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Deligne范畴$\underline{Rep}(GL_t)$中的范畴作用与多重性

Categorical actions and multiplicities in the Deligne category $\underline{Rep}(GL_t)$

Inna Entova-Aizenbud

备注：
ver2: fixed Section 10 and some typos. ver3: fixed support acknowledgement
发表信息：
Journal of Algebra 504 (2018): 391-431
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摘要：

本文研究了在arXiv：1511.07699中构造的Deligne范畴$\mathcal{D}_t=\underline{Rep}(GL_t)$（这里是$t \in \mathbb{C}$）及其“阿贝尔包络”$\mathcal{V}_t$上的范畴类型A作用.对于$t \in \mathbb{Z}$，这个作用将李代数$\mathfrak{sl}_{\mathbb{Z}}$在Fock空间$\mathfrak{F}$与$\mathfrak{F}_t^{\vee}$的张量积上的作用分类，其限制对偶被$t$“移位”，如I.洛瑟夫事实上，这个动作使范畴$\mathcal{V}_t$成为范畴$\mathfrak{sl}_{\mathbb Z}$ -模块$Pol$和$Pol_t^{\vee}$的张量积（在Losev和Webster的意义上，arXiv：1303.1336）。后者分别对$\mathfrak{F}$和$\mathfrak{F}_t^{\vee}$进行了分类，这两种情况下的基本范畴都是稳定多项式表示的范畴（也称为舒尔函子的范畴），如Hong和Yacobi，arXiv：1101.2456（也参见Losev，arXiv：1209.1067）所述。当$t \notin \mathbb Z$时，Deligne范畴$\mathcal{D}_t$是交换半单的，并且A型作用诱导$\mathfrak{sl}_{\mathbb{Z}} \times \mathfrak{sl}_{\mathbb{Z}}$的范畴作用。这个动作将$\mathfrak{sl}_{\mathbb{Z}} \times \mathfrak{sl}_{\mathbb{Z} }$ -模块$\mathfrak{F} \boxtimes \mathfrak{F}^{\vee}$分类，使$\mathcal{D}_t$成为范畴$\mathfrak{sl}_{\mathbb Z}$ -模块$Pol$、$Pol^{\vee}$的外张量积。在此过程中，我们建立了一个新的Kazhdan-Lusztig系数和$\mathcal{V}_t$倾斜对象的标准过滤中的多重性之间的关系。

英文摘要：

We study the categorical type A action on the Deligne category $\mathcal{D}_t=\underline{Rep}(GL_t)$ (here $t \in \mathbb{C}$) and its "abelian envelope" $\mathcal{V}_t$ constructed in arXiv:1511.07699. For $t \in \mathbb{Z}$, this action categorifies an action of the Lie algebra $\mathfrak{sl}_{\mathbb{Z}}$ on the tensor product of the Fock space $\mathfrak{F}$ with $\mathfrak{F}_t^{\vee}$, its restricted dual "shifted" by $t$, as was suggested by I. Losev. In fact, this action makes the category $\mathcal{V}_t$ the tensor product (in the sense of Losev and Webster, arXiv:1303.1336) of categorical $\mathfrak{sl}_{\mathbb Z}$-modules $Pol$ and $Pol_t^{\vee}$. The latter categorify $\mathfrak{F}$ and $\mathfrak{F}_t^{\vee}$ respectively, the underlying category in both cases being the category of stable polynomial representations (also known as the category of Schur functors), as described by Hong and Yacobi, arXiv:1101.2456 (see also Losev, arXiv:1209.1067). When $t \notin \mathbb Z$, the Deligne category $\mathcal{D}_t$ is abelian semisimple, and the type A action induces a categorical action of $\mathfrak{sl}_{\mathbb{Z}} \times \mathfrak{sl}_{\mathbb{Z}}$. This action categorifies the $\mathfrak{sl}_{\mathbb{Z}} \times \mathfrak{sl}_{\mathbb{Z} }$-module $\mathfrak{F} \boxtimes \mathfrak{F}^{\vee}$, making $\mathcal{D}_t$ the exterior tensor product of the categorical $\mathfrak{sl}_{\mathbb Z}$-modules $Pol$, $Pol^{\vee}$. Along the way we establish a new relation between the Kazhdan-Lusztig coefficients and the multiplicities in the standard filtrations of tilting objects in $\mathcal{V}_t$.

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