秩 $1$ 李群的某些不变式、协变式和辛商的多级 Hilbert 级数
Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups
AI总结：
本文计算圆群和正交群 O₂ 表示的不变式与协变式的单变量及多级 Hilbert 级数，涵盖最大分级和双分级，并给出算法及示例，推广至半直积情形。
摘要：
我们计算了圆群和正交群 $\operatorname{O}_2$ 的表示的不变式和协变式的单变量和多级 Hilbert 级数。所考虑的多级包括与表示分解为不可约表示相关的最大分级，以及与余切提升表示相关的双分级，或等价地，与实不变式和协变式的全纯与反全纯部分相关的双分级。该双分级在辛商的在壳不变式代数上诱导一个双分级，相应的 Hilbert 级数也被计算。我们还计算了单变量 Hilbert 级数的前几个 Laurent 系数，给出了多级 Laurent 系数的示例计算，并给出一个例子说明这些技术对圆群与其他有限群的半直积的推广。我们描述了一个算法来计算每个相关的 Hilbert 级数。
英文摘要：
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.
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