Representation Theory of Symmetric Groups
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Symmetry Preserving Interpolation Erick Rodriguez Bazan, Evelyne Hubert
Symmetry Preserving Interpolation Erick Rodriguez Bazan, Evelyne Hubert To cite this version: Erick Rodriguez Bazan, Evelyne Hubert. Symmetry Preserving Interpolation. ISSAC 2019 - International Symposium on Symbolic and Algebraic Computation, Jul 2019, Beijing, China. 10.1145/3326229.3326247. hal-01994016 HAL Id: hal-01994016 https://hal.archives-ouvertes.fr/hal-01994016 Submitted on 25 Jan 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Symmetry Preserving Interpolation Erick Rodriguez Bazan Evelyne Hubert Université Côte d’Azur, France Université Côte d’Azur, France Inria Méditerranée, France Inria Méditerranée, France [email protected] [email protected] ABSTRACT general concept. An interpolation space for a set of linear forms is The article addresses multivariate interpolation in the presence of a subspace of the polynomial ring that has a unique interpolant for symmetry. Interpolation is a prime tool in algebraic computation each instantiated interpolation problem. We show that the unique while symmetry is a qualitative feature that can be more relevant interpolants automatically inherit the symmetry of the problem to a mathematical model than the numerical accuracy of the pa- when the interpolation space is invariant (Section 3). -
Chapter 1 BASICS of GROUP REPRESENTATIONS
Chapter 1 BASICS OF GROUP REPRESENTATIONS 1.1 Group representations A linear representation of a group G is identi¯ed by a module, that is by a couple (D; V ) ; (1.1.1) where V is a vector space, over a ¯eld that for us will always be R or C, called the carrier space, and D is an homomorphism from G to D(G) GL (V ), where GL(V ) is the space of invertible linear operators on V , see Fig. ??. Thus, we ha½ve a is a map g G D(g) GL(V ) ; (1.1.2) 8 2 7! 2 with the linear operators D(g) : v V D(g)v V (1.1.3) 2 7! 2 satisfying the properties D(g1g2) = D(g1)D(g2) ; D(g¡1 = [D(g)]¡1 ; D(e) = 1 : (1.1.4) The product (and the inverse) on the righ hand sides above are those appropriate for operators: D(g1)D(g2) corresponds to acting by D(g2) after having appplied D(g1). In the last line, 1 is the identity operator. We can consider both ¯nite and in¯nite-dimensional carrier spaces. In in¯nite-dimensional spaces, typically spaces of functions, linear operators will typically be linear di®erential operators. Example Consider the in¯nite-dimensional space of in¯nitely-derivable functions (\functions of class 1") Ã(x) de¯ned on R. Consider the group of translations by real numbers: x x + a, C 7! with a R. This group is evidently isomorphic to R; +. As we discussed in sec ??, these transformations2 induce an action D(a) on the functions Ã(x), de¯ned by the requirement that the transformed function in the translated point x + a equals the old original function in the point x, namely, that D(a)Ã(x) = Ã(x a) : (1.1.5) ¡ 1 Group representations 2 It follows that the translation group admits an in¯nite-dimensional representation over the space V of 1 functions given by the di®erential operators C d d a2 d2 D(a) = exp a = 1 a + + : : : : (1.1.6) ¡ dx ¡ dx 2 dx2 µ ¶ In fact, we have d a2 d2 dà a2 d2à D(a)Ã(x) = 1 a + + : : : Ã(x) = Ã(x) a (x) + (x) + : : : = Ã(x a) : ¡ dx 2 dx2 ¡ dx 2 dx2 ¡ · ¸ (1.1.7) In most of this chapter we will however be concerned with ¯nite-dimensional representations. -
Symmetries on the Lattice
Symmetries On The Lattice K.Demmouche January 8, 2006 Contents Background, character theory of finite groups The cubic group on the lattice Oh Representation of Oh on Wilson loops Double group 2O and spinor Construction of operator on the lattice MOTIVATION Spectrum of non-Abelian lattice gauge theories ? Create gauge invariant spin j states on the lattice Irreducible operators Monte Carlo calculations Extract masses from time slice correlations Character theory of point groups Groups, Axioms A set G = {a, b, c, . } A1 : Multiplication ◦ : G × G → G. A2 : Associativity a, b, c ∈ G ,(a ◦ b) ◦ c = a ◦ (b ◦ c). A3 : Identity e ∈ G , a ◦ e = e ◦ a = a for all a ∈ G. −1 −1 −1 A4 : Inverse, a ∈ G there exists a ∈ G , a ◦ a = a ◦ a = e. Groups with finite number of elements → the order of the group G : nG. The point group C3v The point group C3v (Symmetry group of molecule NH3) c ¡¡AA ¡ A ¡ A Z ¡ A Z¡ A Z O ¡ Z A ¡ Z A ¡ Z A Z ¡ Z A ¡ ZA a¡ ZA b G = {Ra(π),Rb(π),Rc(π),E(2π),R~n(2π/3),R~n(−2π/3)} noted G = {A, B, C, E, D, F } respectively. Structure of Groups Subgroups: Definition A subset H of a group G that is itself a group with the same multiplication operation as G is called a subgroup of G. Example: a subgroup of C3v is the subset E, D, F Classes: Definition An element g0 of a group G is said to be ”conjugate” to another element g of G if there exists an element h of G such that g0 = hgh−1 Example: on can check that B = DCD−1 Conjugacy Class Definition A class of a group G is a set of mutually conjugate elements of G. -
Lie Algebras of Generalized Quaternion Groups 1 Introduction 2
Lie Algebras of Generalized Quaternion Groups Samantha Clapp Advisor: Dr. Brandon Samples Abstract Every finite group has an associated Lie algebra. Its Lie algebra can be viewed as a subspace of the group algebra with certain bracket conditions imposed on the elements. If one calculates the character table for a finite group, the structure of its associated Lie algebra can be described. In this work, we consider the family of generalized quaternion groups and describe its associated Lie algebra structure completely. 1 Introduction The Lie algebra of a group is a useful tool because it is a vector space where linear algebra is available. It is interesting to consider the Lie algebra structure associated to a specific group or family of groups. A Lie algebra is simple if its dimension is at least two and it only has f0g and itself as ideals. Some examples of simple algebras are the classical Lie algebras: sl(n), sp(n) and o(n) as well as the five exceptional finite dimensional simple Lie algebras. A direct sum of simple lie algebras is called a semi-simple Lie algebra. Therefore, it is also interesting to consider if the Lie algebra structure associated with a particular group is simple or semi-simple. In fact, the Lie algebra structure of a finite group is well known and given by a theorem of Cohen and Taylor [1]. In this theorem, they specifically describe the Lie algebra structure using character theory. That is, the associated Lie algebra structure of a finite group can be described if one calculates the character table for the finite group. -
Equivariant and Isovariant Function Spaces
UC Riverside UC Riverside Electronic Theses and Dissertations Title Equivariant and Isovariant Function Spaces Permalink https://escholarship.org/uc/item/7fk0p69b Author Safii, Soheil Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA RIVERSIDE Equivariant and Isovariant Function Spaces A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics by Soheil Safii March 2015 Dissertation Committee: Professor Reinhard Schultz, Chairperson Professor Julia Bergner Professor Fred Wilhelm Copyright by Soheil Safii 2015 The Dissertation of Soheil Safii is approved: Committee Chairperson University of California, Riverside Acknowledgments I would like to thank Professor Reinhard Schultz for all of the time and effort he put into being a wonderful advisor. Without his expansive knowledge and unending patience, none of our work would be possible. Having the opportunity to work with him has been a privilege. I would like to acknowledge Professor Julia Bergner and Professor Fred Wilhelm for their insight and valuable feedback. I am grateful for the unconditional support of my classmates: Mathew Lunde, Jason Park, and Jacob West. I would also like to thank my students, who inspired me and taught me more than I could have ever hoped to teach them. Most of all, I am appreciative of my friends and family, with whom I have been blessed. iv To my parents. v ABSTRACT OF THE DISSERTATION Equivariant and Isovariant Function Spaces by Soheil Safii Doctor of Philosophy, Graduate Program in Mathematics University of California, Riverside, March 2015 Professor Reinhard Schultz, Chairperson The Browder{Straus Theorem, obtained independently by S. -
Math 123—Algebra II
Math 123|Algebra II Lectures by Joe Harris Notes by Max Wang Harvard University, Spring 2012 Lecture 1: 1/23/12 . 1 Lecture 19: 3/7/12 . 26 Lecture 2: 1/25/12 . 2 Lecture 21: 3/19/12 . 27 Lecture 3: 1/27/12 . 3 Lecture 22: 3/21/12 . 31 Lecture 4: 1/30/12 . 4 Lecture 23: 3/23/12 . 33 Lecture 5: 2/1/12 . 6 Lecture 24: 3/26/12 . 34 Lecture 6: 2/3/12 . 8 Lecture 25: 3/28/12 . 36 Lecture 7: 2/6/12 . 9 Lecture 26: 3/30/12 . 37 Lecture 8: 2/8/12 . 10 Lecture 27: 4/2/12 . 38 Lecture 9: 2/10/12 . 12 Lecture 28: 4/4/12 . 40 Lecture 10: 2/13/12 . 13 Lecture 29: 4/6/12 . 41 Lecture 11: 2/15/12 . 15 Lecture 30: 4/9/12 . 42 Lecture 12: 2/17/16 . 17 Lecture 31: 4/11/12 . 44 Lecture 13: 2/22/12 . 19 Lecture 14: 2/24/12 . 21 Lecture 32: 4/13/12 . 45 Lecture 15: 2/27/12 . 22 Lecture 33: 4/16/12 . 46 Lecture 17: 3/2/12 . 23 Lecture 34: 4/18/12 . 47 Lecture 18: 3/5/12 . 25 Lecture 35: 4/20/12 . 48 Introduction Math 123 is the second in a two-course undergraduate series on abstract algebra offered at Harvard University. This instance of the course dealt with fields and Galois theory, representation theory of finite groups, and rings and modules. These notes were live-TEXed, then edited for correctness and clarity. -
21 Schur Indicator
21 Schur indicator Complex representation theory gives a good description of the homomorphisms from a group to a special unitary group. The Schur indicator describes when these are orthogonal or quaternionic, so it is also easy to describe all homomor- phisms to the compact orthogonal and symplectic groups (and a little bit of fiddling around with central extensions gives the homomorphisms to compact spin groups). So the homomorphisms to any compact classical group are well understood. However there seems to be no easy way to describe the homomor- phisms of a group to the exceptional compact Lie groups. We have several closely related problems: • Which irreducible representations have symmetric or alternating forms? • Classify the homomorphisms to orthogonal and symplectic groups • Classify the real and quaternionic representations given the complex ones. Suppose we have an irreducible representation V of a compact group G. We study the space of invariant bilinear forms on V . Obviously the bilinear forms correspond to maps from V to its dual, so the space of such forms is at most 1-dimensional, and is non-zero if and only if V is isomorphic to its dual, in other words if and only if the character of V is real. We also want to know when the bilinear form is symmetric or alternating. These two possibilities correspond to the symmetric or alternating square containing the 1-dimensional irreducible representation. So (1, χS2V − χΛ2V ) is 1, 0, or −1 depending on whether V has a symmetric bilinear form, no bilinear form, or an alternating bilinear form. By 2 the formulas for the symmetric and alternating squares this is given by G χg , and is called the Schur indicator. -
Definition 1.1. a Group Is a Quadruple (G, E, ⋆, Ι)
GROUPS AND GROUP ACTIONS. 1. GROUPS We begin by giving a definition of a group: Definition 1.1. A group is a quadruple (G, e, ?, ι) consisting of a set G, an element e G, a binary operation ?: G G G and a map ι: G G such that ∈ × → → (1) The operation ? is associative: (g ? h) ? k = g ? (h ? k), (2) e ? g = g ? e = g, for all g G. ∈ (3) For every g G we have g ? ι(g) = ι(g) ? g = e. ∈ It is standard to suppress the operation ? and write gh or at most g.h for g ? h. The element e is known as the identity element. For clarity, we may also write eG instead of e to emphasize which group we are considering, but may also write 1 for e where this is more conventional (for the group such as C∗ for example). Finally, 1 ι(g) is usually written as g− . Remark 1.2. Let us note a couple of things about the above definition. Firstly clo- sure is not an axiom for a group whatever anyone has ever told you1. (The reason people get confused about this is related to the notion of subgroups – see Example 1.8 later in this section.) The axioms used here are not “minimal”: the exercises give a different set of axioms which assume only the existence of a map ι without specifying it. We leave it to those who like that kind of thing to check that associa- tivity of triple multiplications implies that for any k N, bracketing a k-tuple of ∈ group elements (g1, g2, . -
Master Thesis Written at the Max Planck Institute of Quantum Optics
Ludwig-Maximilians-Universitat¨ Munchen¨ and Technische Universitat¨ Munchen¨ A time-dependent Gaussian variational description of Lattice Gauge Theories Master thesis written at the Max Planck Institute of Quantum Optics September 2017 Pablo Sala de Torres-Solanot Ludwig-Maximilians-Universitat¨ Munchen¨ Technische Universitat¨ Munchen¨ Max-Planck-Institut fur¨ Quantenoptik A time-dependent Gaussian variational description of Lattice Gauge Theories First reviewer: Prof. Dr. Ignacio Cirac Second reviewer: Dr. Tao Shi Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science within the programme Theoretical and Mathematical Physics Munich, September 2017 Pablo Sala de Torres-Solanot Abstract In this thesis we pursue the novel idea of using Gaussian states for high energy physics and explore some of the first necessary steps towards such a direction, via a time- dependent variational method. While these states have often been applied to con- densed matter systems via Hartree-Fock, BCS or generalized Hartree-Fock theory, this is not the case in high energy physics. Their advantage is the fact that they are completely characterized, via Wick's theorem, by their two-point correlation func- tions (and one-point averages in bosonic systems as well), e.g., all possible pairings between fermionic operators. These are collected in the so-called covariance matrix which then becomes the most relevant object in their description. In order to reach this goal, we set the theory on a lattice by making use of Lat- tice Gauge Theories (LGT). This method, introduced by Kenneth Wilson, has been extensively used for the study of gauge theories in non-perturbative regimes since it allows new analytical methods as for example strong coupling expansions, as well as the use of Monte Carlo simulations and more recently, Tensor Networks techniques as well. -
20 Representation Theory
20 Representation theory A representation of a group is an action of a group on something, in other words a homomorphism from the group to the group of automorphisms of something. The most obvious representations are permutation representations: actions of a group on a (discrete) set. For example the group of rotations of a cube has order 24, and has several obvious permutation representations: it can act on faces, edges, vertices, diagonals, pairs of opposite faces, and so on. For another example, the group SL2(R) acts on the real projective line, and on the upper half plane. These sets carry topologies, which we should take into account when defining representations. Let us try to classify all permutation representations of some group G . First of all, if the group G acts on some set X, then we can write X as the disjoint union of the orbits of G on X, and G is transitive on each of these orbits. So this reduces the classification of all permutation representations to that of transitive permutation representations. Next, given a transitive permutation representation of G on a set X, fix a point x ∈ X and write Gx for the subgroup fixing x. Then as a permutation representation, X is isomorphic to the action of G on the set of cosets G/Gx. If we choose a different point y, then Gx is conjugate to Gy (using some element of G taking x to y), so we see that transitive permutation representations up to isomorphism correspond to conjugacy classes of subgroups of G. -
GROUP REPRESENTATIONS and CHARACTER THEORY Contents 1
GROUP REPRESENTATIONS AND CHARACTER THEORY DAVID KANG Abstract. In this paper, we provide an introduction to the representation theory of finite groups. We begin by defining representations, G-linear maps, and other essential concepts before moving quickly towards initial results on irreducibility and Schur's Lemma. We then consider characters, class func- tions, and show that the character of a representation uniquely determines it up to isomorphism. Orthogonality relations are introduced shortly afterwards. Finally, we construct the character tables for a few familiar groups. Contents 1. Introduction 1 2. Preliminaries 1 3. Group Representations 2 4. Maschke's Theorem and Complete Reducibility 4 5. Schur's Lemma and Decomposition 5 6. Character Theory 7 7. Character Tables for S4 and Z3 12 Acknowledgments 13 References 14 1. Introduction The primary motivation for the study of group representations is to simplify the study of groups. Representation theory offers a powerful approach to the study of groups because it reduces many group theoretic problems to basic linear algebra calculations. To this end, we assume that the reader is already quite familiar with linear algebra and has had some exposure to group theory. With this said, we begin with a preliminary section on group theory. 2. Preliminaries Definition 2.1. A group is a set G with a binary operation satisfying (1) 8 g; h; i 2 G; (gh)i = g(hi)(associativity) (2) 9 1 2 G such that 1g = g1 = g; 8g 2 G (identity) (3) 8 g 2 G; 9 g−1 such that gg−1 = g−1g = 1 (inverses) Definition 2.2. -
Representation Theory
M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23.