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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 . -
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) . -
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. -
To Handle the Inevitable Ambiguity Of
594 BOOK REVIEWS To handle the inevitable ambiguity of sign in SL(2, C), the author orients his lines so that if / has ends u and u taken in that order it corresponds to the half-turn matrix 1 = -7 {u + u -2uud -u-u ). u - u Then trace relations for products of these matrices based on the formula tratrb = trab + tra" b give the desired trigonometrical relations without ambiguity of sign. Full details, including conventions to handle special posi tion and degeneracy and additional machinery to handle opposite isometries, must await the reader's own study of this intriguing book. For a first perusal that quickly reaches the most accessible parts of the main results, I recommend §§1.3, V.3, VI.2, and VI.5 and 6. REFERENCES 1. Lars V. Ahlfors, Möbius transformations in several dimensions, Ordway Professorship Lectures in Mathematics, University of Minnesota, 1981. 2. Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Math., vol. 91, Springer-Verlag, Berlin and New York, 1983. 3. H. S. M. Coxeter, Inversive distance, Ann. Mat. Pura Appl. (4) 71 (1966), 73-83. 4. , The inversive plane and hyperbolic space, Abh. Math. Sem. Univ. Hamburg 29 (1966), 217-242. 5. J. B. Wilker, Inversive geometry, The Geometric Vein (Coxeter Festschrift), Springer-Verlag, New York, 1981, pp. 379-442. 6. , Möbius transformations in dimension n , Period. Math. Hungar. 14 (1983), 93-99. J. B. WILKER UNIVERSITY OF TORONTO BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 23, Number 2, October 1990 ©1990 American Mathematical Society 0273-0979/90 $1.00+ $.25 per page The classical groups and K-theory, by A. -
Remarks on Classical Invariant Theory 549
transactions of the american mathematical society Volume 313, Number 2, June 1989 REMARKS ON CLASSICAL INVARIANTTHEORY ROGER HOWE Abstract. A uniform formulation, applying to all classical groups simultane- ously, of the First Fundamental Theory of Classical Invariant Theory is given in terms of the Weyl algebra. The formulation also allows skew-symmetric as well as symmetric variables. Examples illustrate the scope of this formulation. 0. Introduction (N.B. This introductory discussion is somewhat breezy. We will be more careful beginning in §1.) In Hermann Weyl's wonderful and terrible book, The classical groups [W], one may discern two main themes: first, the study of the polynomial invariants for an arbitrary number of (contravariant or covariant) variables for a standard classical group action; second, the isotypic decomposition of the full tensor algebra for such an action. It may be observed that the two questions are more or less equivalent. (Indeed, Weyl exploits one direction of this equivalence.) Further, it is not hard to see that both questions are equivalent to an apparently simpler one, namely the description of the invariants in the full (mixed) tensor algebra of such an action. (These equivalences are not a priori clear. It is the cleanness of the answers that allow one to make the connections.) Thus if Weyl had first established the result on tensor invariants, then proceeded to the other two topics, his presentation would have been considerably streamlined and presumably more acceptable to modern taste. A plausible explanation for his failure to proceed along that route is that he saw no way of establishing the result on tensor invariants except through the result on polynomial invariants. -
The Capelli Identity, Tube Domains, and the Generalized Laplace Transform*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ADVANCES IN MATHEMATICS 87, 7 l-92 (199 1) The Capelli Identity, Tube Domains, and the Generalized Laplace Transform* BERTRAM KOSTANT Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 AND SIDDHARTHA SAHI Department of Mathematics, Princeton University. Princeton, New Jersey 08544 INTRODUCTION The key result in this paper is a proof of a large class of identities, generalizing the Capelli identity [Cl. The Capelli identity is a centerpiece of 19th century invariant theory. It asserts the equality of two differential operators on the n2-dimensional space M(n, R) of n x n real matrices. Let E, denote the ijth elementary matrix, let xii denote the linear functions on M(n, R) dual to E,, and write 3, for the partial differential operator a/ax,. Then this identity says det(rcU + (n - i) 6,) = det(xV) det(a,), (0.1) where 6, is the Kronecker delta, and rcij is the vector field (0.2) By the determinant of an n x n matrix (A,) of (possibly) noncommuting variables, we mean the expression C wb4 &,,,, , -4.(,,..; (0.3) wE .s” and the left side of (0.1) is to be understood in this sense. * The authors were supported in part by NSF grants at MIT and at Princeton University. 71 OOOl-8708/91 $7.50 Copyright 0 1991 by Academic Press, Inc. All nghfs of reproduction m any Iorm reserved 72 KOSTANT AND SAHI The vector field in (0.2) arises as follows. -
Notices of the American Mathematical Society ABCD Springer.Com
ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Highlights in Springer’s eBook Collection of the American Mathematical Society September 2009 Volume 56, Number 8 Bôcher, Osgood, ND and the Ascendance NEW NEW 2 EDITION EDITION of American Mathematics forms bridges between From the reviews of the first edition The theory of elliptic curves is Mathematics at knowledge, tradition, and contemporary 7 Chorin and Hald provide excellent distinguished by the diversity of the Harvard life. The continuous development and explanations with considerable insight methods used in its study. This book growth of its many branches permeates and deep mathematical understanding. treats the arithmetic theory of elliptic page 916 all aspects of applied science and 7 SIAM Review curves in its modern formulation, technology, and so has a vital impact on through the use of basic algebraic our society. The book will focus on these 2nd ed. 2009. X, 162 p. 7 illus. number theory and algebraic geometry. aspects and will benefit from the (Surveys and Tutorials in the Applied contribution of world-famous scientists. Mathematical Sciences) Softcover 2nd ed. 2009. XVIII, 514 p. 14 illus. Invariant Theory ISBN 978-1-4419-1001-1 (Graduate Texts in Mathematics, 2009. XI, 263 p. (Modeling, Simulation & 7 Approx. $39.95 Volume 106) Hardcover of Tensor Product Applications, Volume 3) Hardcover ISBN 978-0-387-09493-9 7 $59.95 Decompositions of ISBN 978-88-470-1121-2 7 $59.95 U(N) and Generalized Casimir Operators For access check with your librarian page 931 Stochastic Partial Linear Optimization Mathematica in Action Riverside Meeting Differential Equations The Simplex Workbook The Power of Visualization A Modeling, White Noise Approach G. -
The Classical Groups
THE CLASSICAL GROUPS KEVIN MCGERTY Date: March, 2006. 1 2 KEVIN MCGERTY 1. INTRODUCTION These notes are the result of teaching Math 241 “Topics in Geometry” in the Spring of 2006 at the University of Chicago. They are study of matrix groups and some of the geometry attached to them. Of course “geometry” is not a technical term, and in order to keep the prerequisites to a minimum the word is used in an essentially combinatorial sense here – the “geometry” of projective space for example is the poset of subspaces, not anything more advanced, such as the struc- ture of a manifold or algebraic variety (though we describe things in enough detail that a more knowledgeable student should easily be able to find such a structure if they know the appropriate defintions). We begin by studying isometries of Rn, focusing on the low dimensional cases of n = 2, 3. We then discuss the quaternions and explain the connection with SO(R3). Motivated by this, we classify composition algebras over R, and define the compact classical groups as matrix groups of R, C and H preserving the ap- propriate Hermitian form. We then formulate a description of these groups as subgroups of GL(Cn), allowing us to obtain the noncompact analogs in terms of bilinear forms, which make sense over any field. We then briefly study projective spaces, which have a rich enough structure that in the next section we can use the action of PGLn on them to show that the projective special linear group is almost always simple. We end with a brief discussion of bilinear forms and the symplectic group. -
CLASSICAL INVARIANT THEORY a Primer
Hanspeter Kraft Claudio Procesi CLASSICAL INVARIANT THEORY A Primer Preliminary Version, July 1996 Contents i Table of Contents Introduction . ................................ 1 § 1Invariant Functions and Covariants . .............. 1 1.1 Polynomial functions . ..................... 1 1.2 Invariants . ......................... 1 1.3 Zariski-dense subsets . ..................... 6 1.4 Covariants . ......................... 8 1.5 Classical Invariant Theory . .............. 11 1.6 Some Finiteness Theorems . .............. 12 § 2 First Fundamental Theorem for the General Linear Group . 16 2.1 Invariants of vectors and covectors . .............. 16 2.2 Geometric interpretation . .............. 17 2.3 Invariants of conjugacy classes . .............. 18 2.4 Traces of powers ......................... 20 2.5 Invariants under simultaneous conjugation .......... 21 § 3 Endomorphisms of Tensors . ..................... 23 3.1 Centralizers and endomorphism rings . .......... 23 3.2 Double Centralizer Theorem . .............. 25 3.3 Decomposition of V ⊗m ..................... 27 § 4Polarization and Restitution . ..................... 30 4.1 Multihomogeneous invariants . .............. 30 4.2 Multilinear invariants of vectors and covectors . 30 4.3 Multilinear invariants of matrices . .............. 32 4.4 Polarization . ......................... 34 4.5 Restitution . ......................... 34 4.6 Generalization to several representations . .......... 35 4.7 Proof of the First Fundamental Theorems .......... 36 4.8 Example: Invariants of forms and vectors -
Index Theory with Applications to Mathematics and Physics
Index Theory with Applications to Mathematics and Physics Index Theory with Applications to Mathematics and Physics David D. Bleecker University of Hawaii at Manoa Bernhelm Booß-Bavnbek Roskilde Universitet, Denmark International Press www.intlpress.com Index Theory with Applications to Mathematics and Physics David D. Bleecker (University of Hawaii at Manoa) Bernhelm Booß-Bavnbek (Roskilde Universitet, Denmark) 2010 Mathematics Subject Classification. Primary: 14D21, 47A53, 57R57, 58J05, 58J20, 58J35, 81T13. Secondary: 01A55, 01A60, 01A61, 01A65, 01A67, 14C40, 14J28, 14J80, 15A75, 15Bxx, 19K56, 19L10, 19L47, 20C33, 22E60, 30E25, 32Q15, 35A02, 35J48, 35J56, 35K08, 35Q41, 35S05, 41A60, 47B35, 46E35, 53C05, 53C07, 53C27, 53Z05, 55N05, 55N15, 55Q40, 55R45, 55R50, 58C15, 58C30, 58C40, 58D27, 58J32, 58J40, 58J50, 58J52, 58Z05, 83C05, 83C47. Copyright © 2013 by International Press of Boston, Inc. Somerville, Massachusetts, U.S.A. All rights reserved. Individual readers of this publication, and non-profit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publica- tion in reviews, provided the customary acknowledgement of the source is given. Repub- lication, systematic copying, or mass reproduction of any material in this publication is permitted only under license from International Press. Excluded from these provisions is any material herein to which another party holds the copyright. In such a case, appropri- ate copyright notice is given with the material, and requests for permission to use or re- print should be addressed to the copyright holder. ISBN: 978-1-57146-264-0 Printed in the United States of America. -
The Classical Groups As a Source of Algebraic Problems C.W
THE CLASSICAL GROUPS AS A SOURCE OF ALGEBRAIC PROBLEMS C.W. CURTIS, University of Oregon Introduction. Rather than attempting a general survey of what has been accomplished in algebra, I shall try to describe the influence on the develop- ment of algebra of one of the great sources of algebraic problems and ideas-the classical linear groups. I intend to show that these groups have provided an ex- perimental, empirical basis for large parts of algebra. They are a family of ex- amples which serve as raw material from which problems, conjectures, abstract ideas and general theorems have emerged, and which have provided direction and continuity to the research based on them. Surveys of recent develop~nents and problems of current interest have been given by Brauer [S], [6], Carter [ll], DieudonnE [24], M. Hall [26], Kaplansky [35], and MacEane [36]. It is impossible in this lecture to give an account of all the major results which involve the classical groups. In malring a selection, I have simply in- cluded the material that has been most helpful to me. 1. Desnitions and background. The story begins with Galois, who proved that a polynomial equation was solvable by radicals if and only if a certain finite group of automorphisms of a field was solvable. Thus he raised the prob- lem of investigating groups in terms of their structure, and observed, as an ex- ample, that the set of all 2 by 2 invertible matrices with coefficients in a finite fie!d of more than 3 elements, was a finite group that was not solvable. -
Stable Branching Rules for Classical Symmetric Pairs 1
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9947(XX)0000-0 STABLE BRANCHING RULES FOR CLASSICAL SYMMETRIC PAIRS ROGER HOWE, ENG-CHYE TAN, JEB F. WILLENBRING 1. Introduction Given completely reducible representations, V and W of complex algebraic groups G and H respectively, together with an embedding H,→ G, we let [V, W ] = dim HomH (W, V ) where V is regarded as a representation of H by restriction. If W is irreducible, then [V, W ] is the multiplicity of W in V . This number may of course be infinite if V or W is infinite dimensional. A description of the numbers [V, W ] is referred in the mathematics and physics literature as a branching rule. The context of this paper has its origins in the work of D. Littlewood. In [Li2], Littlewood describes two classical branching rules from a combinatorial perspective (see also [Li1]). Specifically, Littlewood’s results are branching multiplicities for GLn to On and GL2n to Sp2n. These pairs of groups are significant in that they are examples of symmetric pairs. A symmetric pair is a pair of groups (H, G) such that G is a reductive algebraic group and H is the fixed point set of a regular involution defined on G. It follows that H is a closed, reductive algebraic subgroup of G. The goal of this paper is to put the formula into the context of the first named author’s theory of dual reductive pairs. The advantage of this point of view is that it relates branching from one symmetric pair to another and as a consequence Littlewood’s formula may be generalized to all classical symmetric pairs.