Algorithms in Invariant Theory
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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. -
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. -
The Algebraic Geometry of Stresses in Frameworks
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/244505791 The Algebraic Geometry of Stresses in Frameworks Article in SIAM Journal on Algebraic and Discrete Methods · December 1983 DOI: 10.1137/0604049 CITATIONS READS 95 199 2 authors, including: Walter Whiteley York University 183 PUBLICATIONS 4,791 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Students’ Understanding of Quadrilaterals View project All content following this page was uploaded by Walter Whiteley on 15 April 2020. The user has requested enhancement of the downloaded file. SIAM J. ALG. DISC. METH. 1983 Society for Industrial and Applied Mathematics Vol. 4, No. 4, December 1983 0196-5212/83/0404-0008 $01.25/0 THE ALGEBRAIC GEOMETRY OF STRESSES IN FRAMEWORKS* NEIL L. WHITEr AND WALTER WHITELEY: Abstract. A bar-and-joint framework, with rigid bars and flexible joints, is said to be generically isostatic if it has just enough bars to be infinitesimally rigid in some realization in Euclidean n-space. We determine the equation that must be satisfied by the coordinates of the joints in a given realization in order to have a nonzero stress, and hence an infinitesimal motion, in the framework. This equation, called the pure condition, is expressed in terms of certain determinants, called brackets. The pure condition is obtained by choosing a way to tie down the framework to eliminate the Euclidean motions, computing a bracket expression by a method due to Rosenberg and then factoring out part of the expression related to the tie-down. -
THE WHITNEY ALGEBRA of a MATROID 1. Introduction The
THE WHITNEY ALGEBRA OF A MATROID HENRY CRAPO AND WILLIAM SCHMITT 1. introduction The concept of matroid, with its companion concept of geometric lattice, was distilled by Hassler Whitney [19], Saunders Mac Lane [10] and Garrett Birkhoff [2] from the common properties of linear and algebraic dependence. The inverse prob- lem, how to represent a given abstract matroid as the matroid of linear dependence of a specified set of vectors over some field (or as the matroid of algebraic depen- dence of a specified set of algebraic functions) has already prompted fifty years of intense effort by the leading researchers in the field: William Tutte, Dominic Welsh, Tom Brylawski, Neil White, Bernt Lindstrom, Peter Vamos, Joseph Kung, James Oxley, and Geoff Whittle, to name only a few. (A goodly portion of this work aimed to provide a proof or refutation of what is now, once again, after a hundred or so years, the 4-color theorem.) One way to attack this inverse problem, the representation problem for matroids, is first to study the ‘play of coordinates’ in vector representations. In a vector representation of a matroid M, each element of M is assigned a vector in such a way that dependent (resp., independent) subsets of M are assigned dependent (resp., independent) sets of vectors. The coefficients of such linear dependencies are computable as minors of the matrix of coordinates of the dependent sets of vectors; this is Cramer’s rule. For instance, if three points a, b, c are represented in R4 (that is, in real projective 3-space) by the dependent vectors forming the rows of the matrix 1 2 3 4 a 1 4 0 6 C = b −2 3 1 −5 , c −4 17 3 −3 they are related by a linear dependence, which is unique up to an overall scalar multiple, and is computable by forming ‘complementary minors’ in any pair of independent columns of the matrix C. -
Asymptotic Behavior of the Length of Local Cohomology
ASYMPTOTIC BEHAVIOR OF THE LENGTH OF LOCAL COHOMOLOGY STEVEN DALE CUTKOSKY, HUY TAI` HA,` HEMA SRINIVASAN, AND EMANOIL THEODORESCU Abstract. Let k be a field of characteristic 0, R = k[x1, . , xd] be a polynomial ring, and m its maximal homogeneous ideal. Let I ⊂ R be a homogeneous ideal in R. In this paper, we show that λ(H0 (R/In)) λ(Extd (R/In,R(−d))) lim m = lim R n→∞ nd n→∞ nd e(I) always exists. This limit has been shown to be for m-primary ideals I in a local Cohen Macaulay d! ring [Ki, Th, Th2], where e(I) denotes the multiplicity of I. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extention modules may not have polynomial growth. Introduction Let R = k[x1, . , xd] be a polynomial ring over a field k, with graded maximal ideal m, and I ⊂ R d n a proper homogeneous ideal. We investigate the asymptotic growth of λ(ExtR(R/I ,R)) as a function of n. When R is a local Gorenstein ring and I is an m-primary ideal, then this is easily seen to be equal to λ(R/In) and hence is a polynomial in n. A theorem of Theodorescu and Kirby [Ki, Th, Th2] extends this to m-primary ideals in local Cohen Macaulay rings R. We consider homogeneous ideals in a polynomial ring which are not m-primary and show that a limit exists asymptotically although it can be irrational. -
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]. -
Arxiv:1703.06832V3 [Math.AC]
REGULARITY OF FI-MODULES AND LOCAL COHOMOLOGY ROHIT NAGPAL, STEVEN V SAM, AND ANDREW SNOWDEN Abstract. We resolve a conjecture of Ramos and Li that relates the regularity of an FI- module to its local cohomology groups. This is an analogue of the familiar relationship between regularity and local cohomology in commutative algebra. 1. Introduction Let S be a standard-graded polynomial ring in finitely many variables over a field k, and let M be a non-zero finitely generated graded S-module. It is a classical fact in commutative algebra that the following two quantities are equal (see [Ei, §4B]): S • The minimum integer α such that Tori (M, k) is supported in degrees ≤ α + i for all i. i • The minimum integer β such that Hm(M) is supported in degrees ≤ β − i for all i. i Here Hm is local cohomology at the irrelevant ideal m. The quantity α = β is called the (Castelnuovo–Mumford) regularity of M, and is one of the most important numerical invariants of M. In this paper, we establish the analog of the α = β identity for FI-modules. To state our result precisely, we must recall some definitions. Let FI be the category of finite sets and injections. Fix a commutative noetherian ring k. An FI-module over k is a functor from FI to the category of k-modules. We write ModFI for the category of FI-modules. We refer to [CEF] for a general introduction to FI-modules. Let M be an FI-module. Define Tor0(M) to be the FI-module that assigns to S the quotient of M(S) by the sum of the images of the M(T ), as T varies over all proper subsets of S. -
Computations in Algebraic Geometry with Macaulay 2
Computations in algebraic geometry with Macaulay 2 Editors: D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels Preface Systems of polynomial equations arise throughout mathematics, science, and engineering. Algebraic geometry provides powerful theoretical techniques for studying the qualitative and quantitative features of their solution sets. Re- cently developed algorithms have made theoretical aspects of the subject accessible to a broad range of mathematicians and scientists. The algorith- mic approach to the subject has two principal aims: developing new tools for research within mathematics, and providing new tools for modeling and solv- ing problems that arise in the sciences and engineering. A healthy synergy emerges, as new theorems yield new algorithms and emerging applications lead to new theoretical questions. This book presents algorithmic tools for algebraic geometry and experi- mental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. A wide range of mathematical scientists should find these expositions valuable. This includes both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all. -
Prizes and Awards Session
PRIZES AND AWARDS SESSION Wednesday, July 12, 2021 9:00 AM EDT 2021 SIAM Annual Meeting July 19 – 23, 2021 Held in Virtual Format 1 Table of Contents AWM-SIAM Sonia Kovalevsky Lecture ................................................................................................... 3 George B. Dantzig Prize ............................................................................................................................. 5 George Pólya Prize for Mathematical Exposition .................................................................................... 7 George Pólya Prize in Applied Combinatorics ......................................................................................... 8 I.E. Block Community Lecture .................................................................................................................. 9 John von Neumann Prize ......................................................................................................................... 11 Lagrange Prize in Continuous Optimization .......................................................................................... 13 Ralph E. Kleinman Prize .......................................................................................................................... 15 SIAM Prize for Distinguished Service to the Profession ....................................................................... 17 SIAM Student Paper Prizes .................................................................................................................... -
Arxiv:2010.06953V2 [Math.RA] 23 Apr 2021
IDENTITIES AND BASES IN THE HYPOPLACTIC MONOID ALAN J. CAIN, ANTÓNIO MALHEIRO, AND DUARTE RIBEIRO Abstract. This paper presents new results on the identities satisfied by the hypoplactic monoid. We show how to embed the hypoplactic monoid of any rank strictly greater than 2 (including infinite rank) into a direct product of copies of the hypoplactic monoid of rank 2. This confirms that all hypoplactic monoids of rank greater than or equal to 2 satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the variety generated by the hypoplactic monoid has finite axiomatic rank, by giving a finite basis for it. 1. Introduction A (non-trivial) identity is a formal equality u ≈ v, where u and v are words over some alphabet of variables, which is not of the form u ≈ u. If a monoid is known to satisfy an identity, an important question is whether the set of identities it satisfies is finitely based, that is, if all these identities are consequences of those in some finite subset (see [37, 42]). The plactic monoid plac, also known as the monoid of Young tableaux, is an important algebraic structure, first studied by Schensted [38] and Knuth [23], and later studied in depth by Lascoux and Schützenberger [27]. It is connected to many different areas of Mathematics, such as algebraic combinatorics, symmetric functions [29], crystal bases [4] and representation theory [14]. In particular, the question of identities in the plactic monoid has received a lot of attention recently [26, 20]. Finitely-generated polynomial-growth groups are virtually nilpotent and so sat- isfy identities [16]. -
Vita BERND STURMFELS Department of Mathematics, University of California, Berkeley, CA 94720 Phone
Vita BERND STURMFELS Department of Mathematics, University of California, Berkeley, CA 94720 Phone: (510) 642 4687, Fax: (510) 642 8204, [email protected] M.A. [Diplom] TH Darmstadt, Germany, Mathematics and Computer Science, 1985 Ph.D. [Dr. rer. nat.] TH Darmstadt, Germany, Mathematics, 1987 Ph.D. University of Washington, Seattle, Mathematics, 1987 Professional Experience: 1987–1988 Postdoctoral Fellow, I.M.A., University of Minnesota, Minneapolis 1988–1989 Assistant Professor, Research Institute for Symbolic Computation, (RISC-Linz), Linz, Austria 1989–1991 Assistant Professor, Department of Mathematics, Cornell University 1992–1996 Associate Professor, Department of Mathematics, Cornell University 1994–2001 Professor, Department of Mathematics, University of California, Berkeley 2001– Professor, Department of Mathematics and Computer Science, UC Berkeley Academic Honors: 1986 - 1987 Alfred P. Sloan Doctoral Dissertation Fellowship 1991 - 1993 Alfred P. Sloan Research Fellow 1992 - 1997 National Young Investigator (NSF) 1992 - 1997 David and Lucile Packard Fellowship 1999 Lester R. Ford Prize for Expository Writing (MAA) 2000-2001 Miller Research Professorship, UC Berkeley Spring 2003 John von Neumann Professor, Technical University M¨unchen 2003-2004 Hewlett-Packard Research Professor at MSRI Berkeley July 2004 Clay Mathematics Institute Senior Scholar Research Interests: Computational Algebra, Combinatorics, Algebraic Geometry Selected Professional Activities: Visiting Positions: D´epartment de Math´ematiques, Universit´ede Nice, France, Spring 1989 Mathematical Sciences Research Institute, Berkeley, Fall 1992 Courant Institute, New York University, 1994–95 RIMS, Kyoto University, Japan, 1997–98 Current Editorial Board Membership: Journal of the American Mathematical Society, Duke Mathematical Journal, Collecteana Mathematica, Beitr¨agezur Geometrie und Algebra, Order, Discrete and Computational Geometry, Applicable Algebra (AAECC) Journal of Combinatorial Theory (Ser. -
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.