A length function for the complex reflection
|
|
- Eileen Atkins
- 5 years ago
- Views:
Transcription
1 A length function for the complex reflection group G(r, r, n) Eli Bagno and Mordechai Novick SLC 78, March 28, 2017
2 General Definitions S n is the symmetric group on {1,..., n}. Z r is the cyclic group of order r. ζ r is the primitive r th root of unity.
3 Complex reflection groups G(r, n) = group of all matrices π = (σ, k), where: σ = a 1 a n S n. k = (k 1,..., k n ) Z n r. (k-vector) π = (σ, k) is the n n monomial matrix with non-zero entries in the (a i, i) positions. ζ k i r Example (n = 3, r = 4) π(312, (1, 3, 3)) = 0 i i i 0 0
4 For p r, G(r, p, n) is the subgroup of G(r, n) consisting of matrices (σ, k) satisfying n (ζ k i r ) r p = 1. i=1 Hence G(r, r, n) is the group of such matrices satisfying: n (ζ k i r ) = 1 i=1
5 One-line notation We denote an element of G(r, p, n) in a more concise manner: (σ, k) = a k 1 1 akn n for σ = a 1 a n and k = (k 1,..., k n ). Example π(312, (1, 3, 3)) =
6 Our goal Various sets of generators have been defined for complex reflection groups but (as far as we know), no length function has been formulated. We provide such a function for the case of G(r, r, n) with a specific choice of generating set proposed by Shi.
7 Shi s Generators for G(r, r, n) For each i {1,..., n 1} let s i = (i, i + 1) be the familiar adjacent transpositions generating S n. Theorem Define t 0 = (1 r 1, n 1 ). The set {t 0, s 1,..., s n 1 } generates G(r, r, n).
8 Example of generators acting from the right Applying s 1 from the right: Applying t 0 from the right: π = π = Remark Places are exchanged, the k vector is not preserved.
9 Example of generators acting from the left Applying s 1 from the left: Applying t 0 from the left: π = π = Remark Numbers are exchanged and the k-vector is preserved.
10 The affine Weyl group S n is defined as follows: S n = {w : Z Z w(i+n) = w(i)+n, i {1,..., n}, n ( ) n + 1 w(i) = }. 2 i=1
11 Each affine permutation can be written in integer window notation in the form: π = (π(1),..., π(n)) = (b 1,..., b n ). By writing b i = n k i + a i, we can use the residue window notation: π = a k 1 1 akn n. where {a 1,..., a n } = {1,..., n}.
12 Generators for the affine group For each i {1,..., n 1} let s i = (i, i + 1) be the known adjacent transpositions generating S n. Define s 0 = (1, n 1 ).
13 Theorem Let π = a k 1 1 akn n S n. Then l(π) = 1 i<j n a i <a j k j k i + 1 i<j n a i >a j k j k i 1 Example If π = then: l(π) = 1 ( 1) ( 1) ( 1) = 5
14 Another presentation of S n Each affine permutation π = a k 1 1 akn n can also be written as a monomial matrix: { 0 i σ(j) M π = (m ij ) = x k i i = σ(j) Example (n = 4) π = = 0 x x 0 x x 1 0
15 Mapping S n to G(r, r, n) Shi defines a homomorphism η : S n G(r, r, n) by substituting a primitive r-th root of unity ζ r in place of x. He tried to adapt his length function for the affine groups to the case of G(r, r, n) but did not obtain a closed formula. Here we provide such a formula.
16 Difficulties in adapting Shi s formula In G(r, r, n) each element does not have a uniquely defined k- vector, as adding a multiple of r to any k i does not change π as an element of G(r, r, n). Example The permutations and represent the same element of G(5, 5, 4).
17 The normal form Definition A permutation (p, k 0 ) G(r, r, n) is said to be in normal form if the following conditions are met: n 1 ki 0 = 0 i=1 2 max(k 0 ) min(k 0 ) r 3 If there exist i < j such that k 0 j k 0 i = r then k0 j k 0 i = r. If (p, k 0 ) is in normal form and is equivalent to (p, k) then we say that (p, k 0 ) is a normal form of (p, k).
18 Example The normal form of G(7, 7, 4) is Theorem For each π G(r, r, n) a normal form exists and is unique. Shi s length function, when applied to all representatives of a permutation in G(r, r, n), attains its minimum on the normal form representative.
19 Decomposition Into Right Cosets of S n Let π = (k, σ) G(r, r, n). As we have seen, for each generator τ of S n, π and τπ have the same k-vector. Hence, it is natural and straightforward to decompose G(r, r, n) into right cosets. Each right coset has a unique representative π = (k, σ) which has minimal length. This leads us to a new length function for G(r, r, n).
20 The length function for G(r, r, n) Let π = a k 1 1 akn n G(r, r, n). Write π = u σ where u S n and σ is the minimal length representative. Then: Theorem l(π) = where 1 i<j n and (as usual) k j k i noninv(k) + inv(u) noninv(k) = #{(i, j) i < j, k(i) < k(j)} inv(u) = #{(i, j) i < j, u(i) > u(j)}.
21 Length Example Let π = G(4, 4, 4). Then σ = , and u = π σ 1 = Hence: 1 i<j n k j k i = ( 2) + 1 ( 2) = 10 And: while noninv(k) = 3 inv(u) = 5 so that l(π) = = 12
22 Finding the minimal-length representative The minimal-length element σ = a k 1 1 akn n G(r, r, n) for the k-vector (k 1,..., k n ) (abbreviated a 1 a 2 a n S n ) is the unique one with the following property: a i < a j iff: Example k(i) > k(j), or k(i) = k(j) and i < j If k = ( 2, 1, 1, 1, 2, 1) then σ =
23 Open question: What is the generating function? Let G r,r,n (q) = q l(π). π G r,r,n From the coset decomposition it is clear that G r,r,n (q) has [n] q! as a factor. Example G 4,4,4 (q) = [4] q!(1+2q 2 +3q 3 +4q 4 +5q 5 +7q 6 +8q 7 +10q 8 +12q 9 +7q 10 +3q 11 ) G 6,6,3 (q) = [3] q!(1+q+2q 2 +2q 3 +3q 4 +3q 5 +4q 6 +4q 7 +5q 8 +5q 9 +6q 10 )
24 A possible direction... There is a bijection between left cosets of S n in the affine group and certain types of partitions (see Bjorner and Brenti (1996) and Eriksson and Eriksson (1998)). In B-B, each partition is the inversion table of the corresponding left coset (i.e., of its ascending minimal-length representative). The bijection in E-E maps each left coset to the conjugate of its inversion table.
25 A possible direction... There is a bijection between left cosets of S n in the affine group and certain types of partitions (see Bjorner and Brenti (1996) and Eriksson and Eriksson (1998)). In B-B, each partition is the inversion table of the corresponding left coset (i.e., of its ascending minimal-length representative). The bijection in E-E maps each left coset to the conjugate of its inversion table.
26 A possible direction... There is a bijection between left cosets of S n in the affine group and certain types of partitions (see Bjorner and Brenti (1996) and Eriksson and Eriksson (1998)). In B-B, each partition is the inversion table of the corresponding left coset (i.e., of its ascending minimal-length representative). The bijection in E-E maps each left coset to the conjugate of its inversion table.
27 This correspondence yields the following generating function for length in the affine group: S n (q) = [n] q! (1 q)(1 q 2 ) (1 q n ) A similar approach may work in our case of right cosets in G(r, r, n).
28 This correspondence yields the following generating function for length in the affine group: S n (q) = [n] q! (1 q)(1 q 2 ) (1 q n ) A similar approach may work in our case of right cosets in G(r, r, n).
29 This correspondence yields the following generating function for length in the affine group: S n (q) = [n] q! (1 q)(1 q 2 ) (1 q n ) A similar approach may work in our case of right cosets in G(r, r, n).
30 This correspondence yields the following generating function for length in the affine group: S n (q) = [n] q! (1 q)(1 q 2 ) (1 q n ) A similar approach may work in our case of right cosets in G(r, r, n).
31 Thank you!!
ALGEBRA 11: Galois theory
Galois extensions Exercise 11.1 (!). Consider a polynomial P (t) K[t] of degree n with coefficients in a field K that has n distinct roots in K. Prove that the ring K[t]/P of residues modulo P is isomorphic
More informationExercises on chapter 1
Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G
More informationInvolutions of the Symmetric Group and Congruence B-Orbits (Extended Abstract)
FPSAC 010, San Francisco, USA DMTCS proc. AN, 010, 353 364 Involutions of the Symmetric Group and Congruence B-Orbits (Extended Abstract) Eli Bagno 1 and Yonah Cherniavsky 1 Jerusalem College of Technology,
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationBASIC GROUP THEORY : G G G,
BASIC GROUP THEORY 18.904 1. Definitions Definition 1.1. A group (G, ) is a set G with a binary operation : G G G, and a unit e G, possessing the following properties. (1) Unital: for g G, we have g e
More informationHigher Bruhat order on Weyl groups of Type B
Mentor: Daniel Thompson Massachusetts Institute of Technology PRIMES-USA, 2013 Outline 1 Purpose 2 Symmetric Group Inversions Bruhat Graph 3 Results/Conjectures Purpose Purpose For Weyl Groups of Type
More informationPermutation groups/1. 1 Automorphism groups, permutation groups, abstract
Permutation groups Whatever you have to do with a structure-endowed entity Σ try to determine its group of automorphisms... You can expect to gain a deep insight into the constitution of Σ in this way.
More informationSupplementary Notes: Simple Groups and Composition Series
18.704 Supplementary Notes: Simple Groups and Composition Series Genevieve Hanlon and Rachel Lee February 23-25, 2005 Simple Groups Definition: A simple group is a group with no proper normal subgroup.
More informationChapter 5 Groups of permutations (bijections) Basic notation and ideas We study the most general type of groups - groups of permutations
Chapter 5 Groups of permutations (bijections) Basic notation and ideas We study the most general type of groups - groups of permutations (bijections). Definition A bijection from a set A to itself is also
More informationLecture 7 Cyclic groups and subgroups
Lecture 7 Cyclic groups and subgroups Review Types of groups we know Numbers: Z, Q, R, C, Q, R, C Matrices: (M n (F ), +), GL n (F ), where F = Q, R, or C. Modular groups: Z/nZ and (Z/nZ) Dihedral groups:
More informationMATH 101A: ALGEBRA I, PART D: GALOIS THEORY 11
MATH 101A: ALGEBRA I, PART D: GALOIS THEORY 11 3. Examples I did some examples and explained the theory at the same time. 3.1. roots of unity. Let L = Q(ζ) where ζ = e 2πi/5 is a primitive 5th root of
More information1 Fields and vector spaces
1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary
More informationPolynomial Properties in Unitriangular Matrices 1
Journal of Algebra 244, 343 351 (2001) doi:10.1006/jabr.2001.8896, available online at http://www.idealibrary.com on Polynomial Properties in Unitriangular Matrices 1 Antonio Vera-López and J. M. Arregi
More informationVersal deformations in generalized flag manifolds
Versal deformations in generalized flag manifolds X. Puerta Departament de Matemàtica Aplicada I Escola Tècnica Superior d Enginyers Industrials de Barcelona, UPC Av. Diagonal, 647 08028 Barcelona, Spain
More informationAn improved tableau criterion for Bruhat order
An improved tableau criterion for Bruhat order Anders Björner and Francesco Brenti Matematiska Institutionen Kungl. Tekniska Högskolan S-100 44 Stockholm, Sweden bjorner@math.kth.se Dipartimento di Matematica
More informationDefinition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson
Definition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson On almost every Friday of the semester, we will have a brief quiz to make sure you have memorized the definitions encountered in our studies.
More informationMathematics for Cryptography
Mathematics for Cryptography Douglas R. Stinson David R. Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, N2L 3G1, Canada March 15, 2016 1 Groups and Modular Arithmetic 1.1
More informationSchool of Mathematics and Statistics. MT5824 Topics in Groups. Problem Sheet I: Revision and Re-Activation
MRQ 2009 School of Mathematics and Statistics MT5824 Topics in Groups Problem Sheet I: Revision and Re-Activation 1. Let H and K be subgroups of a group G. Define HK = {hk h H, k K }. (a) Show that HK
More informationSince G is a compact Lie group, we can apply Schur orthogonality to see that G χ π (g) 2 dg =
Problem 1 Show that if π is an irreducible representation of a compact lie group G then π is also irreducible. Give an example of a G and π such that π = π, and another for which π π. Is this true for
More informationProblem 1. Let I and J be ideals in a ring commutative ring R with 1 R. Recall
I. Take-Home Portion: Math 350 Final Exam Due by 5:00pm on Tues. 5/12/15 No resources/devices other than our class textbook and class notes/handouts may be used. You must work alone. Choose any 5 problems
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2017) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal
More informationGroups Subgroups Normal subgroups Quotient groups Homomorphisms Cyclic groups Permutation groups Cayley s theorem Class equations Sylow theorems
Group Theory Groups Subgroups Normal subgroups Quotient groups Homomorphisms Cyclic groups Permutation groups Cayley s theorem Class equations Sylow theorems Groups Definition : A non-empty set ( G,*)
More informationINTRODUCTION TO THE GROUP THEORY
Lecture Notes on Structure of Algebra INTRODUCTION TO THE GROUP THEORY By : Drs. Antonius Cahya Prihandoko, M.App.Sc e-mail: antoniuscp.fkip@unej.ac.id Mathematics Education Study Program Faculty of Teacher
More informationSolutions for Homework Assignment 5
Solutions for Homework Assignment 5 Page 154, Problem 2. Every element of C can be written uniquely in the form a + bi, where a,b R, not both equal to 0. The fact that a and b are not both 0 is equivalent
More informationAlgebraic Structures Exam File Fall 2013 Exam #1
Algebraic Structures Exam File Fall 2013 Exam #1 1.) Find all four solutions to the equation x 4 + 16 = 0. Give your answers as complex numbers in standard form, a + bi. 2.) Do the following. a.) Write
More informationMATH 433 Applied Algebra Lecture 19: Subgroups (continued). Error-detecting and error-correcting codes.
MATH 433 Applied Algebra Lecture 19: Subgroups (continued). Error-detecting and error-correcting codes. Subgroups Definition. A group H is a called a subgroup of a group G if H is a subset of G and the
More informationCourse 2316 Sample Paper 1
Course 2316 Sample Paper 1 Timothy Murphy April 19, 2015 Attempt 5 questions. All carry the same mark. 1. State and prove the Fundamental Theorem of Arithmetic (for N). Prove that there are an infinity
More informationGalois Theory TCU Graduate Student Seminar George Gilbert October 2015
Galois Theory TCU Graduate Student Seminar George Gilbert October 201 The coefficients of a polynomial are symmetric functions of the roots {α i }: fx) = x n s 1 x n 1 + s 2 x n 2 + + 1) n s n, where s
More information6 Permutations Very little of this section comes from PJE.
6 Permutations Very little of this section comes from PJE Definition A permutation (p147 of a set A is a bijection ρ : A A Notation If A = {a b c } and ρ is a permutation on A we can express the action
More informationq xk y n k. , provided k < n. (This does not hold for k n.) Give a combinatorial proof of this recurrence by means of a bijective transformation.
Math 880 Alternative Challenge Problems Fall 2016 A1. Given, n 1, show that: m1 m 2 m = ( ) n+ 1 2 1, where the sum ranges over all positive integer solutions (m 1,..., m ) of m 1 + + m = n. Give both
More informationCHAPTER 2 -idempotent matrices
CHAPTER 2 -idempotent matrices A -idempotent matrix is defined and some of its basic characterizations are derived (see [33]) in this chapter. It is shown that if is a -idempotent matrix then it is quadripotent
More informationThe Waring rank of the Vandermonde determinant
The Waring rank of the Vandermonde determinant Alexander Woo (U. Idaho) joint work with Zach Teitler(Boise State) SIAM Conference on Applied Algebraic Geometry, August 3, 2014 Waring rank Given a polynomial
More informationϕ : Z F : ϕ(t) = t 1 =
1. Finite Fields The first examples of finite fields are quotient fields of the ring of integers Z: let t > 1 and define Z /t = Z/(tZ) to be the ring of congruence classes of integers modulo t: in practical
More informationStab(t) = {h G h t = t} = {h G h (g s) = g s} = {h G (g 1 hg) s = s} = g{k G k s = s} g 1 = g Stab(s)g 1.
1. Group Theory II In this section we consider groups operating on sets. This is not particularly new. For example, the permutation group S n acts on the subset N n = {1, 2,...,n} of N. Also the group
More informationAn Additive Characterization of Fibers of Characters on F p
An Additive Characterization of Fibers of Characters on F p Chris Monico Texas Tech University Lubbock, TX c.monico@ttu.edu Michele Elia Politecnico di Torino Torino, Italy elia@polito.it January 30, 2009
More informationSF2729 GROUPS AND RINGS LECTURE NOTES
SF2729 GROUPS AND RINGS LECTURE NOTES 2011-03-01 MATS BOIJ 6. THE SIXTH LECTURE - GROUP ACTIONS In the sixth lecture we study what happens when groups acts on sets. 1 Recall that we have already when looking
More informationCSIR - Algebra Problems
CSIR - Algebra Problems N. Annamalai DST - INSPIRE Fellow (SRF) Department of Mathematics Bharathidasan University Tiruchirappalli -620024 E-mail: algebra.annamalai@gmail.com Website: https://annamalaimaths.wordpress.com
More informationGroup Actions Definition. Let G be a group, and let X be a set. A left action of G on X is a function θ : G X X satisfying:
Group Actions 8-26-202 Definition. Let G be a group, and let X be a set. A left action of G on X is a function θ : G X X satisfying: (a) θ(g,θ(g 2,x)) = θ(g g 2,x) for all g,g 2 G and x X. (b) θ(,x) =
More informationHW2 Solutions Problem 1: 2.22 Find the sign and inverse of the permutation shown in the book (and below).
Teddy Einstein Math 430 HW Solutions Problem 1:. Find the sign and inverse of the permutation shown in the book (and below). Proof. Its disjoint cycle decomposition is: (19)(8)(37)(46) which immediately
More informationGroup rings of finite strongly monomial groups: central units and primitive idempotents
Group rings of finite strongly monomial groups: central units and primitive idempotents joint work with E. Jespers, G. Olteanu and Á. del Río Inneke Van Gelder Vrije Universiteit Brussel Recend Trends
More informationSOLVING SOLVABLE QUINTICS. D. S. Dummit
D. S. Dummit Abstract. Let f(x) = x 5 + px 3 + qx + rx + s be an irreducible polynomial of degree 5 with rational coefficients. An explicit resolvent sextic is constructed which has a rational root if
More informationMATH 223A NOTES 2011 LIE ALGEBRAS 35
MATH 3A NOTES 011 LIE ALGEBRAS 35 9. Abstract root systems We now attempt to reconstruct the Lie algebra based only on the information given by the set of roots Φ which is embedded in Euclidean space E.
More informationTC10 / 3. Finite fields S. Xambó
TC10 / 3. Finite fields S. Xambó The ring Construction of finite fields The Frobenius automorphism Splitting field of a polynomial Structure of the multiplicative group of a finite field Structure of the
More informationM3/4/5P12 PROBLEM SHEET 1
M3/4/5P12 PROBLEM SHEET 1 Please send any corrections or queries to jnewton@imperialacuk Exercise 1 (1) Let G C 4 C 2 s, t : s 4 t 2 e, st ts Let V C 2 with the stard basis Consider the linear transformations
More informationPresentation of Normal Bases
Presentation of Normal Bases Mohamadou Sall mohamadou1.sall@ucad.edu.sn University Cheikh Anta Diop, Dakar (Senegal) Pole of Research in Mathematics and their Applications in Information Security (PRMAIS)
More informationMath 3121, A Summary of Sections 0,1,2,4,5,6,7,8,9
Math 3121, A Summary of Sections 0,1,2,4,5,6,7,8,9 Section 0. Sets and Relations Subset of a set, B A, B A (Definition 0.1). Cartesian product of sets A B ( Defintion 0.4). Relation (Defintion 0.7). Function,
More informationThe Outer Automorphism of S 6
Meena Jagadeesan 1 Karthik Karnik 2 Mentor: Akhil Mathew 1 Phillips Exeter Academy 2 Massachusetts Academy of Math and Science PRIMES Conference, May 2016 What is a Group? A group G is a set of elements
More informationResolvability of a cyclic orbit of a subset of Z v and spread decomposition of Singer cycles of projective lines
ACA2015, Augst 26-30, 2015, Michigan Tech. Univ. Resolvability of a cyclic orbit of a subset of Z v and spread decomposition of Singer cycles of projective lines Masakazu Jimbo (Chubu University) jimbo@isc.chubu.ac.jp
More information18.702: Quiz 1 Solutions
MIT MATHEMATICS 18.702: Quiz 1 Solutions February 28 2018 There are four problems on this quiz worth equal value. You may quote without proof any result stated in class or in the assigned reading, unless
More informationSUPPLEMENT ON THE SYMMETRIC GROUP
SUPPLEMENT ON THE SYMMETRIC GROUP RUSS WOODROOFE I presented a couple of aspects of the theory of the symmetric group S n differently than what is in Herstein. These notes will sketch this material. You
More informationMath 430 Exam 2, Fall 2008
Do not distribute. IIT Dept. Applied Mathematics, February 16, 2009 1 PRINT Last name: Signature: First name: Student ID: Math 430 Exam 2, Fall 2008 These theorems may be cited at any time during the test
More informationHyperplane Arrangements & Diagonal Harmonics
Hyperplane Arrangements & Diagonal Harmonics Drew Armstrong arxiv:1005.1949 Coinvariants Coinvariants Theorems (Newton-Chevalley-etc): Let S n act on S = C[x 1,..., x n ] by permuting variables. Coinvariants
More informationNonnegative Matrices I
Nonnegative Matrices I Daisuke Oyama Topics in Economic Theory September 26, 2017 References J. L. Stuart, Digraphs and Matrices, in Handbook of Linear Algebra, Chapter 29, 2006. R. A. Brualdi and H. J.
More informationREU 2007 Discrete Math Lecture 2
REU 2007 Discrete Math Lecture 2 Instructor: László Babai Scribe: Shawn Drenning June 19, 2007. Proofread by instructor. Last updated June 20, 1 a.m. Exercise 2.0.1. Let G be an abelian group and A G be
More informationEnumerative Combinatorics 7: Group actions
Enumerative Combinatorics 7: Group actions Peter J. Cameron Autumn 2013 How many ways can you colour the faces of a cube with three colours? Clearly the answer is 3 6 = 729. But what if we regard two colourings
More informationButson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4
Butson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4 Akihiro Munemasa Tohoku University (joint work with Takuya Ikuta) November 17, 2017 Combinatorics Seminar
More informationDETERMINANTS. , x 2 = a 11b 2 a 21 b 1
DETERMINANTS 1 Solving linear equations The simplest type of equations are linear The equation (1) ax = b is a linear equation, in the sense that the function f(x) = ax is linear 1 and it is equated to
More informationCounting Equivalence Classes for Monomial Rotation Symmetric Boolean Functions with Prime Dimension
Counting Equivalence Classes for Monomial Rotation Symmetric Boolean Functions with Prime Dimension Thomas W. Cusick, Pantelimon Stănică Abstract Recently much progress has been made on the old problem
More informationExercises on chapter 4
Exercises on chapter 4 Always R-algebra means associative, unital R-algebra. (There are other sorts of R-algebra but we won t meet them in this course.) 1. Let A and B be algebras over a field F. (i) Explain
More informationEXERCISES ON THE OUTER AUTOMORPHISMS OF S 6
EXERCISES ON THE OUTER AUTOMORPHISMS OF S 6 AARON LANDESMAN 1. INTRODUCTION In this class, we investigate the outer automorphism of S 6. Let s recall some definitions, so that we can state what an outer
More informationPROBLEMS FROM GROUP THEORY
PROBLEMS FROM GROUP THEORY Page 1 of 12 In the problems below, G, H, K, and N generally denote groups. We use p to stand for a positive prime integer. Aut( G ) denotes the group of automorphisms of G.
More informationAbstract Algebra, HW6 Solutions. Chapter 5
Abstract Algebra, HW6 Solutions Chapter 5 6 We note that lcm(3,5)15 So, we need to come up with two disjoint cycles of lengths 3 and 5 The obvious choices are (13) and (45678) So if we consider the element
More informationSolutions of exercise sheet 6
D-MATH Algebra I HS 14 Prof. Emmanuel Kowalski Solutions of exercise sheet 6 1. (Irreducibility of the cyclotomic polynomial) Let n be a positive integer, and P Z[X] a monic irreducible factor of X n 1
More informationGroup Theory
Group Theory 2014 2015 Solutions to the exam of 4 November 2014 13 November 2014 Question 1 (a) For every number n in the set {1, 2,..., 2013} there is exactly one transposition (n n + 1) in σ, so σ is
More informationENTRY GROUP THEORY. [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld.
ENTRY GROUP THEORY [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld Group theory [Group theory] is studies algebraic objects called groups.
More informationMATH 433 Applied Algebra Lecture 22: Review for Exam 2.
MATH 433 Applied Algebra Lecture 22: Review for Exam 2. Topics for Exam 2 Permutations Cycles, transpositions Cycle decomposition of a permutation Order of a permutation Sign of a permutation Symmetric
More informationMA441: Algebraic Structures I. Lecture 26
MA441: Algebraic Structures I Lecture 26 10 December 2003 1 (page 179) Example 13: A 4 has no subgroup of order 6. BWOC, suppose H < A 4 has order 6. Then H A 4, since it has index 2. Thus A 4 /H has order
More informationA conjugacy criterion for Hall subgroups in finite groups
MSC2010 20D20, 20E34 A conjugacy criterion for Hall subgroups in finite groups E.P. Vdovin, D.O. Revin arxiv:1004.1245v1 [math.gr] 8 Apr 2010 October 31, 2018 Abstract A finite group G is said to satisfy
More information2) e = e G G such that if a G 0 =0 G G such that if a G e a = a e = a. 0 +a = a+0 = a.
Chapter 2 Groups Groups are the central objects of algebra. In later chapters we will define rings and modules and see that they are special cases of groups. Also ring homomorphisms and module homomorphisms
More informationInsolvability of the Quintic (Fraleigh Section 56 Last one in the book!)
Insolvability of the Quintic (Fraleigh Section 56 Last one in the book!) [I m sticking to Freleigh on this one except that this will be a combination of part of the section and part of the exercises because
More informationCombinatorial Method in the Coset Enumeration. of Symmetrically Generated Groups II: Monomial Modular Representations
International Journal of Algebra, Vol. 1, 2007, no. 11, 505-518 Combinatorial Method in the Coset Enumeration of Symmetrically Generated Groups II: Monomial Modular Representations Mohamed Sayed Department
More informationSpectra of Semidirect Products of Cyclic Groups
Spectra of Semidirect Products of Cyclic Groups Nathan Fox 1 University of Minnesota-Twin Cities Abstract The spectrum of a graph is the set of eigenvalues of its adjacency matrix A group, together with
More informationQ N id β. 2. Let I and J be ideals in a commutative ring A. Give a simple description of
Additional Problems 1. Let A be a commutative ring and let 0 M α N β P 0 be a short exact sequence of A-modules. Let Q be an A-module. i) Show that the naturally induced sequence is exact, but that 0 Hom(P,
More informationAbstract Algebra Study Sheet
Abstract Algebra Study Sheet This study sheet should serve as a guide to which sections of Artin will be most relevant to the final exam. When you study, you may find it productive to prioritize the definitions,
More informationTitle: Equidistribution of negative statistics and quotients of Coxeter groups of type B and D
Title: Euidistribution of negative statistics and uotients of Coxeter groups of type B and D Author: Riccardo Biagioli Address: Université de Lyon, Université Lyon 1 Institut Camille Jordan - UMR 5208
More informationQuadrature for the Finite Free Convolution
Spectral Graph Theory Lecture 23 Quadrature for the Finite Free Convolution Daniel A. Spielman November 30, 205 Disclaimer These notes are not necessarily an accurate representation of what happened in
More informationREPRESENTATION THEORY OF THE SYMMETRIC GROUP (FOLLOWING [Ful97])
REPRESENTATION THEORY OF THE SYMMETRIC GROUP (FOLLOWING [Ful97]) MICHAEL WALTER. Diagrams and Tableaux Diagrams and Tableaux. A (Young) diagram λ is a partition of a natural number n 0, which we often
More informationButson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4
Butson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4 Akihiro Munemasa Tohoku University (joint work with Takuya Ikuta) November 17, 2017 Combinatorics Seminar
More informationINTRODUCTION TO GROUP THEORY
INTRODUCTION TO GROUP THEORY LECTURE NOTES BY STEFAN WANER Contents 1. Complex Numbers: A Sketch 2 2. Sets, Equivalence Relations and Functions 5 3. Mathematical Induction and Properties of the Integers
More informationON THE BRUHAT ORDER OF THE SYMMETRIC GROUP AND ITS SHELLABILITY
ON THE BRUHAT ORDER OF THE SYMMETRIC GROUP AND ITS SHELLABILITY YUFEI ZHAO Abstract. In this paper we discuss the Bruhat order of the symmetric group. We give two criteria for comparing elements in this
More informationAlgebra I. Randall R. Holmes Auburn University
Algebra I Randall R. Holmes Auburn University Copyright c 2008 by Randall R. Holmes Last revision: June 8, 2009 Contents 0 Introduction 2 1 Definition of group and examples 4 1.1 Definition.............................
More informationSolutions of exercise sheet 8
D-MATH Algebra I HS 14 Prof. Emmanuel Kowalski Solutions of exercise sheet 8 1. In this exercise, we will give a characterization for solvable groups using commutator subgroups. See last semester s (Algebra
More informationbut no smaller power is equal to one. polynomial is defined to be
13. Radical and Cyclic Extensions The main purpose of this section is to look at the Galois groups of x n a. The first case to consider is a = 1. Definition 13.1. Let K be a field. An element ω K is said
More information5 Group theory. 5.1 Binary operations
5 Group theory This section is an introduction to abstract algebra. This is a very useful and important subject for those of you who will continue to study pure mathematics. 5.1 Binary operations 5.1.1
More informationGroup. Orders. Review. Orders: example. Subgroups. Theorem: Let x be an element of G. The order of x divides the order of G
Victor Adamchik Danny Sleator Great Theoretical Ideas In Computer Science Algebraic Structures: Group Theory II CS 15-251 Spring 2010 Lecture 17 Mar. 17, 2010 Carnegie Mellon University Group A group G
More informationGroups and Symmetries
Groups and Symmetries Definition: Symmetry A symmetry of a shape is a rigid motion that takes vertices to vertices, edges to edges. Note: A rigid motion preserves angles and distances. Definition: Group
More informationΦ + θ = {φ + θ : φ Φ}
74 3.7 Cyclic designs Let Θ Z t. For Φ Θ, a translate of Φ is a set of the form Φ + θ {φ + θ : φ Φ} for some θ in Θ. Of course, Φ is a translate of itself. It is possible to have Φ + θ 1 Φ + θ 2 even when
More informationD-MATH Algebra I HS 2013 Prof. Brent Doran. Solution 3. Modular arithmetic, quotients, product groups
D-MATH Algebra I HS 2013 Prof. Brent Doran Solution 3 Modular arithmetic, quotients, product groups 1. Show that the functions f = 1/x, g = (x 1)/x generate a group of functions, the law of composition
More informationCitation Osaka Journal of Mathematics. 43(2)
TitleIrreducible representations of the Author(s) Kosuda, Masashi Citation Osaka Journal of Mathematics. 43(2) Issue 2006-06 Date Text Version publisher URL http://hdl.handle.net/094/0396 DOI Rights Osaka
More information1 Finite abelian groups
Last revised: May 16, 2014 A.Miller M542 www.math.wisc.edu/ miller/ Each Problem is due one week from the date it is assigned. Do not hand them in early. Please put them on the desk in front of the room
More informationIntroduction to Groups
Introduction to Groups Hong-Jian Lai August 2000 1. Basic Concepts and Facts (1.1) A semigroup is an ordered pair (G, ) where G is a nonempty set and is a binary operation on G satisfying: (G1) a (b c)
More informationTeddy Einstein Math 4320
Teddy Einstein Math 4320 HW4 Solutions Problem 1: 2.92 An automorphism of a group G is an isomorphism G G. i. Prove that Aut G is a group under composition. Proof. Let f, g Aut G. Then f g is a bijective
More informationPermutation-like matrix groups
Linear Algebra and its Applications 422 (2007) 486 505 www.elsevier.com/locate/laa Permutation-like matrix groups Grega Cigler University of Ljubljana, Faculty for Mathematics and Physics, Jadranska 9,
More informationChapter 16 MSM2P2 Symmetry And Groups
Chapter 16 MSM2P2 Symmetry And Groups 16.1 Symmetry 16.1.1 Symmetries Of The Square Definition 1 A symmetry is a function from an object to itself such that for any two points a and b in the object, the
More informationDISCRETE MATH (A LITTLE) & BASIC GROUP THEORY - PART 3/3. Contents
DISCRETE MATH (A LITTLE) & BASIC GROUP THEORY - PART 3/3 T.K.SUBRAHMONIAN MOOTHATHU Contents 1. Cayley s Theorem 1 2. The permutation group S n 2 3. Center of a group, and centralizers 4 4. Group actions
More informationDirectional Field. Xiao-Ming Fu
Directional Field Xiao-Ming Fu Outlines Introduction Discretization Representation Objectives and Constraints Outlines Introduction Discretization Representation Objectives and Constraints Definition Spatially-varying
More informationSupplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.
Glossary 1 Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.23 Abelian Group. A group G, (or just G for short) is
More informationOrbitopes. Marc Pfetsch. joint work with Volker Kaibel. Zuse Institute Berlin
Orbitopes Marc Pfetsch joint work with Volker Kaibel Zuse Institute Berlin What this talk is about We introduce orbitopes. A polyhedral way to break symmetries in integer programs. Introduction 2 Orbitopes
More informationProof of a Conjecture on Monomial Graphs
Proof of a Conjecture on Monomial Graphs Xiang-dong Hou Department of Mathematics and Statistics University of South Florida Joint work with Stephen D. Lappano and Felix Lazebnik New Directions in Combinatorics
More informationA study of permutation groups and coherent configurations. John Herbert Batchelor. A Creative Component submitted to the graduate faculty
A study of permutation groups and coherent configurations by John Herbert Batchelor A Creative Component submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER
More information