Journal of Pure and Applied Algebra ( IF 0.8 ) Pub Date : 2020-08-25 , DOI: 10.1016/j.jpaa.2020.106524 Chris McDaniel , Larry Smith
Modeled on work of Bott–Samelson, Soergel, and Elias–Williamson we introduce the Bott–Samelson ring associated to reflections where is a field. We show that for semisimple reflections it is a complete intersection algebra with a triangular pattern of relations and has the strong Lefschetz property. For a finite nonmodular reflection group generated by reflections of order two we produce a degree one embedding of the coinvariant algebra into a Bott–Samelson algebra where are reflections of order two generating G answering a question of J. Watanabe. We show by example that the strong Lefschetz property of need not be inherited by the embedded answering another question of J. Watanabe. Finally we investigate the decomposition map where we start with a representation and its associated equivariant coinvariant algebra , and for each the w-twisted multiplication map is defined by . These fit together to define the decomposition map into the direct sum of copies of indexed by the elements of W and we show this map is a monomorphism if the representation of W is nonmodular. For a reflection representation ρ of W and a family of reflections generating W we also define a map and characterize for which families of reflections it is a degree one embedding.
中文翻译:
等变协变环,Bott–Samelson环和渡边的大胆猜想
以Bott–Samelson,Soergel和Elias–Williamson的工作为模型,我们介绍了Bott–Samelson环 与反思相关 哪里 是一个领域。我们表明,对于半简单反射,它是具有三角关系的完整交点代数,并具有很强的Lefschetz性质。对于由二阶反射生成的有限非模反射群,我们产生协变代数的一阶嵌入 进入博特-萨默森 代数在哪里 是阶次生成G的反映,回答了渡边J.的问题。我们通过示例表明,Lefschetz的强属性 不需要被嵌入式继承 回答渡边J.的另一个问题。最后我们研究分解图
我们从表示开始 及其相关的等变协变代数 ,并且每个 在W¯¯ -twisted乘法地图 由定义 。这些适合在一起以定义分解图 直接成 的副本 由W的元素索引,并且如果W的表示形式是非模块化的,则表明该映射是单态的。用于反射表示ρ的W¯¯和家庭反射生成W我们还定义了一张地图 并说明它是针对哪一类反射的嵌入。