论文标题
多式希尔伯特一系列不变的人，协变量和某些等级的符合人物$ 1 $ lie groupp
Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups
论文作者
论文摘要
我们计算单变量和多面化的希尔伯特系列不变的系列和圆圈和正交组$ \ operatatorName {o} _2 $的协变。所考虑的跨层包括与代表分解为不可减数的最大分级以及与固定型代表形式相关的大型分级，或者等效地，与真实不变和协方差群的全体形态和抗黑色晶状体相关的大型级别。这种巨大的诱导诱导了符号商的壳体不变的代数，也计算了相应的希尔伯特系列。我们还计算单变量希尔伯特系列的前几个劳伦系数，对多层的月朗特系数进行了示例计算，并举例说明了这些技术将这些技术扩展到其他有限组的圆圈的半程乘积。我们描述了一种计算每个相关希尔伯特系列的算法。
We compute univariate and multigraded Hilbert series of invariants and covariants of representations of the circle and orthogonal group $\operatorname{O}_2$. The multigradings considered include the maximal grading associated to the decomposition of the representation into irreducibles as well as the bigrading associated to a cotangent-lifted representation, or equivalently, the bigrading associated to the holomorphic and antiholomorphic parts of the real invariants and covariants. This bigrading induces a bigrading on the algebra of on-shell invariants of the symplectic quotient, and the corresponding Hilbert series are computed as well. We also compute the first few Laurent coefficients of the univariate Hilbert series, give sample calculations of the multigraded Laurent coefficients, and give an example to illustrate the extension of these techniques to the semidirect product of the circle by other finite groups. We describe an algorithm to compute each of the associated Hilbert series.
