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Mumford-Tate Groups and DomainsTheir Geometry and Arithmetic (AM-183)$
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Mark Green, Phillip A. Griffiths, and Matt Kerr

Print publication date: 2012

Print ISBN-13: 9780691154244

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691154244.001.0001

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Hodge Representations and Hodge Domains

Hodge Representations and Hodge Domains

(p.85) Chapter IV Hodge Representations and Hodge Domains
Mumford-Tate Groups and Domains

Mark Green

Phillip Griffiths

Matt Kerr

Princeton University Press

This chapter deals with Hodge representations and Hodge domains. For general polarized Hodge structures, it considers which semi-simple ℚ-algebraic groups M can be Mumford-Tate groups of polarized Hodge structures, the different realizations of M as a Mumford-Tate group, and the relationship among the corresponding Mumford-Tate domains. The chapter uses standard material from the structure theory of semisimple Lie algebras and their representation theory. The discussion covers the adjoint representation and characterization of which weights give faithful Hodge representations, the classical groups and the exceptional groups, and Mumford-Tate domains as particular homogeneous complex manifolds. The examples concerning the classical groups illustrate both the linear algebra and Vogan diagram methods.

Keywords:   polarized Hodge structure, Hodge representation, Hodge domain, Mumford-Tate group, Mumford-Tate domain, semisimple Lie algebra, classical group, exceptional group, homogeneous complex manifold, Vogan diagram method

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