论文标题
一般线性超级群
Levi-type Schur-Sergeev duality for general linear super groups
论文作者
论文摘要
在本说明中，我们研究了一种通用线性超级组的双重核心属性。对于超级空间，$ v = \ mathbb {k}^{m \ mid n} $在代数封闭的字段$ \ mathbb {k} $上的特征不等于$ 2 $超组$ \ tilde g：= \ text {gl}（v）\ times \ textbf {g} _m $上的自然动作。然后，我们将张量产品supermodule（$ \ useverline {v}^{\ otimes r} $，$ρ_r$）和$ \ tilde g $。我们提出了一种广义的Schur-Sergeev二元性，据说$ \ tilde g $的Schur Superalgebras $ s'（m | n，r）$和所谓的弱变性dementer double double hecke代数$ \下划线{\ Mathcal {\ Mathcal {h} _ r $ r $是双中央库。弱脱位双Hecke代数是无限尺寸代数，在张量产品空间上具有自然表示。这个概念来自\ cite {b-y-y2020}，并进行了一些修改。
In this note, we investigate a kind of double centralizer property for general linear supergroups. For the super space $V=\mathbb{K}^{m\mid n}$ over an algebraically closed field $\mathbb{K}$ whose characteristic is not equal to $2$, we consider its $\mathbb{Z}_2$-homogeneous one-dimensional extension $\underline V=V\oplus\mathbb{K}v$, and the natural action of the supergroup $\tilde G:=\text{GL}(V)\times \textbf{G}_m$ on $\underline V$. Then we have the tensor product supermodule ($\underline{V}^{\otimes r}$, $ρ_r$) of $\tilde G$. We present a kind of generalized Schur-Sergeev duality which is said that the Schur superalgebras $S'(m|n,r)$ of $\tilde G$ and a so-called weak degenerate double Hecke algebra $\underline{\mathcal{H}}_r$ are double centralizers. The weak degenerate double Hecke algebra is an infinite dimensional algebra, which has a natural representation on the tensor product space. This notion comes from \cite{B-Y-Y2020}, with a little modification.
