论文标题
对称代数上不变的图形方法和环
Graphical methods and rings of invariants on the symmetric algebra
论文作者
论文摘要
让$ g $成为一个复杂的古典群体，让$ v $为其定义表示形式（可能是二元的副本）。经典不变理论中的一个基本问题是，在空间上写下$ g $ invariant多项式函数的发电机和关系，$ \ mathcal {p}^m（v）$ $ m $ - $ m $同质的多项式函数的$ v $。在本文中，我们用完整的多项式代数$ \ MATHCAL {p}（v）$替换$ \ Mathcal {p}^m（v）$。结果，不变环不再是有限生成的。因此，我们旨在写下大型组件的线性基础，而不是寻求发电机。确实，当$ g $的排名足够高时，我们将这些基础视为具有规定数量的顶点和边缘的图集。当$ g $的等级很小时，图表之间存在复杂的线性依赖性，但是我们通过表示理论来纠正此挫折：特别是，我们确定了任意组件的维度，该维度是从一般线性组到对称组的分支倍增性方面的分支倍数。因此，我们在$ \ Mathcal {p}（v）$上获得了Bigraded Hilbert系列的Bigraded Hilbert系列的表达。我们以图形符号为例，以示例为例，其中一些恢复了经典结果。
Let $G$ be a complex classical group, and let $V$ be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of $G$-invariant polynomial functions on the space $\mathcal{P}^m(V)$ of degree-$m$ homogeneous polynomial functions on $V$. In this paper, we replace $\mathcal{P}^m(V)$ with the full polynomial algebra $\mathcal{P}(V)$. As a result, the invariant ring is no longer finitely generated. Hence instead of seeking generators, we aim to write down linear bases for bigraded components. Indeed, when $G$ is of sufficiently high rank, we realize these bases as sets of graphs with prescribed number of vertices and edges. When the rank of $G$ is small, there arise complicated linear dependencies among the graphs, but we remedy this setback via representation theory: in particular, we determine the dimension of an arbitrary component in terms of branching multiplicities from the general linear group to the symmetric group. We thereby obtain an expression for the bigraded Hilbert series of the ring of invariants on $\mathcal{P}(V)$. We conclude with examples using our graphical notation, several of which recover classical results.
