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广义Rook幺半群的Jucys-Murphy元和Grothendieck群

Jucys-Murphy elements and Grothendieck groups for generalized rook monoids

Volodymyr Mazorchuk, Shraddha Srivastava

备注：
Minor changes and added a few more references. Comments welcome!
发表信息：
J. Comb. Algebra, 2022
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摘要：

我们考虑复数域$\mathbb{C}$上的广义车幺半群代数的塔，并观察到与此塔相关联的Bratteli图是一个简单图.本文构造了广义车幺半群的单模，并刻画了广义车幺半群的Jucys-Murphy元.在正特征$p$的代数闭域$\Bbbk$上，利用车幺半群代数的Jucys-Murphy元，对$0\leq i\leq p-1$定义了相应的$i$ -限制函子和$i$ -诱导函子以及两个额外的函子.在$\Bbbk$上的车幺半群代数上的模范畴的Grothendieck群的直和$\mathcal{G}_{\mathbb{C}}$上，这些函子诱导出泛包络代数$U(\hat{\mathfrak{sl}}_p(\mathbb{C}))$与双圈幺半群$\mathcal{B}$的幺半群代数$\mathbb{C}[\mathcal{B}]$的张量积的作用。进一步证明了$\mathcal{G}_{\mathbb{C}}$同构于$U(\hat{\mathfrak{sl}}_{p}(\mathbb{C}))$的基本表示与$\mathbb{C}[\mathcal{B}]$上唯一无限维单模的张量积，并证明了$\mathcal{G}_{\mathbb{C}}$是一个双代数.在对$\Bbbk$的特征的某些自然限制下，我们给出了广义车幺半群的相应结果.

英文摘要：

We consider a tower of generalized rook monoid algebras over the field $\mathbb{C}$ of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys-Murphy elements for generalized rook monoid algebras. Over an algebraically closed field $\Bbbk$ of positive characteristic $p$, utilizing Jucys-Murphy elements of rook monoid algebras, for $0\leq i\leq p-1$ we define the corresponding $i$-restriction and $i$-induction functors along with two extra functors. On the direct sum $\mathcal{G}_{\mathbb{C}}$ of the Grothendieck groups of module categories over rook monoid algebras over $\Bbbk$, these functors induce an action of the tensor product of the universal enveloping algebra $U(\hat{\mathfrak{sl}}_p(\mathbb{C}))$ and the monoid algebra $\mathbb{C}[\mathcal{B}]$ of the bicyclic monoid $\mathcal{B}$. Furthermore, we prove that $\mathcal{G}_{\mathbb{C}}$ is isomorphic to the tensor product of the basic representation of $U(\hat{\mathfrak{sl}}_{p}(\mathbb{C}))$ and the unique infinite-dimensional simple module over $\mathbb{C}[\mathcal{B}]$, and also exhibit that $\mathcal{G}_{\mathbb{C}}$ is a bialgebra. Under some natural restrictions on the characteristic of $\Bbbk$, we outline the corresponding result for generalized rook monoids.

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