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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 Representations Theory Structures Studied Linear Representations Character Theory Characters Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups What is Representation Theory? 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) 8s; t 2 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 2 3 Representations x11 x12 ::: x1n Character Theory 6x21 x22 ::: x2n 7 Characters Tr(X ) = Tr( ) Orthogonality of 6 7 Characters 4::::::::::::::::::5 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 fα; βg := α(g)β(g) (1) G Character Theory g2G 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 2 Gj9g 2 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: f0¯g,f1¯g,f2¯g,...,fn −¯ 1g. 2πi Let !n=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 !n=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: f0¯g,f1¯g,f2¯g,...,fn −¯ 1g. Representations Theory Structures Studied Linear Representations Character Theory Characters Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups 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: f0¯g,f1¯g,f2¯g,...,fn −¯ 1g. Representations Theory 2πi n Structures Studied Let !n=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) = !n where 0 ≤ j ≤ n − 1 and 0 ≤ m ≤ n − 1. Examples: Generalized Cyclic Group
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