表示的容量与Brascamp-Lieb常数
The capacity of quiver representations and Brascamp-Lieb constants
摘要：
设$Q$是二分图，$V$是$Q$的实表示，$σ$是$Q$的与$V$的维数向量正交的整数权。在不变量理论的指导下，引入了与$(V,σ)$相关的Brascamp-Lieb算子$T_{V,σ}$，并研究了它的容量，记为$\mathbf{D}_Q(V, σ)$ .当$Q$是$m$ -子空间不等式时，分析中的$m$ -多线性Brascamp-Lieb不等式中的Brascamp-Lieb常数与数据容量密切相关.我们证明了$\mathbf{D}_Q(V, σ)$的正性等价于$V$的$σ$ -半稳定性。当$\mathbf{D}_Q(V, σ)$为正时，我们还找到了它的特征公式.我们的主要工具是一个著名的结果Kempf-Ness在不变量理论中的闭轨道的一个修正版本。这一结果使我们考虑某些真正的代数簇，携带相关的信息，我们的主要研究对象。它使我们能够表达的能力，由$σ$诱导的字符和样本点的品种所涉及的数据。此外，我们还利用这个特征公式证明了可重构数据容量的一个因子分解。我们还证明了$(V, σ)$的高斯极值的存在性等价于$V$是$σ$ -多稳定的，并且高斯极值的唯一性意味着$V$是$σ$ -稳定的.最后，我们解释了如何找到高斯极值的高斯极值数据$(V, σ)$使用代数簇相关的$(V,σ)$。
英文摘要：
Let $Q$ be a bipartite quiver, $V$ a real representation of $Q$, and $σ$ an integral weight of $Q$ orthogonal to the dimension vector of $V$. Guided by quiver invariant theoretic considerations, we introduce the Brascamp-Lieb operator $T_{V,σ}$ associated to $(V,σ)$ and study its capacity, denoted by $\mathbf{D}_Q(V, σ)$. When $Q$ is the $m$-subspace quiver, the capacity of quiver data is intimately related to the Brascamp-Lieb constants that occur in the $m$-multilinear Brascamp-Lieb inequality in analysis. We show that the positivity of $\mathbf{D}_Q(V, σ)$ is equivalent to the $σ$-semi-stability of $V$. We also find a character formula for $\mathbf{D}_Q(V, σ)$ whenever it is positive. Our main tool is a quiver version of a celebrated result of Kempf-Ness on closed orbits in invariant theory. This result leads us to consider certain real algebraic varieties that carry information relevant to our main objects of study. It allows us to express the capacity of quiver data in terms of the character induced by $σ$ and sample points of the varieties involved. Furthermore, we use this character formula to prove a factorization of the capacity of quiver data. We also show that the existence of gaussian extremals for $(V, σ)$ is equivalent to $V$ being $σ$-polystable, and that the uniqueness of gaussian extremals implies that $V$ is $σ$-stable. Finally, we explain how to find the gaussian extremals of a gaussian-extremisable datum $(V, σ)$ using the algebraic variety associated to $(V,σ)$.
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