论文标题
\ vsapovalov元素的明确表达式
Explicit expressions for \vSapovalov elements in Type A
论文作者
论文摘要
我们为A型代数和超级甲壳虫中的\ vSapovalov元素提供明确的表达式。使用非共同的决定因素，在Arxiv：1710.10528第9节中已经给出了显式表达式，实际上，我们的第一个主要结果，定理2.3和2.6可以看作是这些决定因素的完整扩展。但是我们提供了新的证据，这似乎更容易，因为它们避免了诱导和辅助因子的扩展。我们还描述了FGL（M，N）在定理3.7中的任意Borel子代甲虫方面的\ vSapovalov元素，并解释了A型A型的\ vSapovalov元素作为Hessenberg矩阵的确定因素4.5和4.9。订购。具有\ vsapovalov元素的明确表达式，使我们可以简单地证明表示理论的几个结果，请参见第6节和第4.6节。
We give explicit expressions for \vSapovalov elements in Type A Lie algebras and superalgebras. Explicit expressions were already given in arXiv:1710.10528 Section 9, using non-commutative determinants, and in fact our first main results, Theorems 2.3 and 2.6 can be viewed as complete expansions of these determinants. But we give new proofs, which seem easier because they avoid induction and cofactor expansion. We also describe \vSapovalov elements for fgl(m,n)with respect to an arbitrary Borel subalgebra in Theorem 3.7, and interpret \vSapovalov elements in Type A as determinants of Hessenberg matrices in Theorems 4.5 and 4.9.The exact form of the explicit expressions depends on an ordering on the set of positive roots, and Hessenberg matrices are useful in changing the ordering. Having explicit expressions for \vSapovalov elements allows us to give easy proofs of several results in representation theory, see Section 6 and Subsection 4.6.
