论文标题
$ a $ a $ and $ c $的Cyclotomic KLR代数的内容系统和变形
Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$
论文作者
论文摘要
本文启动了一项系统的研究，对Aggine型$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a和$ c $的系统研究。首先，使用内容系统引入了这些代数的分级变形，并使用内容系统和对称组的年轻人的eminoral形式的概括构建了变形环体klr代数的所有不可约表示。令人惊讶的是，该理论同时捕获了$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a和$ c $的代表理论，主要区别在于tableaux残留物序列的定义。然后，我们使用半神经变形构建两个“双重”蜂窝基础，用于非偏见的KLR代数$ a $ a $ a $ and $ c $。作为该理论的应用，我们从类型$ a $中的表示理论中恢复了许多主要功能，同时证明了它们用于$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a和$ c $的环体klr代数。这些结果在类型$ c $中是全新的，我们通常在$ $ a $中更直接的证明。特别是，我们表明这些代数对相应的Kac-Moody代数的不可约合的最高权重模块进行了分类，我们构建和分类了它们的简单模块，我们研究了与规范底座的链接，并将Kleshchchev的模块化分支规则推广到这些代数。
This paper initiates a systematic study of the cyclotomic KLR algebras of affine types $A$ and $C$. We start by introducing a graded deformation of these algebras and the constructing all of the irreducible representations of the deformed cyclotomic KLR algebras using content systems and a generalisation of the Young's seminormal forms for the symmetric groups. Quite amazingly, this theory simultaneously captures the representation theory of the cyclotomic KLR algebras of types $A$ and $C$, with the main difference being the definition of residue sequences of tableaux. We then use our semisimple deformations to construct two "dual" cellular bases for the non-semisimple KLR algebras of affine types $A$ and $C$. As applications of this theory we recover many of the main features from the representation theory in type $A$, simultaneously proving them for the cyclotomic KLR algebras of types $A$ and $C$. These results are completely new in type $C$ and we, usually, more direct proofs in type $A$. In particular, we show that these algebras categorify the irreducible integrable highest weight modules of the corresponding Kac-Moody algebras, we construct and classify their simple modules, we investigate links with canonical bases and we generalise Kleshchev's modular branching rules to these algebras.
