论文标题
Schur Superalgebras的对称器
Symmetrizers for Schur superalgebras
论文作者
论文摘要
对于schur superalgebra $ s = s（m | n，r）$在地面上的特征零，我们定义了symmetrizers $ t^λ[i：j] $ tableaux $ t_i，t_i，t_j $的形状$λ$，并显示出$ k $ k $ -span $ -span $ a __-s $ a___ __ _ _^us k {λ， $ t^λ[i：j] $具有$ t^λ[i：j] $的基础，$ t_i，t_j $ semistandard。 $ s $ -superbimodule $ a_ {λ，k} $被标识为％$δ（λ）^*\ otimes_k \ nabla（λ）$，其中$δ（λ）^*$是标准的supermodule％和$ \ nabla and $ \ nabla（λ）$ cost of costprive the Costprive $ cosspryule the the Standard supermodule％和$ nable。 $d_λ\ otimes_k d^o_λ$，其中$d_λ$和$ d^o_λ$是左而右的，不可约$ s $ -supermodules，最高权重$λ$。 我们定义修改的对称器$ t^λ\ {i：j \} $，并证明他们的$ \ mathbb {z} $ - span形式a $ \ mathbb {z} $ - form $ a_ {λ{λ，\ mathbb {z}} $ $ a_ {z}} $我们表明，每个修改的对称器$ t^λ\ {i：j \} $都是$ \ mathbb {z} $ - symemetrizers $ t^λ\ {i：j \} $的线性组合，用于$ t_i，t_j j $ semistandard。使用模块化减少到特征$ p> 2 $的字段$ k $，我们得到$ a_ {λ，k} $具有由修改的symmetrizers $ t^λ\ {i：j \} $组成的基础。
For the Schur superalgebra $S=S(m|n,r)$ over a ground field $K$ of characteristic zero, we define symmetrizers $T^λ[i:j]$ of the ordered pairs of tableaux $T_i, T_j$ of the shape $λ$ and show that the $K$-span $A_{λ,K}$ of all symmetrizers $T^λ[i:j]$ has a basis consisting of $T^λ[i:j]$ for $T_i,T_j$ semistandard. The $S$-superbimodule $A_{λ,K}$ is identified as %$Δ(λ)^*\otimes_K \nabla(λ)$, where $Δ(λ)^*$ is the dual of the standard supermodule %and $\nabla(λ)$ is the costandard supermodule of the highest weight $λ$. $D_λ\otimes_K D^o_λ$, where $D_λ$ and $D^o_λ$ are left and right irreducible $S$-supermodules of the highest weight $λ$. We define modified symmetrizers $T^λ\{i:j\}$ and show that their $\mathbb{Z}$-span form a $\mathbb{Z}$-form $A_{λ,\mathbb{Z}}$ of $A_{λ, \mathbb{Q}}$. We show that every modified symmetrizer $T^λ\{i:j\}$ is a $\mathbb{Z}$-linear combination of symmetrizers $T^λ\{i:j\}$ for $T_i, T_j$ semistandard. Using modular reduction to a field $K$ of characteristic $p>2$, we obtain that $A_{λ,K}$ has a basis consisting of modified symmetrizers $T^λ\{i:j\}$ for $T_i, T_j$ semistandard.
