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Conjugacy Classes and Group Representations

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Conjugacy Classes and Group Representations Conjugacy classes and group representations David Vogan Introduction Conjugacy classes and group Groups representations Conj classes Repn theory Symmetric groups David Vogan Groups of matrices Conclusion Department of Mathematics Massachusetts Institute of Technology John U. Free Seminar Eastern Nazarene College February 26, 2016 Conjugacy classes Outline and group representations David Vogan What’s representation theory about? Introduction Groups Abstract symmetry and groups Conj classes Repn theory Symmetric groups Conjugacy classes Groups of matrices Conclusion Representation theory Symmetric groups and partitions Matrices and eigenvalues Conclusion Conjugacy classes The talk in one slide and group representations David Vogan Introduction Three topics. Groups Conj classes 1. GROUPS: abstract way to think about symmetry. Repn theory 2. CONJUGACY CLASSES: organizing group elements. Symmetric groups 3. REPRESENTATIONS: linear algebra and group theory. Groups of matrices crude Representations of G ! conjugacy classes in G. Conclusion Better: relation is like duality for vector spaces. dim(reps), size(conj classes) ! noncommutativity. ?? dim(representation) ! (size)1=2(conjugacy class). Talk is about examples of all these things. Conjugacy classes Two cheers for linear algebra and group representations David Vogan My favorite mathematics is linear algebra. Introduction Complicated enough to describe interesting stuff. Groups Simple enough to calculate with. Conj classes Repn theory Linear map T : V ! V eigenvalues, eigenvectors. Symmetric groups First example: V = fns on R, S = chg vars x 7! −x. Groups of matrices Eigvals: ±1. Eigenspace for +1: even fns (like cos(x)). Conclusion Eigenspace for −1: odd functions (like sin(x)). Linear algebra says: to study sign changes in x, write fns using even and odd fns. d Second example: V = functions on R, T = dx . Eigenvals: λ 2 C. Eigenspace for λ: multiples of eλx . d Linear algebra says: to study dx , write functions using exponentials eλx . Conjugacy classes The third cheer for linear algebra and group representations David Vogan Introduction Best part about linear algebra is noncommutativity... Groups 7! − d Third example: V = fns on R, S = (x x), T = dx . Conj classes S and T don’t commute; can’t diagonalize both. Repn theory Symmetric groups Only common eigenvectors are constant fns. Groups of matrices Representation theory idea: look at smallest subspaces Conclusion preserved by both S and T. λx −λx W±λ = h e ; e i = hcosh(λx); sinh(λx)i. | {z } | {z } | {z } eigenfns of d=dt even odd These two bases of W±λ are good for different things. First for solving diff eqs, second for describing bridge cable. No one basis is good for everything. Conjugacy classes Six symmetries of a triangle and group representations Basic idea in mathematics is symmetry. David Vogan A symmetry of something is a way of rearranging it so that nothing you care about changes. Introduction Groups B C Conj classes J J Repn theory s J s J¨¨ ¨ Symmetric groups J ¨ J J ¨ ¨ J Groups of matrices A C A B Conclusion B C B s s s s J J J s J s J s J J J J J J J A C B A C A A A s s s s s s J J s J HH s J J HH J J HHJ C B B C s s s s Conjugacy classes Composing symmetries and group representations David Vogan What you can do with symmetries is compose them. Introduction Groups If g and h are symmetries, so is Conj classes Repn theory ◦ = g h def first do h, then do g Symmetric groups Groups of matrices 4 example: if r = rotate 240◦, r = rotate 120◦, 240 120 Conclusion ◦ ◦ r240 ◦ r120 = rotate (240 + 120 ) = do nothing = r0: ◦ Harder: if r240 = rotate 240 , sA = reflection fixing A, r240 ◦ sA = exchange B and C, then A ! B ! C ! A = (A ! B; B ! A; C ! C) = sC : Conjugacy classes Composition law for triangle symmetries and group representations David Vogan We saw that the triangle has six symmetries: Introduction Groups r0 r120 r240 rotations Conj classes sA sB sC reflections: Repn theory Symmetric groups Here is how you compose them. Groups of matrices Conclusion ◦ r0 r120 r240 sA sB sC r0 r0 r120 r240 sA sB sC r120 r120 r240 r0 sB sC sA r240 r240 r0 r120 sC sA sB sA sA sC sB r0 r240 r120 sB sB sA sC r120 r0 r240 sC sC sB sA r240 r120 r0 This is the multiplication table for triangle symmetries. Conjugacy classes Abstract groups and group representations David Vogan An abstract group is a multiplication table: a set G with a product ◦ taking g; h 2 G and giving g ◦ h 2 G. Introduction Product ◦ is required to have some properties (that are Groups automatic for composition of symmetries. ) Conj classes 1. ASSOCIATIVITY: g ◦ (h ◦ k)=( g ◦ h) ◦ k (g; h; k 2 G); Repn theory 2. there’s an IDENTITY e 2 G: e ◦ g = g (g 2 G); Symmetric groups 3. each g 2 G has INVERSE g−1 2 G, g−1 ◦ g = e. Groups of matrices Conclusion For symmetries, these properties are always true: 1. first doing( k then h), then doing g, is the same as first doing k, then doing( h then g); 2. doing g then doing nothing is the same as just doing g; 3. undoing a symmetry (putting things back where you found them) is also a symmetry. Here’s an example of a group with ◦ e s just two elements e and s. In fact e e s it’s the only example. s s e Conjugacy classes Approaching symmetry and group representations David Vogan Normal person’s approach to symmetry: Introduction 1. look at something interesting; Groups 2. find the symmetries. Conj classes This approach standard model in physics. Repn theory Explains everything that you can see without LIGO. Symmetric groups Groups of matrices Mathematician’s approach to symmetry: Conclusion 1. find all multiplication tables for abstract groups; 2. pick an interesting abstract group; 3. find something it’s the symmetry group of; 4. decide that something must be interesting. This approach Conway group (which has 8,315,553,613,086,720,000 elements) and Leech lattice (critical for packing 24-dimensional cannonballs). Anyway, I’m a mathematician. Conjugacy classes Which symmetries are really different? and group representations Here are some of the symmetries of a triangle: David Vogan B C B Introduction J J J sJ sJ ¨ sJ Groups J ¨¨J J ¨ Conj classes J ¨ J J A C A B C A Repn theory e = do nothing s = flip (B; C) s = flip (A; C) s s sA s sB s Symmetric groups sA and sB are “same thing” from different points of view. Groups of matrices Conclusion Can accomplish sB in three steps: (A; B; C) 1. flip (A; B) (apply sC ); (B; A; C) 2. flip (B; C) (apply sA ); (C; A; B) −1 3. unflip (A; B) (apply sC ). (C; B; A) −1 Summary: sB = sC sA sC . Defn. g; h conjugate if there’s k 2 G so h = k −1gk. Three conjugacy classes of symmetries of triangle: three reflections sA , sB , sC (exchange two vertices); two rotations r120, r240 (cyclically permute vertices); one trivial symmetry r0 (do nothing). Conjugacy classes Conjugacy classes and group representations David Vogan Introduction G any group; elements g and h in G are conjugate if Groups there’s k in G so h = k −1gk. Conj classes Repn theory Conjugacy class in G is an equivalence class. Symmetric groups G = disjoint union of conjugacy classes Groups of matrices Example: 4 symms = freflsg [ frotnsg [ fidentityg. Conclusion | {z } |{z} | {z } | {z } 6 elts 3 elts 2 elts 1 elt 6 = 3 + 2 + 1 is class eqn for triangle symms. G is abelian if gh = hg (g; h 2 G ). G is abelian ! each conjugacy class is one element. Size of conjugacy classes ! how non-abelian G is. Conjugacy classes Conjugacy classes in Sn and group representations David Vogan Sn =def all( n!) rearrangements of f1; 2;:::; ng. Introduction = Symmetries of (n − 1)-simplex: join n equidistant pts. Groups Conj classes S3 = 6 symms of triangle; S4 = 24 symms of reg tetrahedron. 1 2 3 4 5 Repn theory Typical rearrangement for n = 5: g = 3 5 4 1 2 . Symmetric groups What g does: (1 ! 3 ! 4 ! 1)(2 ! 5 ! 2). Groups of matrices Shorthand: g = (134)(25): cycle (134) and (25) Conclusion 1 2 3 4 5 This g is conjugate to h = (125)(34) = 2 5 4 3 1 . Theorem Any elt of Sn is a product of disjt cycles of P sizes p1 ≥ p2 ≥ · · · ≥ pr , pj = n. Two elts are conjugate ! have same cycle sizes. P Definition Partition of n is p1 ≥ p2 ≥ · · · ≥ pr , pj = n. Corollary Conj classes in Sn ! partitions of n. Conjugacy classes Gelfand program. and group representations David Vogan . for using groups to do other math. Introduction Groups Say G is a group of symmetries of X. Conj classes Repn theory Step 1: LINEARIZE. X V(X) vec space of fns on X. Symmetric groups Now G acts by linear maps. Groups of matrices Step 2: DIAGONALIZE. Decompose V(X) into minimal Conclusion G-invariant subspaces. Step 3: REPRESENTATION THEORY. Understand all ways that G can act by linear maps. Step 4: PRETENDING TO BE SMART. Use understanding of V(X) to answer questions about X. One hard step is3: how can G act by linear maps? Conjugacy classes Definition of representation and group representations David Vogan Introduction G group; representation of G is Groups 1. (complex) vector space V, and Conj classes 2. collection of linear maps fπ(g): V ! V j g 2 Gg Repn theory subject to π(g)π(h) = π(gh); π(e) = identity.
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