Lawrence-Krammer representations are an important family of linear representations of Artin-Tits groups of small type, which are known, under some assumptions on the parameters, to be faithful when the type is spherical (or more generally when they are restricted to the Artin-Tits monoid) and irreducible when the type is connected. Here, we investigate an analogue of these representations --- introduced by Digne in the spherical cases --- for every Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under a group of graph automorphisms, and for the corresponding Artin-Tits group. Under the same assumptions on the parameters as in the small type cases, we first show that these so-called "twisted Lawrence-Krammer representations" are faithful, and we then prove, by computing their formulas when the group of graph automorphisms is of order two or three, their irreducibility in all the spherical and connected cases but one.

中文翻译：

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扭曲的 Lawrence-Krammer 表示

Lawrence-Krammer 表示是小类型 Artin-Tits 群的一个重要的线性表示系列，在参数的一些假设下，当类型是球形时（或更一般地说，当它们被限制在 Artin -Tits monoid) 并且当类型连接时不可约。在这里，我们研究了这些表示的类似物 --- 由 Digne 在球形情况下引入 --- 对于每个 Artin-Tits 幺半群，它表现为一组图下的小类型 Artin-Tits 幺半群的不动点的亚幺半群自同构，以及相应的 Artin-Tits 群。在与小类型情况相同的参数假设下，我们首先证明这些所谓的“扭曲的劳伦斯-克雷默表示”是忠实的，然后我们证明，