DOCSLIB.ORG

Invariant Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Invariant Theory Invariant Theory Notes by Bernd Sturmfels for the lecture on June 19, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra We fix the polynomial ring K[x] = K[x1; : : : ; xn] over a field K of characteristic zero. The group GL(n; K) of invertible n × n matrices acts on Kn. This induces an action by G on n the ring of polynomial functions on K . Namely, if σ = (σij) is a matrix in GL(n; K) and f is a polynomial in K[x] then σf is the polynomial that is obtained from f by replacing the Pn variable xi by the linear form j=1 σijxj for i = 1; : : : ; n. Let G be a subgroup of GL(n; K). A polynomial f 2 K[x] is an invariant of the group G if σf = f for all σ 2 G. We write K[x]G for the set of all such invariants. This set is a subring because the sum of two invariants is again an invariant, and same for the product. In this lecture we discuss two scenarios. First we consider finite groups G, and later we consider representations of nice infinite groups like SL(d; K) and SO(d; K). Such groups are called reductive. A celebrated theorem of Hilbert shows that the invariant ring is finitely generated in this case. After two initial examples, we begin by proving this for finite groups G. Example 1. Let G be the group of n × n permutation matrices. The invariant ring K[x]G consists of all polynomials f that are invariant under permuting the coordinates, i.e. f(xπ1 ; xπ2 ; : : : ; xπn ) = f(x1; x2; : : : ; xn) for all permutations π of f1; 2; : : : ; ng. The invariant ring K[x]G is generated by the n elementary symmetric polynomials E1;:::;En. These are the coefficients of the following auxiliary polynomial in one variable z: n n X n−i (z + x1)(z + x2) ··· (z + xn) = z + Ei(x)z : (1) i=1 G We also set E0 = 1. Alternatively, K[x] can also be generated by the power sums j j j Pj(x) = x1 + x2 + ··· + xn for j = 1; 2; : : : ; n: The formulas which translate between the Ei and the Pj are known as Newton's Identities: kE = Pk (−1)i−1E P k i=1 k−i i (2) k−1 Pk−1 k−1−i and Pk = (−1) kEk + i=1 (−1) Ek−iPi for 1 ≤ k ≤ n: 1 Invariants are polynomial functions that are constant along G-orbits on Kn. They offer an algebraic view on the space of orbits. Namely, we think of the spectrum of K[x]G as a quotient space Kn==G, whose points are these orbits. This interpretation is only informal, as the details are very subtle. Making it all precise is the aim of Geometric Invariant Theory. Example 2. For n = 2, consider the following representation of the cyclic group of order 4: 1 0 −1 0 0 1 0 −1 G = ; ; ; : (3) 0 1 0 −1 −1 0 1 0 These are the rotational symmetries of the square. Its invariant ring is generated by 2 2 2 2 3 3 I1 = x1 + x2 ;I2 = x1x2 ;I3 = x1x2 − x1x2: These three invariants are algebraically dependent. Using their relation we can write G 2 2 2 K[x1; x2] = K[I1;I2;I3] ' K[y1; y2; y3]= y1y2 − 4y2 − y3 : (4) 3 2 2 2 The spectrum of the ring (4) is the cubic surface in K defined by the equation y1y2 = 4y2+y3. The points on this surface are in one-to-one correspondence with the G-orbits on K2. In what follows, we assume that G is a finite subgroup of GL(n; K). One can create invariants by averaging polynomials. The Reynolds operator, denoted by a star, is defined as G 1 X ∗ : K[x] ! K[x] ; f 7! f ∗ := σf: (5) jGj σ2G Each of the following properties of the Reynolds operator is easily verified: Lemma 3. The Reynolds operators ∗ has the following three properties: (a) The map ∗ is K-linear, i.e. (λf + νg)∗ = λf ∗ + νg∗ for all f; g 2 K[x] and λ, ν 2 K. (b) The map ∗ restricts to the identity on K[x]G, i.e. I∗ = I for all invariant polynomials I. (c) The map ∗ is a K[x]G-module homomorphism, i.e. (fI)∗ = f ∗I for all f 2 K[x] and I 2 K[x]G. The following result from 1890 is seen as the beginning of modern Commutative Algebra. Theorem 4 (Hilbert's Finiteness Theorem). The invariant ring K[x]G of any finite matrix group G ⊂ GL(n; K) is finitely generated as a K-algebra. We present the proof under the hypothesis that K has characteristic zero. However, the result holds for every field K. For a proof see [3]. This is known as modular invariant theory. 2 G Proof. Let IG = hK[x]+i be the ideal in K[x] that is generated by all homogeneous in- variants of positive degree. By Lemma 3 (a), every invariant is a K-linear combination of symmetrized monomials (xa)∗. These homogeneous invariants are the images of monomials a ∗ n under the Reynolds operator. Thus IG is generated by the set f (x ) : a 2 N nf0gg. By n Hilbert's Basis Theorem, the ideal IG is finitely generated, so that a finite subset of a in N suffices. In conclusion, there exist invariants I1;I2;:::;Im such that IG = hI1;I2;:::;Imi. We claim that these m invariants generate the invariant ring K[x]G as a K-algebra. Suppose the contrary, and let I be a homogeneous element of minimal degree in G Pm K[x] nK[I1;I2;:::;Im]. Since I 2 IG, we have I = j=1 fjIj for some homogeneous polynomials fj 2 K[x] whose degrees are all strictly less than deg(I). Pm Applying the Reynolds operator on both sides of the equation I = j=1 fjIj, we obtain m m ∗ X ∗ X ∗ I = I = fjIj = fj Ij: j=1 j=1 ∗ Here we are using the properties (b) and (c) in Lemma 3. The new coefficients fj are homogeneous invariants whose degrees are less than deg(I). From the minimality assumption ∗ on the degree of I, we get fj 2 K[I1;:::;Im] for j = 1; : : : ; m. This implies I 2 K[I1;:::;Im], which is a contradiction to our assumption. This completes the proof of Theorem 4. Hilbert's Finiteness Theorem also holds for an infinite group G ⊂ GL(n; K) that has a Reynolds operator ∗ satisfying the properties (a), (b) and (c) in Lemma 3. Such G are known as reductive groups. We shall return to this important point later in the lecture. Corollary 5. Consider any reductive group G of n×n-matrices. If fg1; g2; : : : ; gmg is any col- ∗ ∗ ∗ lection of homogeneous polynomials that generates the ideal IG then its image fg1; g2; : : : ; gmg under the Reynolds operator generates the invariant ring K[x]G as a K-algebra. Proof. Let M = hx1; : : : ; xni be the homogeneous maximal ideal in K[x], and consider the finite-dimensional vector space IG=MIG. It has a basis of invariants since IG is generated by invariants. This means that the Reynolds operator acts as the identity on IG=MIG. The images of g1; g2; : : : ; gm also span IG=MIG as a vector space, and hence so do the invariants ∗ ∗ ∗ ∗ ∗ ∗ g1; g2; : : : ; gm. By Nakayama's Lemma, we find that g1; g2; : : : ; gm generate the ideal IG. As ∗ ∗ ∗ G in the proof of Theorem 4, we conclude that g1; g2; : : : ; gm generate the K-algebra K[x] . Theorem 6 (Noether's Degree Bound). If G is finite and char(K) = 0 then the invariant ring K[x]G is generated by homogeneous invariants of degree at most jGj. Proof. Let u = (u1; : : : ; un) be new variables. For any d 2 N, we consider the expression d ∗ Sd(u; x) = (u1x1 + ··· + unxn) 1 P d = jGj σ2G u1(σx1) + ··· + un(σxn) : This is a polynomial in u whose coefficients are polynomials in x. Up to a multiplicative constant, they are the invariants (xa)∗ where jaj = d. All polynomials in u are fixed under ∗. 3 Consider the jGj expressions u1(σx1) + ··· + un(σxn) , one for each group element σ 2 G. The polynomial Sd(u; x) is the dth power sum of these expressions. The power sums for d > jGj are polynomials in the first jGj power sums. Such a representation is derived from Newton's Identities (2). It implies that all u-coefficients of Sd(u; x) for d > jGj are a ∗ polynomial functions in the u-coefficients of Sd(u; x) for d ≤ jGj. Hence all invariants (x ) with jaj > jGj are polynomial functions (over K) in the invariants (xb)∗ with jbj ≤ jGj. This proves the claim. We note that Example 2 attains Noether's degree bound. The cyclic group in that example has order 4, and the invariant ring requires a generator of degree 4. Our next theorem is a useful tool for constructing the invariant ring. It says that we can count invariants by averaging the reciprocal characteristic polynomials of the group elements. Theorem 7 (Molien series). The Hilbert series of the invariant ring K[x]G equals 1 X 1 X 1 dim K[x]Gzd = : (6) K d jGj det(id − zσ) d=0 σ2G The coefficient of zd in this series is the number of linearly independent invariants of degree d. Proof. See [4, Theorem 2.2.1].
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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].
    [Show full text]
  • The Transfer in Modular Invariant Theory1
    Journal of Pure and Applied Algebra 142 (1999) 63–77 www.elsevier.com/locate/jpaa The transfer in modular invariant theory1 R. James Shank a;∗, David L. Wehlau a;b aDepartment of Mathematics and Statistics, Queen’s University, Kingston, Ont., Canada K7L 3N6 bDepartment of Mathematics and Computer Science, Royal Military College, Kingston, Ont., Canada K7K 7B4 Communicated by A.V. Geramita; received 8 July 1997; received in revised form 29 November 1997 Abstract We study the transfer homomorphism in modular invariant theory paying particular attention to the image of the transfer which is a proper non-zero ideal in the ring of invariants. We prove that, for a p-group over Fp whose ring of invariants is a polynomial algebra, the image of the transfer is a principal ideal. We compute the image of the transfer for SLn(Fq) and GLn(Fq) showing that both ideals are principal. We prove that, for a permutation group, the image of the transfer is a radical ideal and for a cyclic permutation group the image of the transfer is a prime ideal. c 1999 Elsevier Science B.V. All rights reserved. MSC: 13A50 1. Introduction We let V be a vector space of dimension n over a ÿeld k and we choose a basis, ∗ {x1;:::;xn}, for the dual, V ,ofV . Consider a ÿnite subgroup G of GL(V ). The action of G on V induces an action on V ∗ which extends to an action by algebra ∗ automorphisms on the symmetric algebra of V , k[V ]=k[x1;:::;xn]. Speciÿcally, for g ∈ G; f ∈ k[V ] and v ∈ V ,(g·f)(v)=f(g−1 ·v).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Birds and Frogs Equation
    Notices of the American Mathematical Society ISSN 0002-9920 ABCD springer.com New and Noteworthy from Springer Quadratic Diophantine Multiscale Principles of Equations Finite Harmonic of the American Mathematical Society T. Andreescu, University of Texas at Element Analysis February 2009 Volume 56, Number 2 Dallas, Richardson, TX, USA; D. Andrica, Methods A. Deitmar, University Cluj-Napoca, Romania Theory and University of This text treats the classical theory of Applications Tübingen, quadratic diophantine equations and Germany; guides readers through the last two Y. Efendiev, Texas S. Echterhoff, decades of computational techniques A & M University, University of and progress in the area. The presenta- College Station, Texas, USA; T. Y. Hou, Münster, Germany California Institute of Technology, tion features two basic methods to This gently-paced book includes a full Pasadena, CA, USA investigate and motivate the study of proof of Pontryagin Duality and the quadratic diophantine equations: the This text on the main concepts and Plancherel Theorem. The authors theories of continued fractions and recent advances in multiscale finite emphasize Banach algebras as the quadratic fields. It also discusses Pell’s element methods is written for a broad cleanest way to get many fundamental Birds and Frogs equation. audience. Each chapter contains a results in harmonic analysis. simple introduction, a description of page 212 2009. Approx. 250 p. 20 illus. (Springer proposed methods, and numerical 2009. Approx. 345 p. (Universitext) Monographs in Mathematics) Softcover examples of those methods. Softcover ISBN 978-0-387-35156-8 ISBN 978-0-387-85468-7 $49.95 approx. $59.95 2009. X, 234 p. (Surveys and Tutorials in The Strong Free Will the Applied Mathematical Sciences) Solving Softcover Theorem Introduction to Siegel the Pell Modular Forms and ISBN: 978-0-387-09495-3 $44.95 Equation page 226 Dirichlet Series Intro- M.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Lectures on Representation Theory and Invariant Theory
    LECTURES ON REPRESENTATION THEORY AND INVARIANT THEORY These are the notes for a lecture course on the symmetric group, the general linear group and invariant theory. The aim of the course was to cover as much of the beautiful classical theory as time allowed, so, for example, I have always restricted to working over the complex numbers. The result is a course which requires no previous knowledge beyond a smattering of rings and modules, character theory, and affine varieties. These are certainly not the first notes on this topic, but I hope they may still be useful, for [Dieudonné and Carrell] has a number of flaws, and [Weyl], although beautifully written, requires a lot of hard work to read. The only new part of these notes is our treatment of Gordan's Theorem in §13: the only reference I found for this was [Grace and Young] where it was proved using the symbolic method. The lectures were given at Bielefeld University in the winter semester 1989-90, and this is a more or less faithful copy of the notes I prepared for that course. I have, however, reordered some of the parts, and rewritten the section on semisimple algebras. The references I found most useful were: [H. Boerner] "Darstellungen von Gruppen" (1955). English translation "Representations of groups" (North-Holland, 1962,1969). [J. A. Dieudonné and J. B. Carrell] "Invariant Theory, Old and New," Advances in Mathematics 4 (1970) 1-80. Also published as a book (1971). [J. H. Grace and A. Young] "The algebra of invariants" (1903, Reprinted by Chelsea).
    [Show full text]
  • If You Have Passion and Focus You Can Accomplish Anything
    The UK’s European university UNIVERSITY OF KENT OPEN DAYS IN 2013/14 Postgraduate events Wed 27 November 2013 Sat 1 February 2014 General Open Days / Canterbury Medway GRADUATE PROSPECTUS Wed 18 September 2013 Sat 12 October 2013 Sat 5 October 2013 Sat 21 June 2014 Sat 12 July 2014 www.kent.ac.uk/opendays University of Kent, The Registry, Canterbury, Kent CT2 7NZ T: +44 (0)1227 764000 E: [email protected] / 2014 IF YOU HAVE PASSION AND FOCUS YOU CAN ACCOMPLISH ANYTHING Graduate 2014 Prospectus By rail The nearest train station to the AUR campus is Trastevere Station on the Pisa-Livorno-Rome, Rome- NOW IT GETS Capranica-Viterbo and Rome-Fiumicino lines. By bus and tram On leaving Trastevere station, take tram 8 towards Largo Argentina. Get off at Via E Morosini and INTERESTING catch bus 44 (in the direction of Montalcini) or bus 75 (in the direction of Poerio/Marino). The campus is located between two stops: Via Our postgraduate programmes are informed Dandolo/Calandrelli and Via Giacinto Carini; both by a commitment to research excellence. As a stops are less than 100 metres from the AUR campus. highly ranked UK university, Kent provides a Campus map dynamic and challenging academic environment. www.kent.ac.uk/maps/rome The University offers students a wide choice of well-structured and ambitious postgraduate programmes. At every stage, students are supported by inspirational teaching and supervision, first-class library and IT facilities and a diverse schedule of seminars, workshops and events. www.kent.ac.uk 1 2 University of Kent / Graduate Prospectus 2014 WELCOME FROM THE DEAN This is an exciting time to be a postgraduate student at the University of Kent.
    [Show full text]