A Brief Introduction to Characters and Representation Theory

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Linear Representations of Finite Groups William Hargis A Brief Introduction to Characters and Representations Theory Structures Studied Representation Theory Linear Representations Character Theory Characters Orthogonality of William Hargis Characters Character Properties Examples of Mathematics DRP Fall 2016 Characters Cyclic Groups Mentor: Xuqiang (Russell) Qin

Dec. 8, 2016 Overview

Linear Representations of Finite Groups

William Hargis 1 Representations Theory Structures Studied Representations Theory Linear Representations Structures Studied Linear Representations Character Theory 2 Character Theory Characters Characters Orthogonality of Characters Character Properties Orthogonality of Characters Examples of Character Properties Characters Cyclic Groups 3 Examples of Characters Cyclic Groups Linear Representations of Material studied: Linear Representations of Finite Groups Finite Groups by Jean-Pierre Serre William Hargis

Linear Representations of Finite Groups

William Hargis

Representations Representation theory is the study of algebraic structures Theory Structures Studied by representing the structure’s elements as linear Linear Representations transformations of vector spaces. Character Theory Characters Orthogonality of Characters Character Properties This makes abstract structures more concrete by Examples of describing the structure in terms of matrices and their Characters Cyclic Groups algebraic operations as matrix operations. Structures Studied

Linear Representations of Finite Groups Structures studied this way include: William Hargis Groups Representations Associative Algebras Theory Structures Studied Linear Lie Algebras Representations Character Theory Characters Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups Finite Groups

Linear Representations of Finite Groups

William Hargis

Representations Finite Groups Theory Structures Studied Study group actions on structures. Linear Representations especially operations of groups on vector spaces; Character Theory other actions are group action on other groups or Characters Orthogonality of Characters sets. Character Properties Examples of Characters Group elements are represented by invertible matrices Cyclic Groups such that the group operation is matrix multiplication. Linear Representations

Linear Representations of Finite Groups

William Hargis

Representations Theory Let V be a K-vector space and G a ﬁnite group. Structures Studied Linear Representations The linear representation of G is a group homomorphism Character Theory ρ : G → GL(V ). Characters Orthogonality of Characters So, a linear representation is a map Character Properties Examples of ρ : G → GL(V ) s.t. ρ(st) = ρ(s)ρ(t) ∀s, t ∈ G. Characters Cyclic Groups Importance of Linear Representations

Linear Representations of Finite Groups

William Hargis

Representations Theory Structures Studied Linear representations allow us to state group theoretic Linear Representations problems in terms of linear algebra. Character Theory Characters Orthogonality of Linear algebra is well understood; reduces complexity of Characters Character Properties problems. Examples of Characters Cyclic Groups Applications of Linear Representations

Linear Representations of Finite Groups Applications in the study of geometric structures and in

William Hargis the physical sciences.

Representations Space Groups Theory Structures Studied Symmetry groups of the configuration of space. Linear Representations Character Theory Characters Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups Lattice Point Groups Lattice groups define the geometric configuration of crystal structures in materials science and crystallography. Characters

Linear Representations of Finite Groups

William Hargis

Representations Theory The character of a group representation is a function on Structures Studied Linear Representations the group that associates the trace of each group Character Theory element’s matrix to the corresponding group element. Characters Orthogonality of Characters Characters contain all of the essential information of the Character Properties Examples of representation in a more condensed form. Characters Cyclic Groups Trace Review

Linear Representations of Finite Groups William Hargis Note: Trace is the sum of the diagonal entries of the Representations matrix. Theory Structures Studied Linear   Representations x11 x12 ... x1n Character Theory x21 x22 ... x2n  Characters Tr(X ) = Tr( ) Orthogonality of   Characters ......  Character Properties xm1 xm2 ... xmn Examples of Characters Cyclic Groups Pn Tr(X ) = x11 + x22 + ... + xmn = i=1 xii Characters

Linear Representations of Finite Groups

William Hargis

Representations Characters: Theory Structures Studied Linear For a representation ρ : G → GL(V ) of a group G on V . Representations Character Theory the character of ρ is the function χρ : G → F given by Characters Orthogonality of χρ(g) = Tr(ρ(g)). Characters Character Properties Examples of Where F is a ﬁeld that the ﬁnite-dimensional vector Characters space V is over. Cyclic Groups Orthogonality of Characters

Linear Representations of Finite Groups William Hargis The space of complex-valued class functions of a ﬁnite

Representations group G has an inner product given by: Theory Structures Studied 1 Linear X ¯ Representations {α, β} := α(g)β(g) (1) G Character Theory g∈G Characters Orthogonality of Characters From this inner product, the irreducible characters form Character Properties Examples of an orthonormal basis for the space of class functions and Characters Cyclic Groups an orthogonality relation for the rows of the character table. Similarly an orthogonality relation is established for the columns of the character table. Consequences of Orthogonality:

Linear Representations of Finite Groups

William Hargis

Representations Consequences of Orthogonality: Theory Structures Studied Linear An unknown character can be decomposed as a Representations Character Theory linear combination of irreducible characters. Characters Orthogonality of The complete character table can be constructed Characters Character Properties when only a few irreducible characters are known. Examples of Characters The order of the group can be found. Cyclic Groups Character Properties

Linear Representations of Finite Groups William Hargis A character χρ is irreducible if ρ is irreducible.

Representations One can read the dimension of the vector space Theory Structures Studied directly from the character. Linear Representations Characters are class functions; take a constant value Character Theory Characters on a given conjugacy. Orthogonality of Characters Character Properties The number of irreducible characters of G is equal Examples of to the number of conjugacy classes of G. Characters Cyclic Groups The set of irreducible characters of a given group G into a ﬁeld K form a basis of the K-vector space of all class functions G → K. Examples of Characters

Linear Representations of Finite Groups

William Hargis

Representations Note: the elements of any group can be partitioned in to Theory Structures Studied conjugacy classes; classes corresponding to the same Linear Representations conjugate element. Character Theory Characters Orthogonality of −1 Characters Cl(a) = b ∈ G|∃g ∈ G with b = gag (2) Character Properties Examples of From the deﬁnition it follows that Abelian groups have a Characters Cyclic Groups conjugacy class corresponding to each element. with conjugacy classes: {0¯},{1¯},{2¯},...,{n −¯ 1}.

2πi Let ωn=e n be a primitive n root of unity. (any complex number that gives 1 when raised to a positive integer power)

Examples: Generalized Cyclic Group Zn

Linear Representations of Note: All of Zn’s irreducible characters are linear. Finite Groups ¯ ¯ ¯ ¯ William Hargis Zn is an additive group where Zn = 0, 1, 2, ..., n − 1

Representations Theory Structures Studied Linear Representations Character Theory Characters Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups 2πi Let ωn=e n be a primitive n root of unity. (any complex number that gives 1 when raised to a positive integer power)

Examples: Generalized Cyclic Group Zn

Linear Representations of Note: All of Zn’s irreducible characters are linear. Finite Groups ¯ ¯ ¯ ¯ William Hargis Zn is an additive group where Zn = 0, 1, 2, ..., n − 1 with conjugacy classes: {0¯},{1¯},{2¯},...,{n −¯ 1}. Representations Theory 2πi n Structures Studied Let ωn=e be a primitive n root of unity. Linear Representations (any complex number that gives 1 when raised to a Character Theory Characters positive integer power) Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups Let χ0, χ1, χ2, ..., χn−1 be the n irreducible characters ¯ jm of Zn then χm(j) = ωn where 0 ≤ j ≤ n − 1 and 0 ≤ m ≤ n − 1.

Examples: Generalized Cyclic Group Zn

Linear Representations of Finite Groups

William Hargis

Representations As the number of irreducible characters is equal to the Theory Structures Studied number of conjugacy classes, then the number of Linear Representations irreducible characters of Zn is n. Character Theory |Irr( )| = n. Characters Zn Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups Examples: Generalized Cyclic Group Zn

Linear Representations of Finite Groups

William Hargis

Representations As the number of irreducible characters is equal to the Theory Structures Studied number of conjugacy classes, then the number of Linear Representations irreducible characters of Zn is n. Character Theory |Irr( )| = n. Characters Zn Orthogonality of Characters Character Properties Let χ0, χ1, χ2, ..., χn−1 be the n irreducible characters jm Examples of of n then χm(j¯) = ω Characters Z n Cyclic Groups where 0 ≤ j ≤ n − 1 and 0 ≤ m ≤ n − 1. Character Table for Zn

Linear Representations of Finite Groups William Hargis Character Table for Zn

Representations Theory gi 0¯ 1¯ 2¯ ... n −¯ 1 Structures Studied Linear |Cl| 1 1 1 ... 1 Representations Character Theory χ0 1 1 1 ... 1 Characters 2 n−1 Orthogonality of χ1 1 ωn ω ... ω Characters n n Character Properties 2 4 2(n−1) χ2 1 ωn ωn ... ωn Examples of Characters . . . . . Cyclic Groups . . . . . n−1 2(n−1) (n−1)(n−1) χn−1 1 ωn ωn ... ωn There are 6 conjugacy classes: {0¯}, {1¯}, {2¯}, {3¯}, {4¯}, {5¯}}.

2πi Let ω6=e 6 be a primitive 6 root of unity; then ¯ jm χm(j) = ω6 . Where 0 ≤ j ≤ 5 and 0 ≤ m ≤ 5.

Cyclic Group Z6

Linear Representations of Finite Groups

William Hargis We can ﬁnd the character table for Z6 fairly easily. Representations ¯ ¯ ¯ ¯ ¯ ¯ Theory Z6 = {0, 1, 2, 3, 4, 5}. Structures Studied Linear Representations Character Theory Characters Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups 2πi Let ω6=e 6 be a primitive 6 root of unity; then ¯ jm χm(j) = ω6 . Where 0 ≤ j ≤ 5 and 0 ≤ m ≤ 5.

Cyclic Group Z6

Linear Representations of Finite Groups

William Hargis We can ﬁnd the character table for Z6 fairly easily. Representations ¯ ¯ ¯ ¯ ¯ ¯ Theory Z6 = {0, 1, 2, 3, 4, 5}. Structures Studied Linear Representations There are 6 conjugacy classes: Character Theory ¯ ¯ ¯ ¯ ¯ ¯ Characters {0}, {1}, {2}, {3}, {4}, {5}}. Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups Cyclic Group Z6

Linear Representations of Finite Groups

William Hargis We can ﬁnd the character table for Z6 fairly easily. Representations ¯ ¯ ¯ ¯ ¯ ¯ Theory Z6 = {0, 1, 2, 3, 4, 5}. Structures Studied Linear Representations There are 6 conjugacy classes: Character Theory ¯ ¯ ¯ ¯ ¯ ¯ Characters {0}, {1}, {2}, {3}, {4}, {5}}. Orthogonality of Characters 2πi Character Properties Let ω6=e 6 be a primitive 6 root of unity; then Examples of ¯ jm Characters χm(j) = ω6 . Cyclic Groups Where 0 ≤ j ≤ 5 and 0 ≤ m ≤ 5. Character Table: Cyclic Group Z6

Linear Representations of Finite Groups 6 William Hargis ω6 = 1 due to cycle.

Representations ¯ ¯ ¯ ¯ ¯ ¯ Theory gi 0 1 2 3 4 5 Structures Studied Linear |Cl| 1 1 1 1 1 1 Representations Character Theory χ0 1 1 1 1 1 1 Characters 2 3 4 5 Orthogonality of χ1 1 ω6 ω6 ω6 ω6 ω6 Characters 2 4 2 4 Character Properties χ2 1 ω6 ω6 1 ω6 ω6 Examples of χ 1 ω3 1 ω3 1 ω3 Characters 3 6 6 6 4 2 4 2 Cyclic Groups χ4 1 ω6 ω6 1 ω6 ω6 5 4 3 2 1 χ5 1 ω6 ω6 ω6 ω6 ω6 Computing the Roots of Unity

Linear Representations of Finite Groups

William Hargis 6 We know that ω6 = 1. 2πi Representations Calculating the rest from ω6 = e 6 : Theory √ Structures Studied 1 1 3 Linear ω = + i, Representations 6 2 2 √ Character Theory 2 1 3 Characters ω6 = − 2 + 2 i, Orthogonality of Characters 3 Character Properties ω6 = −1, Examples of √ Characters ω4 = − 1 − 3 i, Cyclic Groups 6 2 2 √ 5 1 3 ω6 = 2 − 2 i. Image Rights

Linear Representations of Finite Groups

William Hargis

Representations Theory Structures Studied Linear Representations Images used belong to WikiMedia with the exception of Character Theory Characters the IU logo which belongs to Indiana University. Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups