CLASSICAL INVARIANT THEORY a Primer
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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. -
View This Volume's Front and Back Matter
http://dx.doi.org/10.1090/gsm/039 Selected Titles in This Series 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2001 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2001 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. -
An Elementary Proof of the First Fundamental Theorem of Vector Invariant Theory
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 102, 556-563 (1986) An Elementary Proof of the First Fundamental Theorem of Vector Invariant Theory MARILENA BARNABEI AND ANDREA BRINI Dipariimento di Matematica, Universitri di Bologna, Piazza di Porta S. Donato, 5, 40127 Bologna, Italy Communicated by D. A. Buchsbaum Received March 11, 1985 1. INTRODUCTION For a long time, the problem of extending to arbitrary characteristic the Fundamental Theorems of classical invariant theory remained untouched. The technique of Young bitableaux, introduced by Doubilet, Rota, and Stein, succeeded in providing a simple combinatorial proof of the First Fundamental Theorem valid over every infinite field. Since then, it was generally thought that the use of bitableaux, or some equivalent device-as the cancellation lemma proved by Hodge after a suggestion of D. E. LittlewoodPwould be indispensable in any such proof, and that single tableaux would not suffice. In this note, we provide a simple combinatorial proof of the First Fun- damental Theorem of invariant theory, valid in any characteristic, which uses only single tableaux or “brackets,” as they were named by Cayley. We believe the main combinatorial lemma (Theorem 2.2) used in our proof can be fruitfully put to use in other invariant-theoretic settings, as we hope to show elsewhere. 2. A COMBINATORIAL LEMMA Let K be a commutative integral domain, with unity. Given an alphabet of d symbols x1, x2,..., xd and a positive integer n, consider the d. n variables (xi\ j), i= 1, 2 ,..., d, j = 1, 2 ,..., n, which are supposed to be algebraically independent. -
Arxiv:1810.05857V3 [Math.AG] 11 Jun 2020
HYPERDETERMINANTS FROM THE E8 DISCRIMINANT FRED´ ERIC´ HOLWECK AND LUKE OEDING Abstract. We find expressions of the polynomials defining the dual varieties of Grass- mannians Gr(3, 9) and Gr(4, 8) both in terms of the fundamental invariants and in terms of a generic semi-simple element. We restrict the polynomial defining the dual of the ad- joint orbit of E8 and obtain the polynomials of interest as factors. To find an expression of the Gr(4, 8) discriminant in terms of fundamental invariants, which has 15, 942 terms, we perform interpolation with mod-p reductions and rational reconstruction. From these expressions for the discriminants of Gr(3, 9) and Gr(4, 8) we also obtain expressions for well-known hyperdeterminants of formats 3 × 3 × 3 and 2 × 2 × 2 × 2. 1. Introduction Cayley’s 2 × 2 × 2 hyperdeterminant is the well-known polynomial 2 2 2 2 2 2 2 2 ∆222 = x000x111 + x001x110 + x010x101 + x100x011 + 4(x000x011x101x110 + x001x010x100x111) − 2(x000x001x110x111 + x000x010x101x111 + x000x100x011x111+ x001x010x101x110 + x001x100x011x110 + x010x100x011x101). ×3 It generates the ring of invariants for the group SL(2) ⋉S3 acting on the tensor space C2×2×2. It is well-studied in Algebraic Geometry. Its vanishing defines the projective dual of the Segre embedding of three copies of the projective line (a toric variety) [13], and also coincides with the tangential variety of the same Segre product [24, 28, 33]. On real tensors it separates real ranks 2 and 3 [9]. It is the unique relation among the principal minors of a general 3 × 3 symmetric matrix [18]. -
Dual Pairs in the Pin-Group and Duality for the Corresponding Spinorial
DUAL PAIRS IN THE PIN-GROUP AND DUALITY FOR THE CORRESPONDING SPINORIAL REPRESENTATION CLÉMENT GUÉRIN, GANG LIU, AND ALLAN MERINO Abstract. In this paper, we give a complete picture of Howe correspondence for the setting (O(E, b), Pin(E, b), Π), where O(E, b) is an orthogonal group (real or complex), Pin(E, b) is the two-fold Pin-covering of O(E, b), and Π is the spinorial representation of Pin(E, b). More pre- cisely, for a dual pair (G, G′) in O(E, b), we determine explicitly the nature of its preimages (G, G′) in Pin(E, b), and prove that apart from some exceptions, (G, G′) is always a dual pair in e f e f Pin(E, b); then we establish the Howe correspondence for Π with respect to (G, G′). e f Contents 1. Introduction 1 2. Preliminaries 3 3. Dual pairs in the Pin-group 6 3.1. Pull-back of dual pairs 6 3.2. Identifyingtheisomorphismclassofpull-backs 8 4. Duality for the spinorial representation 14 References 21 1. Introduction The first duality phenomenon has been discovered by H. Weyl who pointed out a correspon- dence between some irreducible finite dimensional representations of the general linear group GL(V) and the symmetric group Sk where V is a finite dimensional vector space over C. In- d deed, considering the joint action of GL(V) and Sd on the space V⊗ , we get the following decomposition: arXiv:1907.09093v1 [math.RT] 22 Jul 2019 d λ V⊗ = M λ σ , [ ⊗ (Vλ,λ) GL(V) ∈ π [ where GL(V)π is the set of equivalence classes of irreducible finite dimensional representations d of GL(V) such that Hom (V , V⊗ ) , 0 and σ is an irreducible representation of S . -
Emmy Noether, Greatest Woman Mathematician Clark Kimberling
Emmy Noether, Greatest Woman Mathematician Clark Kimberling Mathematics Teacher, March 1982, Volume 84, Number 3, pp. 246–249. Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). With more than 100,000 members, NCTM is the largest organization dedicated to the improvement of mathematics education and to the needs of teachers of mathematics. Founded in 1920 as a not-for-profit professional and educational association, NCTM has opened doors to vast sources of publications, products, and services to help teachers do a better job in the classroom. For more information on membership in the NCTM, call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright March 1982 by the National Council of Teachers of Mathematics. All rights reserved. mmy Noether was born over one hundred years ago in the German university town of Erlangen, where her father, Max Noether, was a professor of Emathematics. At that time it was very unusual for a woman to seek a university education. In fact, a leading historian of the day wrote that talk of “surrendering our universities to the invasion of women . is a shameful display of moral weakness.”1 At the University of Erlangen, the Academic Senate in 1898 declared that the admission of women students would “overthrow all academic order.”2 In spite of all this, Emmy Noether was able to attend lectures at Erlangen in 1900 and to matriculate there officially in 1904. -
Sylow P -Subgroups of the Classical Groups Over Finite Fields with Characteristic Prime to P
SYLOW /--SUBGROUPS OF THE CLASSICAL GROUPS OVER FINITE FIELDS WITH CHARACTERISTIC PRIME TO p A. J. WEIR If Sn is a Sylow /--subgroup of the symmetric group of degree pn, then any group of order pn may be imbedded in 5„. We may express Sn as the complete product1 C o C o • • • o C of « cyclic groups of order p and the purpose of this paper is to show that any Sylow p- subgroup of a classical group (see §1) over the finite field GF(q) with q elements, where (q, p) = 1, is expressible as a direct product of basic subgroups 5„= C o C o • • • o C (n factors), where C is cyclic of order pT. (We assume always that p7£2.) Since C may be imbedded in ST, we see that Sn is imbedded in Sn+r-i in a particularly simple way. The above r is defined by the equation q*— \ =pT * where g* is the first power of q which is congruent to 1 mod p and * denotes some unspecified number prime to p. The case r = 1 is therefore of frequent occurrence, and then clearly Sn=Sn. Professor Philip Hall was my research supervisor in Cambridge (England) during the years 1949-1952 and it is a pleasure to acknowl- edge here my indebtedness to him for his generous encouragement. 1. We shall refer to the following groups as the Classical Groups:* I. The general linear group GL(n, q) is the group of all nonsingular (nXn) matrices with coefficients in GF(q). The order of GL(n, q) is qn(n-l)l2fq _ 1)(?2 _ J) . -
The Invariant Theory of Binary Forms
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 10, Number 1, January 1984 THE INVARIANT THEORY OF BINARY FORMS BY JOSEPH P. S. KUNG1 AND GIAN-CARLO ROTA2 Dedicated to Mark Kac on his seventieth birthday Table of Contents 1. Introduction 2. Umbral Notation 2.1 Binary forms and their covariants 2.2 Umbral notation 2.3 Changes of variables 3. The Fundamental Theorems 3.1 Bracket polynomials 3.2 The straightening algorithm 3.3 The second fundamental theorem 3.4 The first fundamental theorem 4. Covariants in Terms of the Roots 4.1 Homogenized roots 4.2 Tableaux in terms of roots 4.3 Covariants and differences 4.4 Hermite's reciprocity law 5. Apolarity 5.1 The apolar covariant 5.2 Forms apolar to a given form 5.3 Sylvester's theorem 5.4 The Hessian and the cubic 6. The Finiteness Theorem 6.1 Generating sets of covariants 6.2 Circular straightening 6.3 Elemental bracket monomials 6.4 Reduction of degree 6.5 Gordan's lemma 6.6 The binary cubic 7. Further Work References 1. Introduction. Like the Arabian phoenix rising out of its ashes, the theory of invariants, pronounced dead at the turn of the century, is once again at the forefront of mathematics. During its long eclipse, the language of modern algebra was developed, a sharp tool now at long last being applied to the very Received by the editors May 26, 1983. 1980 Mathematics Subject Classification. Primary 05-02, 13-02, 14-02, 15-02. 1 Supported by a North Texas State University Faculty Research Grant. -
Representations and Invariants of the Classical Groups, by Roe Goodman and Nolan R
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 4, Pages 533{538 S 0273-0979(99)00795-8 Article electronically published on July 28, 1999 Representations and invariants of the classical groups, by Roe Goodman and Nolan R. Wallach, Cambridge Univ. Press, Cambridge, 1998, xvi + 685 pp., $100.00, ISBN 0-521-58273-3, paperback, $39.95, ISBN 0-521-66348-2 This 685-page text by Goodman and Wallach constitutes a carefully organized exposition, not indeed of the entire finite-dimensional representation-theory over R or C of the classical groups (which would be hard to imagine in a single text three times the size of this one), but rather of a selection of some key topics therein. This selection of topics, especially as concerns invariant-theory, seems (as Goodman and Wallach indicate in the preface to their text) to have been in many ways inspired by the contents of H. Weyl’s The Classical Groups, Their Invariants and Representations [31]. Approximately half of the text under review is purely introductory; the remaining half then utilizes this introductory material to develop the main themes. The introductory material, covered in a total of approximately 280 pages in the first three chapters and in four appendices, introduces the reader to some necessary topics in elementary algebraic and differential geometry and to such basic concepts of representation theory as: linear algebraic groups and Lie groups, Lie algebras associated to such, Jordan decomposition, maximal tori, roots and weights, real forms of classical groups, the PBW theorem, invariant integration over compact Lie groups, etc. -
GEOMETRIC INVARIANT THEORY and FLIPS Ever Since the Invention
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 3, July 1996 GEOMETRIC INVARIANT THEORY AND FLIPS MICHAEL THADDEUS Ever since the invention of geometric invariant theory, it has been understood that the quotient it constructs is not entirely canonical, but depends on a choice: the choice of a linearization of the group action. However, the founders of the subject never made a systematic study of this dependence. In light of its fundamental and elementary nature, this is a rather surprising gap, and this paper will attempt to fill it. In one sense, the question can be answered almost completely. Roughly, the space of all possible linearizations is divided into finitely many polyhedral chambers within which the quotient is constant (2.3), (2.4), and when a wall between two chambers is crossed, the quotient undergoes a birational transformation which, under mild conditions, is a flip in the sense of Mori (3.3). Moreover, there are sheaves of ideals on the two quotients whose blow-ups are both isomorphic to a component of the fibred product of the two quotients over the quotient on the wall (3.5). Thus the two quotients are related by a blow-up followed by a blow-down. The ideal sheaves cannot always be described very explicitly, but there is not much more to say in complete generality. To obtain more concrete results, we require smoothness, and certain conditions on the stabilizers which, though fairly strong, still include many interesting examples. The heart of the paper, 4and 5, is devoted to describing the birational transformations between quotients§§ as explicitly as possible under these hypotheses. -
CONSTRUCTIVE INVARIANT THEORY Contents 1. Hilbert's First
CONSTRUCTIVE INVARIANT THEORY HARM DERKSEN* Contents 1. Hilbert's first approach 1 1.1. Hilbert's Basissatz 2 1.2. Algebraic groups 3 1.3. Hilbert's Finiteness Theorem 5 1.4. An algorithm for computing generating invariants 8 2. Hilbert's second approach 12 2.1. Geometry of Invariant Rings 12 2.2. Hilbert's Nullcone 15 2.3. Homogeneous Systems of Parameters 16 3. The Cohen-Macaulay property 17 4. Hilbert series 20 4.1. Basic properties of Hilbert series 20 4.2. Computing Hilbert series a priori 23 4.3. Degree bounds 27 References 28 In these lecture notes, we will discuss the constructive aspects of modern invariant theory. A detailed discussion on the computational aspect of invariant theory can be found in the books [38, 9]. The invariant theory of the nineteenth century was very constructive in nature. Many invariant rings for classical groups (and SL2 in particular) were explicitly computed. However, we will not discuss this classical approach here, but we will refer to [32, 25] instead. Our starting point will be a general theory of invariant rings. The foundations of this theory were built by Hilbert. For more on invariant theory, see for example [23, 35, 24]. 1. Hilbert's first approach Among the most important papers in invariant theory are Hilbert's papers of 1890 and 1893 (see [15, 16]). Both papers had an enormous influence, not only on invariant theory but also on commutative algebra and algebraic geometry. Much of the lectures *The author was partially supported by NSF, grant DMS 0102193. -
Invariant(Theory- A(Gentle(Introduction For(Computer(Scientists (Optimization(And(Complexity)
Invariant(theory- a(gentle(introduction for(computer(scientists (optimization(and(complexity) Avi$Wigderson,$IAS,$Princeton Ankit$Garg,$Rafael$Oliveira Zeyuan Allen=Zhu,$Peter$Burgisser,$Cole$Franks,$ Leonid$Gurvits,$Yuanzhi Li,$Michael$Walter Prehistory Linial,'Samorodnitsky,'W'2000'''Cool'algorithm Discovered'many'times'before' Kruithof 1937''in'telephone'forecasting,' DemingDStephan'1940'in'transportation'science, Brown'1959''in'engineering,' Wilkinson'1959'in'numerical'analysis, Friedlander'1961,'Sinkhorn 1964 in'statistics. Stone'1964'in'economics,' Matrix'Scaling'algorithm A non1negative'matrix.'Try'making'it'doubly'stochastic. Alternating'scaling'rows'and'columns 1 1 1 1 1 0 1 0 0 Matrix'Scaling'algorithm A non1negative'matrix.'Try'making'it'doubly'stochastic. Alternating'scaling'rows'and'columns 1/3 1/3 1/3 1/2 1/2 0 1 0 0 Matrix'Scaling'algorithm A non1negative'matrix.'Try'making'it'doubly'stochastic. Alternating'scaling'rows'and'columns 2/11 2/5 1 3/11 3/5 0 6/11 0 0 Matrix'Scaling'algorithm A non1negative'matrix.'Try'making'it'doubly'stochastic. Alternating'scaling'rows'and'columns 10/87 22/87 55/87 15/48 33/48 0 1 0 0 Matrix'Scaling'algorithm A non1negative'matrix.'Try'making'it'doubly'stochastic. 0 0 1 Alternating'scaling'rows'and'columns A+very+different+efficient+ 0 1 0 Perfect+matching+algorithm+ 1 0 0 We’ll+understand+it+much+ better+with+Invariant+Theory Converges+(fast)+iff Per(A)+>0 Outline Main%motivations,%questions,%results,%structure • Algebraic%Invariant%theory% • Geometric%invariant%theory% • Optimization%&%Duality • Moment%polytopes • Algorithms • Conclusions%&%Open%problems R = Energy Invariant(Theory = Momentum symmetries,(group(actions, orbits,(invariants 1 Physics,(Math,(CS 5 2 S5 4 3 1 5 2 ??? 4 3 Area Linear'actions'of'groups Group&! acts&linearly on&vector&space&" (= %&).&&&[% =&(] Action:&Matrix7Vector&multiplication ! reductive.