UNIVERSITY LECTURE SERIES VOLUME 69

American Mathematical Society

The Invariant Theory of Matrices

Corrado De ConciniClaudio Procesi

The Invariant Theory of Matrices

The Invariant Theory of Matrices

Corrado De Concini Claudio Procesi

UNIVERSITY LECTURE SERIES VOLUME 69

American Mathematical SocietyProvidence, Rhode Island

10.1090/ulect/069

EDITORIAL COMMITTEE

Jordan S. EllenbergWilliam P. Minicozzi II (Chair)

Robert GuralnickTatiana Toro

2010 Mathematics Subject Classification. Primary 15A72, 14L99, 20G20, 20G05.

For additional information and updates on this book, visitwww.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 | Algebraicgeometry – Algebraic groups – None of the above, but in this section. msc | Group theoryand generalizations – Linear algebraic groups and related topics – Linear algebraic groups overthe reals, the complexes, the quaternions. msc | Group theory and generalizations – Linearalgebraic 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

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Table of Contents

Introduction and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 11. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Part I. The classical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 193. Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204. Algebras with trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Part II. Quasi-hereditary algebras . . . . . . . . . . . . . . . . . . . . . . 395. Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406. Good filtrations and quasi-hereditary algebras . . . . . . . . . . . . . . . . . . 43

Part III. The Schur algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 497. The Schur algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508. Double tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519. Modules for the Schur algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6210. Rational GL(m)-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511. Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Part IV. Matrix functions and invariants . . . . . . . . . . . . . . . . . 8712. A reduction for invariants of several matrices . . . . . . . . . . . . . . . . . 8813. Polarization and specialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9114. Exterior products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9515. Matrix functions and invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Part V. Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10716. Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10817. Describing Km . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11018. Km versus K̃m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Part VI. The Schur algebra of a free algebra . . . . . . . . . . . . . . . 13119. Preliminary facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13220. The Schur algebra of the free algebra . . . . . . . . . . . . . . . . . . . . . . . 135

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

General Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

v

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General Index

Δ-filtration, 44∇-filtration, 44

adjacency, 52affine scheme, 16

group, 16

algebraquasi hereditary, 46Schur, 50trace, 28

algebra with trace, 28

bitableauxsemistandard, 60

canonical tableau, 53

Cayley–Hamilton identity, 30composition, 95cyclic equivalence, 5

dominance order, 51

equivariant maps, 26essential extension, 40

free Schur algebra, 137full polarization, 92

height, 20

injective hull, 40

linear algebraic group, 12

matrix variable, 34module

socle, 42superfluous, 40tilting, 82

top, 42monomial

primitive, 4multilinearization, 24multiplicative map, 11

partial polarization, 92polarization

full, 24polynomial law, 9projective cover, 40

rational representation, 13restitution, 24rim

of a diagram, 76row bitableau, 59

Schur functors, 66Schur module, 65Schur symmetric function, 20shape of the diagram, 51simple degeneration, 103skew diagram, 51specialization, 24Standard filtration, 68straightening relation, 56

traceformal, 28

weight, 51of a bitableau, 63dominant, 51

Weyl module, 69

Young diagram, 51Young superclass, 103

Young tableau, 52standard, 52

149

Symbol Index

SA = SA〈X〉, 37Tm, 38Tstλ ⊂ Tλ, 88A[G], 12

Dμ, 99

F+〈X〉, 5L ⊂ ∇λ, 73Mλ, 88

R ⊂ F [xi,j ], 56Rn, 50

SCg, 103

Sch〈X〉, 110

Sm〈X〉, 108Tμ, 99

Wp, 4

Y := Tβ , 80

Λ′, 89Θc, 99

Ξc, 103

π̄Z,V , 111

λ̌, 51

λ t, 51λ \ μ, 51AdF (R), 140AF 〈X〉, 141S, 37S(c), 99Fν , 80Fd, 135Fm〈X〉 := Sm〈X〉GL(V ), 108P, 58P(μ, s), 79PA(M,N), 9S〈X〉, 137Sm(R), 11SA〈X〉, 137Y, 61Adc(F ), 97

Ac,d(A), 97

Tr(M), 111

∇λ(V ) , 20πd, 136

σ1 = tr, 34

σi, 34Q̃λ := Rm,t(λ), 72εj1,...,jk , 9|λ|, 51bj , 125e(λ), 104

eI = eα, 10ef = σf1,...,fh(M1, . . . ,Mh), 138f∗, 40ht(λ), 20k : [1, . . . , N ] → [1, . . . , r], 95o(λ), 104pI , 54, 55t(R) := {t(a), a ∈ R}, 28W0, 5Tλ, 64deg p, 99

151

Selected Published Titles in This Series

69 Corrado De Concini and Claudio Procesi, The Invariant Theory of Matrices, 2017

67 Sylvie Ruette, Chaos on the Interval, 2017

66 Robert Steinberg, Lectures on Chevalley Groups, 2016

65 Alexander M. Olevskii and Alexander Ulanovskii, Functions with DisconnectedSpectrum, 2016

64 Larry Guth, Polynomial Methods in Combinatorics, 2016

63 Gonçalo Tabuada, Noncommutative Motives, 2015

62 H. Iwaniec, Lectures on the Riemann Zeta Function, 2014

61 Jacob P. Murre, Jan Nagel, and Chris A. M. Peters, Lectures on the Theory ofPure Motives, 2013

60 William H. Meeks III and Joaqúın Pérez, A Survey on Classical Minimal SurfaceTheory, 2012

59 Sylvie Paycha, Regularised Integrals, Sums and Traces, 2012

58 Peter D. Lax and Lawrence Zalcman, Complex Proofs of Real Theorems, 2012

57 Frank Sottile, Real Solutions to Equations from Geometry, 2011

56 A. Ya. Helemskii, Quantum Functional Analysis, 2010

55 Oded Goldreich, A Primer on Pseudorandom Generators, 2010

54 John M. Mackay and Jeremy T. Tyson, Conformal Dimension, 2010

53 John W. Morgan and Frederick Tsz-Ho Fong, Ricci Flow and Geometrization of3-Manifolds, 2010

52 Marian Aprodu and Jan Nagel, Koszul Cohomology and Algebraic Geometry, 2010

51 J. Ben Hough, Manjunath Krishnapur, Yuval Peres, and Bálint Virág, Zeros ofGaussian Analytic Functions and Determinantal Point Processes, 2009

50 John T. Baldwin, Categoricity, 2009

49 József Beck, Inevitable Randomness in Discrete Mathematics, 2009

48 Achill Schürmann, Computational Geometry of Positive Definite Quadratic Forms, 2008

47 Ernst Kunz, David A. Cox, and Alicia Dickenstein, Residues and Duality forProjective Algebraic Varieties, 2008

46 Lorenzo Sadun, Topology of Tiling Spaces, 2008

45 Matthew Baker, Brian Conrad, Samit Dasgupta, Kiran S. Kedlaya, and JeremyTeitelbaum, p-adic Geometry, 2008

44 Vladimir Kanovei, Borel Equivalence Relations, 2008

43 Giuseppe Zampieri, Complex Analysis and CR Geometry, 2008

42 Holger Brenner, Jürgen Herzog, and Orlando Villamayor, Three Lectures onCommutative Algebra, 2008

41 James Haglund, The q, t-Catalan Numbers and the Space of Diagonal Harmonics, 2008

40 Vladimir Pestov, Dynamics of Infinite-dimensional Groups, 2006

39 Oscar Zariski, The Moduli Problem for Plane Branches, 2006

38 Lars V. Ahlfors, Lectures on Quasiconformal Mappings, Second Edition, 2006

37 Alexander Polishchuk and Leonid Positselski, Quadratic Algebras, 2005

36 Matilde Marcolli, Arithmetic Noncommutative Geometry, 2005

35 Luca Capogna, Carlos E. Kenig, and Loredana Lanzani, Harmonic Measure, 2005

34 E. B. Dynkin, Superdiffusions and Positive Solutions of Nonlinear Partial DifferentialEquations, 2004

33 Kristian Seip, Interpolation and Sampling in Spaces of Analytic Functions, 2004

32 Paul B. Larson, The Stationary Tower, 2004

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/ulectseries/.

This book gives a unifi ed, complete, and self-contained exposition of the main algebraic theorems of invariant theory for matrices in a characteristic free approach. More precisely, it contains the descrip-tion of polynomial functions in several variables on the set of m × m matrices with coeffi cients in an infi nite fi eld or even the ring of integers, invariant under simultaneous conjugation.

Following Hermann Weyl’s classical approach, the ring of invariants is described by formulating and proving

• the fi rst fundamental theorem that describes a set of generators in the ring of invariants, and

• the second fundamental theorem that describes relations between these generators.

The authors study both the case of matrices over a fi eld of characteristic 0 and the case of matrices over a fi eld of positive characteristic. While the case of characteristic 0 can be treated following a classical approach, the case of positive characteristic (developed by Donkin and Zubkov) is much harder. A presentation of this case requires the develop-ment of a collection of tools. These tools and their application to the study of invariants are exlained in an elementary, self-contained way in the book.

For additional informationand updates on this book, visit

www.ams.org/bookpages/ulect-69

www.ams.org

ULECT/69