On a Representation of the Group GL (K; R) and Its Spherical Functions
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Physics.Hist-Ph] 15 May 2018 Meitl,Fo Hsadsadr Nepeainlprinciples
Why Be Regular?, Part I Benjamin Feintzeig Department of Philosophy University of Washington JB (Le)Manchak, Sarita Rosenstock, James Owen Weatherall Department of Logic and Philosophy of Science University of California, Irvine Abstract We provide a novel perspective on “regularity” as a property of representations of the Weyl algebra. We first critique a proposal by Halvorson [2004, “Complementarity of representa- tions in quantum mechanics”, Studies in History and Philosophy of Modern Physics 35(1), pp. 45–56], who argues that the non-regular “position” and “momentum” representations of the Weyl algebra demonstrate that a quantum mechanical particle can have definite values for position or momentum, contrary to a widespread view. We show that there are obstacles to such an intepretation of non-regular representations. In Part II, we propose a justification for focusing on regular representations, pace Halvorson, by drawing on algebraic methods. 1. Introduction It is standard dogma that, according to quantum mechanics, a particle does not, and indeed cannot, have a precise value for its position or for its momentum. The reason is that in the standard Hilbert space representation for a free particle—the so-called Schr¨odinger Representation of the Weyl form of the canonical commutation relations (CCRs)—there arXiv:1805.05568v1 [physics.hist-ph] 15 May 2018 are no eigenstates for the position and momentum magnitudes, P and Q; the claim follows immediately, from this and standard interpretational principles.1 Email addresses: [email protected] (Benjamin Feintzeig), [email protected] (JB (Le)Manchak), [email protected] (Sarita Rosenstock), [email protected] (James Owen Weatherall) 1Namely, the Eigenstate–Eigenvalue link, according to which a system has an exact value of a given property if and only if its state is an eigenstate of the operator associated with that property. -
Fourier Analysis Using Representation Theory
FOURIER ANALYSIS USING REPRESENTATION THEORY NEEL PATEL Abstract. In this paper, it is our goal to develop the concept of Fourier transforms by using Representation Theory. We begin by laying basic def- initions that will help us understand the definition of a representation of a group. Then, we define representations and provide useful definitions and the- orems. We study representations and key theorems about representations such as Schur's Lemma. In addition, we develop the notions of irreducible repre- senations, *-representations, and equivalence classes of representations. After doing so, we develop the concept of matrix realizations of irreducible represen- ations. This concept will help us come up with a few theorems that lead up to our study of the Fourier transform. We will develop the definition of a Fourier transform and provide a few observations about the Fourier transform. Contents 1. Essential Definitions 1 2. Group Representations 3 3. Tensor Products 7 4. Matrix Realizations of Representations 8 5. Fourier Analysis 11 Acknowledgments 12 References 12 1. Essential Definitions Definition 1.1. An internal law of composition on a set R is a product map P : R × R ! R Definition 1.2. A group G, is a set with an internal law of composition such that: (i) P is associative. i.e. P (x; P (y; z)) = P (P (x; y); z) (ii) 9 an identity; e; 3 if x 2 G; then P (x; e) = P (e; x) = x (iii) 9 inverses 8x 2 G, denoted by x−1, 3 P (x; x−1) = P (x−1; x) = e: Let it be noted that we shorthand P (x; y) as xy. -
Factorization of Unitary Representations of Adele Groups Paul Garrett [email protected]
(February 19, 2005) Factorization of unitary representations of adele groups Paul Garrett [email protected] http://www.math.umn.edu/˜garrett/ The result sketched here is of fundamental importance in the ‘modern’ theory of automorphic forms and L-functions. What it amounts to is proof that representation theory is in principle relevant to study of automorphic forms and L-function. The goal here is to get to a coherent statement of the basic factorization theorem for irreducible unitary representations of reductive linear adele groups, starting from very minimal prerequisites. Thus, the writing is discursive and explanatory. All necessary background definitions are given. There are no proofs. Along the way, many basic concepts of wider importance are illustrated, as well. One disclaimer is necessary: while in principle this factorization result makes it clear that representation theory is relevant to study of automorphic forms and L-functions, in practice there are other things necessary. In effect, one needs to know that the representation theory of reductive linear p-adic groups is tractable, so that conversion of other issues into representation theory is a change for the better. Thus, beyond the material here, one will need to know (at least) the basic properties of spherical and unramified principal series representations of reductive linear groups over local fields. The fact that in some sense there is just one irreducible representation of a reductive linear p-adic group will have to be pursued later. References and historical notes -
Q-KRAWTCHOUK POLYNOMIALS AS SPHERICAL FUNCTIONS on the HECKE ALGEBRA of TYPE B
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 10, Pages 4789{4813 S 0002-9947(00)02588-5 Article electronically published on April 21, 2000 q-KRAWTCHOUK POLYNOMIALS AS SPHERICAL FUNCTIONS ON THE HECKE ALGEBRA OF TYPE B H. T. KOELINK Abstract. The Hecke algebra for the hyperoctahedral group contains the Hecke algebra for the symmetric group as a subalgebra. Inducing the index representation of the subalgebra gives a Hecke algebra module, which splits multiplicity free. The corresponding zonal spherical functions are calculated in terms of q-Krawtchouk polynomials using the quantised enveloping algebra for sl(2; C). The result covers a number of previously established interpretations of (q-)Krawtchouk polynomials on the hyperoctahedral group, finite groups of Lie type, hypergroups and the quantum SU(2) group. 1. Introduction The hyperoctahedral group is the finite group of signed permutations, and it con- tains the permutation group as a subgroup. The functions on the hyperoctahedral group, which are left and right invariant with respect to the permutation group, are spherical functions. The zonal spherical functions are the spherical functions which are contained in an irreducible subrepresentation of the group algebra under the left regular representation. The zonal spherical functions are known in terms of finite discrete orthogonal polynomials, the so-called symmetric Krawtchouk poly- nomials. This result goes back to Vere-Jones in the beginning of the seventies, and is also contained in the work of Delsarte on coding theory; see [9] for information and references. There are also group theoretic interpretations of q-Krawtchouk polynomials. Here q-Krawtchouk polynomials are q-analogues of the Krawtchouk polynomials in the sense that for q tending to one we recover the Krawtchouk polynomials. -
23 Infinite Dimensional Unitary Representations
23 Infinite dimensional unitary representations Last lecture, we found the finite dimensional (non-unitary) representations of SL2(R). 23.1 Background about infinite dimensional representa- tions (of a Lie group G) What is an infinite dimensional representation? 1st guess Banach space acted on by G? This is no good for the following reasons: Look at the action of G on the functions on G (by left translation). We could use L2 functions, or L1 or Lp. These are completely different Banach spaces, but they are essentially the same representation. 2nd guess Hilbert space acted on by G? This is sort of okay. The problem is that finite dimensional representations of SL2(R) are NOT Hilbert space representations, so we are throwing away some interesting representations. Solution (Harish-Chandra) Take g to be the Lie algebra of G, and let K be the maximal compact subgroup. If V is an infinite dimensional representation of G, there is no reason why g should act on V . The simplest example fails. Let R act on L2(R) by left translation. Then d d 2 R d the Lie algebra is generated by dx (or i dx ) acting on L ( ), but dx of an L2 function is not in L2 in general. Let V be a Hilbert space. Set Vω to be the K-finite vectors of V , which are the vectors contained in a finite dimensional representation of K. The point is that K is compact, so V splits into a Hilbert space direct sum finite dimensional representations of K, at least if V is a Hilbert space. -
Kobe University Repository : Kernel
Kobe University Repository : Kernel タイトル Zonal spherical functions on the quantum homogeneous space Title SUq(n+1)/SUq(n) 著者 Noumi, Masatoshi / Yamada, Hirofumi / Mimachi, Katsuhisa Author(s) 掲載誌・巻号・ページ Proceedings of the Japan Academy. Ser. A, Mathematical Citation sciences,65(6):169-171 刊行日 1989-06 Issue date 資源タイプ Journal Article / 学術雑誌論文 Resource Type 版区分 publisher Resource Version 権利 Rights DOI 10.3792/pjaa.65.169 JaLCDOI URL http://www.lib.kobe-u.ac.jp/handle_kernel/90003238 PDF issue: 2021-10-01 No. 6] Proc. Japan Acad., 65, Ser. A (1989) 169 Zonal Spherical Functions on the Quantum Homogeneous Space SU (n + 1 )/SU (n) By Masatoshi NOUMI,*) Hirofumi YAMADA,**) and Katsuhisa :M:IMACHI***) (Communicated by Shokichi IYANA,GA, M. ff.A., June. 13, 1989) In this note, we give an explicit expression to the zonal spherical func- tions on the quantum homogeneous space SU(n + 1)/SU(n). Details of the following arguments as well as the representation theory of the quantum group SU(n+ 1) will be presented in our forthcoming paper [3]. Through- out this note, we fix a non-zero real number q. 1. Following [4], we first make a brief review on the definition of the quantum groups SLy(n+ 1; C) and its real form SUb(n+ 1). The coordinate ring A(SL,(n+ 1; C)) of SLy(n+ 1; C) is the C-algebra A=C[x,; O_i, ]_n] defined by the "canonical generators" x, (Oi, ]n) and the following fundamental relations" (1.1) xixj=qxx, xixj=qxx or O_i]_n, O_k_n, (1.2) xx=xxt, xx--qxxj=xtx--q-lxx or Oi]_n, Okl_n and (1.3) det-- 1. -
Arxiv:Math/9512225V1
SOME REMARKS ON THE CONSTRUCTION OF QUANTUM SYMMETRIC SPACES Mathijs S. Dijkhuizen Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe 657, Japan 31 May 1995 Abstract. We present a general survey of some recent developments regarding the construction of compact quantum symmetric spaces and the analysis of their zonal spherical functions in terms of q-orthogonal polynomials. In particular, we define a one-parameter family of two-sided coideals in Uq(gl(n, C)) and express the zonal spherical functions on the corresponding quantum projective spaces as Askey-Wilson polynomials containing two continuous and one discrete parameter. 0. Introduction In this paper, we discuss some recent progress in the analysis of compact quantum symmetric spaces and their zonal spherical functions. It is our aim to present a more or less coherent overview of a number of quantum analogues of symmetric spaces that have recently been studied. We shall try to emphasize those aspects of the theory that distinguish quantum symmetric spaces from their classical counterparts. It was recognized not long after the introduction of quantum groups by Drinfeld [Dr], Jimbo [J1], and Woronowicz [Wo], that the quantization of symmetric spaces was far from straightforward. We discuss some of the main problems that came up. First of all, there was the, at first sight rather annoying, lack of interesting quan- tum subgroups. Let us mention, for instance, the well-known fact that there is no analogue of SO(n) inside the quantum group SUq(n). A major step forward in overcoming this hurdle was made by Koornwinder [K2], who was able to construct a quantum analogue of the classical 2-sphere SU(2)/SO(2) and express the zonal arXiv:math/9512225v1 [math.QA] 21 Dec 1995 spherical functions as a subclass of Askey-Wilson polynomials by exploiting the notion of a twisted primive element in the quantized universal enveloping algebra q(sl(2, C)). -
Noncommutative Fourier Transform for the Lorentz Group Via the Duflo Map
PHYSICAL REVIEW D 99, 106005 (2019) Noncommutative Fourier transform for the Lorentz group via the Duflo map Daniele Oriti Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam-Golm, Germany and Arnold-Sommerfeld-Center for Theoretical Physics, Ludwig-Maximilians-Universität, Theresienstrasse 37, D-80333 München, Germany, EU Giacomo Rosati INFN, Sezione di Cagliari, Cittadella Universitaria, 09042 Monserrato, Italy (Received 11 February 2019; published 13 May 2019) We defined a noncommutative algebra representation for quantum systems whose phase space is the cotangent bundle of the Lorentz group, and the noncommutative Fourier transform ensuring the unitary equivalence with the standard group representation. Our construction is from first principles in the sense that all structures are derived from the choice of quantization map for the classical system, the Duflo quantization map. DOI: 10.1103/PhysRevD.99.106005 I. INTRODUCTION of (noncommutative) Fourier transform to establish the (unitary) equivalence between this new representation and Physical systems whose configuration or phase space is the one based on L2 functions on the group configura- endowed with a curved geometry include, for example, tion space. point particles on curved spacetimes and rotor models in The standard way of dealing with this issue is indeed to condensed matter theory, and they present specific math- renounce to a representation that makes direct use ematical challenges in addition to their physical interest. of the noncommutative Lie algebra variables, and to use In particular, many important results have been obtained in instead a representation in terms of irreducible representa- the case in which their domain spaces can be identified tions of the Lie group, i.e., to resort to spectral decom- with (Abelian) group manifolds or their associated homo- position as the proper analogue of the traditional Fourier geneous spaces. -
Lecture 1: Group C*-Algebras and Actions of Finite Groups on C*-Algebras 11–29 July 2016
The Second Summer School on Operator Algebras and Noncommutative Geometry 2016 East China Normal University, Shanghai Lecture 1: Group C*-algebras and Actions of Finite Groups on C*-Algebras 11{29 July 2016 Lecture 1 (11 July 2016): Group C*-algebras and Actions of Finite N. Christopher Phillips Groups on C*-Algebras Lecture 2 (13 July 2016): Introduction to Crossed Products and More University of Oregon Examples of Actions. Lecture 3 (15 July 2016): Crossed Products by Finite Groups; the 11 July 2016 Rokhlin Property. Lecture 4 (18 July 2016): Crossed Products by Actions with the Rokhlin Property. Lecture 5 (19 July 2016): Crossed Products of Tracially AF Algebras by Actions with the Tracial Rokhlin Property. Lecture 6 (20 July 2016): Applications and Problems. N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 1 / 28 N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 2 / 28 A rough outline of all six lectures General motivation The material to be described is part of the structure and classification The beginning: The C*-algebra of a group. theory for simple nuclear C*-algebras (the Elliott program). More Actions of finite groups on C*-algebras and examples. specifically, it is about proving that C*-algebras which appear in other Crossed products by actions of finite groups: elementary theory. parts of the theory (in these lectures, certain kinds of crossed product C*-algebras) satisfy the hypotheses of known classification theorems. More examples of actions. Crossed products by actions of finite groups: Some examples. -
CHAPTER 2 Representations of Finite Groups in This Chapter We Consider
CHAPTER 2 Representations of Finite Groups In this chapter we consider only finite-dimensional representations. 2.1. Unitarity, complete reducibility, orthogonality relations Theorem 1. A representation of a finite group is unitary. c Proof. Let (π, V ) be a (finite- dimensional) representation of a finite group G = { g1, g2, . gn }. Let h·, ·i1 be any inner product on V . Set n X hv, wi = hπ(gj)v, π(gj)wi1, v, w ∈ V. j=1 Then it is clear from the definition that hπ(g)v, π(g)wi = hv, wi, v, w ∈ V, g ∈ G. Pn Note that if v ∈ V , then hv, vi = j=1hπ(gj)v, π(gj)vi1 and if v 6= 0, then π(gj)v 6= 0 for all j implies hπ(gj)v, π(gj)vi1 > 0 for all j. Hence v 6= 0 implies hv, vi > 0. The other properties of inner product are easy to verify for h·, ·i, using the fact that h·, ·i1 is an inner product, and each π(gj) is linear. The details are left as an exercise. qed The following is an immediate consequence of Theorem 1 and a result from Chapter I stating that a finite-dimensional unitary representation is completely reducible. Theorem 2. A representation of a finite group is completely reducible. Example. Let G be a finite group acting on a finite set X. Let V be a complex vector space having a basis { vx1 , . , vxm } indexed by the elements x1, . , xm of X. If g ∈ G, let π(g) be the operator sending vxj to vg·xj , 1 ≤ j ≤ m. -
Wiener's Tauberian Theorem for Spherical Functions on the Automorphism Group of the Unit Disk
http://www.diva-portal.org This is the published version of a paper published in Arkiv för matematik. Citation for the original published paper (version of record): Hedenmalm, H., Benyamini, Y., Ben Natan, Y., Weit, Y. (1996) Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk. Arkiv för matematik, 34: 199-224 Access to the published version may require subscription. N.B. When citing this work, cite the original published paper. Permanent link to this version: http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-182412 Ark. Mat., 34 (1996), 199-224 (~) 1996 by Institut Mittag-Leffier. All rights reserved Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk Yaakov Ben Natan, Yoav Benyamini, Hs Hedenmalm and Yitzhak Weit(1) Abstract. Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spherical integrable functions on the (con- formal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of functions with integral zero. This is then used to prove a generalization of Furstenberg's theorem, which characterizes harmonic functions on the unit disk by a mean value property, and a "two circles" Morera type theorem (earlier announced by AgranovskiY). Introduction If G is a locally compact abelian group, Wiener's tanberian theorem asserts that if the Fourier transforms of the elements of a closed ideal I of the convolution algebra LI(G) have no common zero, then I=LI(G). -
Arxiv:0809.2017V1 [Math.OC] 11 Sep 2008 Lc Ignlzto,Teafnto,Pstv Kernels, Harmonics
LECTURE NOTES: SEMIDEFINITE PROGRAMS AND HARMONIC ANALYSIS FRANK VALLENTIN Abstract. Lecture notes for the tutorial at the workshop HPOPT 2008 — 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands. Contents 1. Introduction 2 2. The theta function for distance graphs 3 2.1. Distance graphs 4 2.2. Original formulations for theta 6 2.3. Positive Hilbert-Schmidt kernels 7 2.4. The theta function for infinite graphs 9 2.5. Exploiting symmetry 10 3. Harmonic analysis 12 3.1. Basic tools 13 3.2. The Peter-Weyl theorem 14 3.3. Bochner’s characterization 15 4. Boolean harmonics 16 4.1. Specializations 16 4.2. Linearprogrammingboundforbinarycodes 17 4.3. Fourier analysis on the Hamming cube 18 5. Spherical harmonics 19 5.1. Specializations 20 arXiv:0809.2017v1 [math.OC] 11 Sep 2008 5.2. Linear programming bound for the unit sphere 21 5.3. Harmonic polynomials 22 5.4. Addition formula 25 Appendix A. Symmetric tensors and polynomials 26 Appendix B. Further reading 27 References 28 Date: September 11, 2008. 1991 Mathematics Subject Classification. 43A35, 52C17, 90C22, 94B65. Key words and phrases. semidefinite programming, harmonic analysis, symmetry reduction, block diagonalization, theta function, positive kernels, Peter-Weyl theorem, Bochner theorem, boolean harmonics, spherical harmonics. The author was partially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant SCHU 1503/4. 1 2 FRANK VALLENTIN 1. Introduction Semidefinite programming is a vast extension of linear programming and has a wide range of applications: combinatorial optimization and control theory are the most famous ones.