(Y=V_m^*\otimes V_n\) is a tensor product of the dual of the natural GL(m)-module \(V_m\) and the natural GL(n)-module \(V_n\), and \(\Lambda (Y)\) is the exterior algebra of Y. For a dominant weight \(\lambda \) of G, we construct explicit \(G_{ev}\)-primitive vectors in \(H^0_G(\lambda )\). Related to this, we give explicit formulas for \(G_{ev}\)-primitive vectors of the supermodules \(H^0_{G_{ev}}(\lambda )\otimes \otimes ^k Y\). Finally, we describe a basis of \(G_{ev}\)-primitive vectors in the largest polynomial subsupermodule \(\nabla (\lambda )\) of \(H^0_G(\lambda )\) (and therefore in the costandard supermodule of the corresponding Schur superalgebra S(m|n)). This yields a description of a basis of \(G_{ev}\)-primitive vectors in arbitrary induced supermodule \(H^0_G(\lambda )\).

中文翻译：

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一般线性超群的诱导超模和Schur超代数的共标准超模中的偶本原矢量

令\（G = GL（m | n）\）为特征为零的代数封闭场K上的一般线性超群，令\（G_ {ev} = GL（m）\ times GL（n）\）为它的偶次超群。感应supermodule \（H ^ 0_G（\拉姆达）\） ，对应于占主导地位的重量\（\拉姆达\）的G ^，可以表示为\（H ^ 0_ {G_ {EV}}（\拉姆达）\ otimes \ Lambda（Y）\），其中\（Y = V_m ^ * \ otimes V_n \）是自然GL（m）-模\（V_m \）和自然GL（n）的对偶的张量积模块\（V_n \）和\（\ Lambda（Y）\）是Y的外代数。了主导重量\（\拉姆达\）的G ^，我们构建明确\（G_ {EV} \）在-primitive矢量\（H ^ 0_G（\拉姆达）\） 。与此相关，我们给出了超模块\（H ^ 0_ {G_ {ev}}（\ lambda）\ otimes \ otimes ^ k Y \）的\（G_ {ev} \）-本原向量的显式公式。最后，我们描述的基础\（G_ {EV} \）在最大多项式subsupermodule -primitive矢量\（\ nabla（\拉姆达）\）的\（H ^ 0_G（\拉姆达）\） （并因此在对应的Schur超代数S（米| n））。这给出了在任意诱导的超模块\（H ^ 0_G（\ lambda）\）中\（G_ {ev} \）-本原矢量的基础的描述。