Group Theoretical Methods in Machine Learning Risi Kondor COLUMBIA
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Finite Generation of Symmetric Ideals
FINITE GENERATION OF SYMMETRIC IDEALS MATTHIAS ASCHENBRENNER AND CHRISTOPHER J. HILLAR In memoriam Karin Gatermann (1965–2005 ). Abstract. Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let SX be the group of permutations of X. The group SX acts on R in a natural way, and this in turn gives R the structure of a left module over the left group ring R[SX ]. We prove that all ideals of R invariant under the action of SX are finitely generated as R[SX ]-modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Gr¨obner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it. 1. Introduction A pervasive theme in invariant theory is that of finite generation. A fundamen- tal example is a theorem of Hilbert stating that the invariant subrings of finite- dimensional polynomial algebras over finite groups are finitely generated [6, Corol- lary 1.5]. In this article, we study invariant ideals of infinite-dimensional polynomial rings. Of course, when the number of indeterminates is finite, Hilbert’s basis theo- rem tells us that any ideal (invariant or not) is finitely generated. Our setup is as follows. Let X be an infinite collection of indeterminates, and let SX be the group of permutations of X. -
Representations of the Symmetric Group and Its Hecke Algebra Nicolas Jacon
Representations of the symmetric group and its Hecke algebra Nicolas Jacon To cite this version: Nicolas Jacon. Representations of the symmetric group and its Hecke algebra. 2012. hal-00731102v1 HAL Id: hal-00731102 https://hal.archives-ouvertes.fr/hal-00731102v1 Preprint submitted on 12 Sep 2012 (v1), last revised 10 Nov 2012 (v2) 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. Representations of the symmetric group and its Hecke algebra N. Jacon Abstract This paper is an expository paper on the representation theory of the symmetric group and its Hecke algebra in arbitrary characteristic. We study both the semisimple and the non semisimple case and give an introduction to some recent results on this theory (AMS Class.: 20C08, 20C20, 05E15) . 1 Introduction Let n N and let S be the symmetric group acting on the left on the set 1, 2, . , n . Let k be a field ∈ n { } and consider the group algebra kSn. This is the k-algebra with: k-basis: t σ S , • { σ | ∈ n} 2 multiplication determined by the following rule: for all (σ, σ′) S , we have t .t ′ = t ′ . • ∈ n σ σ σσ The aim of this survey is to study the Representation Theory of Sn over k. -
Math 263A Notes: Algebraic Combinatorics and Symmetric Functions
MATH 263A NOTES: ALGEBRAIC COMBINATORICS AND SYMMETRIC FUNCTIONS AARON LANDESMAN CONTENTS 1. Introduction 4 2. 10/26/16 5 2.1. Logistics 5 2.2. Overview 5 2.3. Down to Math 5 2.4. Partitions 6 2.5. Partial Orders 7 2.6. Monomial Symmetric Functions 7 2.7. Elementary symmetric functions 8 2.8. Course Outline 8 3. 9/28/16 9 3.1. Elementary symmetric functions eλ 9 3.2. Homogeneous symmetric functions, hλ 10 3.3. Power sums pλ 12 4. 9/30/16 14 5. 10/3/16 20 5.1. Expected Number of Fixed Points 20 5.2. Random Matrix Groups 22 5.3. Schur Functions 23 6. 10/5/16 24 6.1. Review 24 6.2. Schur Basis 24 6.3. Hall Inner product 27 7. 10/7/16 29 7.1. Basic properties of the Cauchy product 29 7.2. Discussion of the Cauchy product and related formulas 30 8. 10/10/16 32 8.1. Finishing up last class 32 8.2. Skew-Schur Functions 33 8.3. Jacobi-Trudi 36 9. 10/12/16 37 1 2 AARON LANDESMAN 9.1. Eigenvalues of unitary matrices 37 9.2. Application 39 9.3. Strong Szego limit theorem 40 10. 10/14/16 41 10.1. Background on Tableau 43 10.2. KOSKA Numbers 44 11. 10/17/16 45 11.1. Relations of skew-Schur functions to other fields 45 11.2. Characters of the symmetric group 46 12. 10/19/16 49 13. 10/21/16 55 13.1. -
Categories with Negation 3
CATEGORIES WITH NEGATION JAIUNG JUN 1 AND LOUIS ROWEN 2 In honor of our friend and colleague, S.K. Jain. Abstract. We continue the theory of T -systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). Prime ground systems are introduced as a way of developing geometry. One basic result is that the polynomial system over a prime system is prime. For module systems, special attention also is paid to tensor products and Hom. Abelian categories are replaced by “semi-abelian” categories (where Hom(A, B) is not a group) with a negation morphism. The theory, summarized categorically at the end, encapsulates general algebraic structures lacking negation but possessing a map resembling negation, such as tropical algebras, hyperfields and fuzzy rings. We see explicitly how it encompasses tropical algebraic theory and hyperfields. Contents 1. Introduction 2 1.1. Objectives 3 1.2. Main results 3 2. Background 4 2.1. Basic structures 4 2.2. Symmetrization and the twist action 7 2.3. Ground systems and module systems 8 2.4. Morphisms of systems 10 2.5. Function triples 11 2.6. The role of universal algebra 13 3. The structure theory of ground triples via congruences 14 3.1. The role of congruences 14 3.2. Prime T -systems and prime T -congruences 15 3.3. Annihilators, maximal T -congruences, and simple T -systems 17 4. The geometry of prime systems 17 4.1. Primeness of the polynomial system A[λ] 17 4.2. -
GRADED LEVEL ZERO INTEGRABLE REPRESENTATIONS of AFFINE LIE ALGEBRAS 1. Introduction in This Paper, We Continue the Study ([2, 3
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 6, June 2008, Pages 2923–2940 S 0002-9947(07)04394-2 Article electronically published on December 11, 2007 GRADED LEVEL ZERO INTEGRABLE REPRESENTATIONS OF AFFINE LIE ALGEBRAS VYJAYANTHI CHARI AND JACOB GREENSTEIN Abstract. We study the structure of the category of integrable level zero representations with finite dimensional weight spaces of affine Lie algebras. We show that this category possesses a weaker version of the finite length property, namely that an indecomposable object has finitely many simple con- stituents which are non-trivial as modules over the corresponding loop algebra. Moreover, any object in this category is a direct sum of indecomposables only finitely many of which are non-trivial. We obtain a parametrization of blocks in this category. 1. Introduction In this paper, we continue the study ([2, 3, 7]) of the category Ifin of inte- grable level zero representations with finite-dimensional weight spaces of affine Lie algebras. The category of integrable representations with finite-dimensional weight spaces and of non-zero level is semi-simple ([16]), and the simple objects are the highest weight modules. In contrast, it is easy to see that the category Ifin is not semi-simple. For instance, the derived Lie algebra of the affine algebra itself provides an example of a non-simple indecomposable object in that category. The simple objects in Ifin were classified in [2, 7] but not much else is known about the structure of the category. The category Ifin can be regarded as the graded version of the category F of finite-dimensional representations of the corresponding loop algebra. -
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. -
BRST Inner Product Spaces and the Gribov Obstruction
PITHA – 98/27 BRST Inner Product Spaces and the Gribov Obstruction Norbert DUCHTING¨ 1∗, Sergei V. SHABANOV2†‡, Thomas STROBL1§ 1 Institut f¨ur Theoretische Physik, RWTH Aachen D-52056 Aachen, Germany 2 Department of Mathematics, University of Florida, Gainesville, FL 32611–8105, USA Abstract A global extension of the Batalin–Marnelius proposal for a BRST inner product to gauge theories with topologically nontrivial gauge orbits is discussed. It is shown that their (appropriately adapted) method is applicable to a large class of mechanical models with a semisimple gauge group in the adjoint and fundamental representa- tion. This includes cases where the Faddeev–Popov method fails. arXiv:hep-th/9808151v1 25 Aug 1998 Simple models are found also, however, which do not allow for a well– defined global extension of the Batalin–Marnelius inner product due to a Gribov obstruction. Reasons for the partial success and failure are worked out and possible ways to circumvent the problem are briefly discussed. ∗email: [email protected] †email: [email protected]fl.edu ‡On leave from the Laboratory of Theoretical Physics, JINR, Dubna, Russia §email: [email protected] 1 Introduction and Overview Canonical quantization of gauge theories leads, in general, to an ill–defined scalar product for physical states. In the Dirac approach [1] physical states are selected by the quantum constraints. Assuming that the theory under consideration is of the Yang–Mills type with its gauge group acting on some configuration space, the total Hilbert space may be realized by square in- tegrable functions on the configuration space and the quantum constraints imply gauge invariance of these wave functions. -
Extending Robustness & Randomization from Consensus To
Extending robustness & randomization from Consensus to Symmetrization Algorithms Luca Mazzarella∗ Francesco Ticozzi† Alain Sarlette‡ June 16, 2015 Abstract This work interprets and generalizes consensus-type algorithms as switching dynamics lead- ing to symmetrization of some vector variables with respect to the actions of a finite group. We show how the symmetrization framework we develop covers applications as diverse as consensus on probability distributions (either classical or quantum), uniform random state generation, and open-loop disturbance rejection by quantum dynamical decoupling. Robust convergence results are explicitly provided in a group-theoretic formulation, both for deterministic and for randomized dynamics. This indicates a way to directly extend the robustness and randomization properties of consensus-type algorithms to more fields of application. 1 Introduction The investigation of randomized and robust algorithmic procedures has been a prominent development of applied mathematics and dynamical systems theory in the last decades [?, ?]. Among these, one of the most studied class of algorithms in the automatic control literature are those designed to reach average consensus [?, ?, ?, ?]: the most basic form entails a linear algorithm based on local iterations that reach agreement on a mean value among network nodes. Linear consensus, despite its simplicity, is used as a subtask for several distributed tasks like distributed estimation [?, ?], motion coordination [?], clock synchronization [?], optimization [?], and of course load balancing; an example is presented later in the paper, while more applications are covered in [?]. A universally appreciated feature of linear consensus is its robustness to parameter values and perfect behavior under time-varying network structure. In the present paper, inspired by linear consensus, we present an abstract framework that pro- duces linear procedures to solve a variety of (a priori apparently unrelated) symmetrization problems with respect to the action of finite groups. -
Notes on Representations of Finite Groups
NOTES ON REPRESENTATIONS OF FINITE GROUPS AARON LANDESMAN CONTENTS 1. Introduction 3 1.1. Acknowledgements 3 1.2. A first definition 3 1.3. Examples 4 1.4. Characters 7 1.5. Character Tables and strange coincidences 8 2. Basic Properties of Representations 11 2.1. Irreducible representations 12 2.2. Direct sums 14 3. Desiderata and problems 16 3.1. Desiderata 16 3.2. Applications 17 3.3. Dihedral Groups 17 3.4. The Quaternion group 18 3.5. Representations of A4 18 3.6. Representations of S4 19 3.7. Representations of A5 19 3.8. Groups of order p3 20 3.9. Further Challenge exercises 22 4. Complete Reducibility of Complex Representations 24 5. Schur’s Lemma 30 6. Isotypic Decomposition 32 6.1. Proving uniqueness of isotypic decomposition 32 7. Homs and duals and tensors, Oh My! 35 7.1. Homs of representations 35 7.2. Duals of representations 35 7.3. Tensors of representations 36 7.4. Relations among dual, tensor, and hom 38 8. Orthogonality of Characters 41 8.1. Reducing Theorem 8.1 to Proposition 8.6 41 8.2. Projection operators 43 1 2 AARON LANDESMAN 8.3. Proving Proposition 8.6 44 9. Orthogonality of character tables 46 10. The Sum of Squares Formula 48 10.1. The inner product on characters 48 10.2. The Regular Representation 50 11. The number of irreducible representations 52 11.1. Proving characters are independent 53 11.2. Proving characters form a basis for class functions 54 12. Dimensions of Irreps divide the order of the Group 57 Appendix A. -
Introduction to the Representation Theory of Finite Groups
Basic Theory of Finite Group Representations over C Adam B Block 1 Introduction We introduce the basics of the representation theory of finite groups in characteristic zero. In the sequel, all groups G will be finite and all vector spaces V will be finite dimensional over C. We first define representa- tions and give some basic examples. Then we discuss morphisms and subrepresentations, followed by basic operations on representations and character theory. We conclude with induced and restricted representa- tions and mention Frobenius Reciprocity. These notes are intended as the background to the UMS summer seminar on the representation theory of symmetric groups from [AMV04]. The author learned this material originally from [FH] and recommends this source for more detail in the following; many of the proofs in the sequel likely come from this source. 2 Definitions and Examples Representation theory is the study of groups acting on vector spaces. As such, we have the following definition: Definition 1. A representation of a group G is a pair (V; ρ) where V is a vector space over C and ρ is a homomorphism ρ : G ! GL(V ). We will often refer to representations by their vector space and assume that the morphism ρ is clear from context. Every representation defines a unique C[G]-module, with the action of G on V being g · v = ρ(g)(v) and vice versa. We refer to the dimension of the representation, defined to be dim V . With the definition in mind, we begin with a few examples. Example 2. Let G be any group and V any vector space. -
Class Numbers of Orders in Quartic Fields
Class Numbers of Orders in Quartic Fields Dissertation der FakultÄat furÄ Mathematik und Physik der Eberhard-Karls-UniversitÄat TubingenÄ zur Erlangung des Grades eines Doktors der Naturwissenschaften vorgelegt von Mark Pavey aus Cheltenham Spa, UK 2006 Mundliche Prufung:Ä 11.05.2006 Dekan: Prof.Dr.P.Schmid 1.Berichterstatter: Prof.Dr.A.Deitmar 2.Berichterstatter: Prof.Dr.J.Hausen Dedicated to my parents: Deryk and Frances Pavey, who have always supported me. Acknowledgments First of all thanks must go to my doctoral supervisor Professor Anton Deit- mar, without whose enthusiasm and unfailing generosity with his time and expertise this project could not have been completed. The ¯rst two years of research for this project were undertaken at Exeter University, UK, with funding from the Engineering and Physical Sciences Re- search Council of Great Britain (EPSRC). I would like to thank the members of the Exeter Maths Department and EPSRC for their support. Likewise I thank the members of the Mathematisches Institut der UniversitÄat Tubingen,Ä where the work was completed. Finally, I wish to thank all the family and friends whose friendship and encouragement have been invaluable to me over the last four years. I have to mention particularly the Palmers, the Haywards and my brother Phill in Seaton, and those who studied and drank alongside me: Jon, Pete, Dave, Ralph, Thomas and the rest. Special thanks to Paul Smith for the Online Chess Club, without which the whole thing might have got done more quickly, but it wouldn't have been as much fun. 5 Contents Introduction iii 1 Euler Characteristics and In¯nitesimal Characters 1 1.1 The unitary duals of SL2(R) and SL4(R) . -
REALIZABILITY of ALGEBRAIC GALOIS EXTENSIONS by STRICTLY COMMUTATIVE RING SPECTRA Introduction We Discuss Some Ideas on Galois T
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 2, February 2007, Pages 827–857 S 0002-9947(06)04201-2 Article electronically published on September 12, 2006 REALIZABILITY OF ALGEBRAIC GALOIS EXTENSIONS BY STRICTLY COMMUTATIVE RING SPECTRA ANDREW BAKER AND BIRGIT RICHTER Abstract. We discuss some of the basic ideas of Galois theory for commuta- tive S-algebras originally formulated by John Rognes. We restrict our attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central rˆoles are played by the notion of strong duality and a trace mapping con- structed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Ga- lois extensions to topological ones by applying obstruction theories of Robinson and Goerss-Hopkins to produce topological models for algebraic Galois exten- sions and the necessary morphisms of commutative S-algebras. Examples such as the complex K-theory spectrum as a KO-algebra indicate that more ex- otic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones, and this leads to computable Harrison groups for such spectra. We end by proving an analogue of Hilbert’s theorem 90 for the units associated with a Galois extension. Introduction We discuss some ideas on Galois theory for commutative S-algebras, also known as brave new (commutative) rings, originally formulated by John Rognes.