论文标题
刚性本地系统和乘法特征值问题
Rigid local systems and the multiplicative eigenvalue problem
论文作者
论文摘要
我们提供了一种结构，该结构在$ \ bbb {p} _ {\ bbb {c}}}}^1- \ {p_1，\ dots，p_s \} $通过Quantum Schubert calculus and Strange duality上产生不可约合的复杂刚性固定系统。这些本地系统是统一的，并源于控制特殊统一组的多重特征值问题$ \ operatatorName {su}（n）$（即确定可能给出了eigenvalue eigenvalue的不单亲矩阵的产物）。粗略地说，我们表明，这些多面体最简单顶点的奇怪双重偶数为所有可能的统一不可还原刚性局部系统提供了。结果，我们可以获得在$ \ bbb {p}^1-s $上以$ \ bbb {p}^1-s $在上面的限制的单位不可约性的本地系统的等级，如果我们修复了集合$ s = \ s = \ {p_1，\ dots，dots，p_s \} $ and p_s \} $ and po_s \} $ and po_s using $ n $ n $ n $ nod $ n $ node $ n $ n $ n $ n n $ node $ n n $ node $ n nive n $ node $ n。回答了N. Katz的问题，我们表明，没有一个不可还原的刚性当地系统大于一个，而有限的全球单曲率，当$ n $都是质量数字时，所有本地单型都有订单划分为$ n $的订单。 我们还表明，在$ \ bbb {p}^1 _ {\ bbb {c}}} -s $上的所有统一不可约的刚性本地系统都带有有限的本地单粒子，作为knizhnik -zamalodchikov方程的解决方案，用于特种线性线性群体上。一路走来，概括了作者和J. Kiers的先前作品，我们提供了一种归纳机制，用于确定$ \ permatatorName {su}（n）$的乘法特征值问题中的所有顶点。
We give a construction which produces irreducible complex rigid local systems on $\Bbb{P}_{\Bbb{C}}^1-\{p_1,\dots,p_s\}$ via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups $\operatorname{SU}(n)$ (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices of these polytopes give all possible unitary irreducible rigid local systems. As a consequence we obtain that the ranks of unitary irreducible rigid local systems, including those with finite global monodromy, on $\Bbb{P}^1-S$ are bounded above if we fix the cardinality of the set $S=\{p_1,\dots,p_s\}$ and require that the local monodromies have orders which divide $n$, for a fixed $n$. Answering a question of N. Katz, we show that there are no irreducible rigid local systems of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing $n$, when $n$ is a prime number. We also show that all unitary irreducible rigid local systems on $\Bbb{P}^1_{\Bbb{C}} -S$ with finite local monodromies arise as solutions to the Knizhnik-Zamalodchikov equations on conformal blocks for the special linear group. Along the way, generalising previous works of the author and J. Kiers, we give an inductive mechanism for determining all vertices in the multiplicative eigenvalue problem for $\operatorname{SU}(n)$.
