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The Invariant Theory of Matrices UNIVERSITY LECTURE SERIES VOLUME 69 The Invariant Theory of Matrices Corrado De Concini Claudio Procesi American Mathematical Society The Invariant Theory of Matrices 10.1090/ulect/069 UNIVERSITY LECTURE SERIES VOLUME 69 The Invariant Theory of Matrices Corrado De Concini Claudio Procesi American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jordan S. Ellenberg Robert Guralnick William P. Minicozzi II (Chair) Tatiana Toro 2010 Mathematics Subject Classification. Primary 15A72, 14L99, 20G20, 20G05. For additional information and updates on this book, visit www.ams.org/bookpages/ulect-69 Library of Congress Cataloging-in-Publication Data Names: De Concini, Corrado, author. | Procesi, Claudio, author. Title: The invariant theory of matrices / Corrado De Concini, Claudio Procesi. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Univer- sity lecture series ; volume 69 | Includes bibliographical references and index. Identifiers: LCCN 2017041943 | ISBN 9781470441876 (alk. paper) Subjects: LCSH: Matrices. | Invariants. | AMS: Linear and multilinear algebra; matrix theory – Basic linear algebra – Vector and tensor algebra, theory of invariants. msc | Algebraic geometry – Algebraic groups – None of the above, but in this section. msc | Group theory and generalizations – Linear algebraic groups and related topics – Linear algebraic groups over the reals, the complexes, the quaternions. msc | Group theory and generalizations – Linear algebraic groups and related topics – Representation theory. msc Classification: LCC QA188 .D425 2017 | DDC 512.9/434–dc23 LC record available at https://lccn. loc.gov/2017041943 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2017 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 222120191817 Table of Contents Introductionandpreliminaries ......................... 1 1. Introduction.......................................... 2 2. Preliminaries......................................... 8 PartI. Theclassicaltheory........................... 19 3. Representationtheory................................... 20 4. Algebraswithtrace..................................... 28 PartII. Quasi-hereditaryalgebras...................... 39 5. Modules............................................. 40 6. Goodfiltrationsandquasi-hereditaryalgebras.................. 43 PartIII. TheSchuralgebra........................... 49 7. TheSchuralgebra...................................... 50 8. Doubletableaux....................................... 51 9. ModulesfortheSchuralgebra............................. 62 10. Rational GL(m)-modules................................ 75 11. Tensorproducts...................................... 78 PartIV. Matrixfunctionsandinvariants................. 87 12. Areductionforinvariantsofseveralmatrices ................. 88 13. Polarizationandspecialization............................ 91 14. Exteriorproducts..................................... 95 15. Matrixfunctionsandinvariants........................... 99 PartV. Relations.................................. 107 16. Relations...........................................108 17. Describing Km .......................................110 18. Km versus K˜m .......................................118 PartVI. TheSchuralgebraofafreealgebra............... 131 19. Preliminaryfacts .....................................132 20. TheSchuralgebraofthefreealgebra.......................135 Bibliography..............................................145 GeneralIndex.............................................149 SymbolIndex..............................................151 v Bibliography [1]E.Aljadeff,A.Giambruno,C.Procesi,A.Regev,Rings with polynomial identities and finite dimensional representations of algebras, book in preparation. [2] S. A. Amitsur, On the characteristic polynomial of a sum of matrices, Linear and Multilinear Algebra 8 (1979/80), no. 3, 177–182, DOI 10.1080/03081088008817315. MR560557 [3] H. Aslaksen, E.-C. Tan, and C.-b. Zhu, Invariant theory of special orthogonal groups,Pacific J. Math. 168 (1995), no. 2, 207–215. MR1339950 [4] I. Assem, D. Simson, and A. 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MR957457 [10] E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99. MR961165 [11] E. Cline, B. Parshall, and L. Scott, Integral and graded quasi-hereditary algebras. I,J.Algebra 131 (1990), no. 1, 126–160, DOI 10.1016/0021-8693(90)90169-O. MR1055002 [12] E. Cline, B. Parshall, and L. Scott, The homological dual of a highest weight category,Proc. London Math. Soc. (3) 68 (1994), no. 2, 294–316, DOI 10.1112/plms/s3-68.2.294. MR1253506 [13] E. Cline, B. Parshall, and L. Scott, Stratifying endomorphism algebras,Mem.Amer.Math. Soc. 124 (1996), no. 591, viii+119, DOI 10.1090/memo/0591. MR1350891 [14] C. de Concini, D. Eisenbud, and C. Procesi, Young diagrams and determinantal varieties, Invent. Math. 56 (1980), no. 2, 129–165, DOI 10.1007/BF01392548. MR558865 [15] V. Dlab and C. M. Ringel, Quasi-hereditary algebras, Illinois J. Math. 33 (1989), no. 2, 280–291. MR987824 [16] V. Dlab, P. Heath, and F. 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