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May 2022 Rigid local systems and the multiplicative eigenvalue problem
Prakash Belkale
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Ann. of Math. (2) 195(3): 911-995 (May 2022). DOI: 10.4007/annals.2022.195.3.3

Abstract

We give a construction that produces irreducible complex rigid local systems on $\mathbb{P}_C^1-\{p_1,\ldots ,p_s\}$ 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 $\mathrm{SU}(n)$ (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 $\mathbb{P}^1-S$ are bounded above if we fix the cardinality of the set $S=\{p_1,\ldots ,p_s\}$ and require that the local monodromies have orders that 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 $\mathbb{P}_C^1-S$ with finite local monodromies arise as solutions to the Knizhnik-Zamalodchikov equations on conformal blocks for the special linear group. Along the way, generalizing previous works of the author and J.~Kiers, we give an inductive mechanism for determining all vertices in the multiplicative eigenvalue problem for $\mathrm{SU}(n)$.

Citation

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Prakash Belkale. "Rigid local systems and the multiplicative eigenvalue problem." Ann. of Math. (2) 195 (3) 911 - 995, May 2022. https://doi.org/10.4007/annals.2022.195.3.3

Information

Published: May 2022
First available in Project Euclid: 29 April 2022

Digital Object Identifier: 10.4007/annals.2022.195.3.3

Subjects:
Primary: 14D05 , 14D20 , 14H60 , 32G34

Keywords: KZ equations , multiplicative eigenvalue problem , quantum Schubert calculus , rigid local systems , strange duality

Rights: Copyright © 2022 Department of Mathematics, Princeton University

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Vol.195 • No. 3 • May 2022
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