COMPUTING CHARACTER TABLES OF FINITE GROUPS. Jay Taylor (Università degli Studi di Padova)
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1 COMPUTING CHARACTER TABLES OF FINITE GROUPS Jay Taylor (Università degli Studi di Padova)
2
3 Symmetry
4 Chemistry Symmetry
5 Symmetry Chemistry Biology
6 Symmetry Chemistry Biology Physics
7 Groups Symmetry Chemistry Biology Physics
8 Number Theory Groups Symmetry Chemistry Biology Physics
9 Number Theory Topology Groups Symmetry Chemistry Biology Physics
10 Number Theory Topology Algebraic Geometry Groups Symmetry Chemistry Biology Physics
11 DEFINITION
12 DEFINITION A group is a pair (G,?) with G a set and? : G G! G a binary operation such that:
13 DEFINITION A group is a pair (G,?) with G a set and? : G G! G a binary operation such that:. there exists an element e G such that x? e = e? x = x for all x G
14 DEFINITION A group is a pair (G,?) with G a set and? : G G! G a binary operation such that:. there exists an element e G such that x? e = e? x = x for all x G. for every g G there exists an element g - G such that g.? g - = g -? g = e
15 DEFINITION A group is a pair (G,?) with G a set and? : G G! G a binary operation such that:. there exists an element e G such that x? e = e? x = x for all x G. for every g G there exists an element g - G such that g.? g - = g -? g = e. a? (b? c) =(a? b)? c for all a, b, c G.
16 DEFINITION A group is a pair (G,?) with G a set and? : G G! G a binary operation such that:. there exists an element e G such that x? e = e? x = x for all x G. for every g G there exists an element g - G such that g.? g - = g -? g = e. a? (b? c) =(a? b)? c for all a, b, c G. Examples with, (Z, +) Z = {...,-, -, 0,,,... }
17 DEFINITION A group is a pair (G,?) with G a set and? : G G! G a binary operation such that:. there exists an element e G such that x? e = e? x = x for all x G. for every g G there exists an element g - G such that g.? g - = g -? g = e. a? (b? c) =(a? b)? c for all a, b, c G. Examples (Z, +) with Z = {...,-, -, 0,,,... }, (R, ) with R = R \{0} where R denote the real numbers.
18 DIHEDRAL GROUPS
19 DIHEDRAL GROUPS
20 DIHEDRAL GROUPS 4
21 DIHEDRAL GROUPS 4
22 DIHEDRAL GROUPS 4
23 DIHEDRAL GROUPS 4
24 DIHEDRAL GROUPS 4
25 DIHEDRAL GROUPS 4
26 DIHEDRAL GROUPS 4
27 DIHEDRAL GROUPS 4
28 DIHEDRAL GROUPS 4
29 DIHEDRAL GROUPS 4
30 DIHEDRAL GROUPS 4 90 o
31 DIHEDRAL GROUPS 4
32 DIHEDRAL GROUPS 4 90 o
33 DIHEDRAL GROUPS 4 90 o
34 DIHEDRAL GROUPS 4 90 o
35 DIHEDRAL GROUPS 4 90 o
36 DIHEDRAL GROUPS 4 90 o
37 DIHEDRAL GROUPS 4 90 o
38 DIHEDRAL GROUPS 4 80 o 90 o
39 DIHEDRAL GROUPS 4 90 o
40 DIHEDRAL GROUPS 4 90 o 80 o
41 DIHEDRAL GROUPS 4 90 o 80 o
42 DIHEDRAL GROUPS 4 90 o 80 o
43 DIHEDRAL GROUPS 4 90 o 80 o
44 DIHEDRAL GROUPS 4 90 o 80 o
45 DIHEDRAL GROUPS 4 90 o 80 o
46 DIHEDRAL GROUPS 70 o 4 90 o 80 o
47 DIHEDRAL GROUPS 90 o 80 o
48 DIHEDRAL GROUPS 90 o 80 o 70 o
49 DIHEDRAL GROUPS 90 o 80 o 70 o
50 DIHEDRAL GROUPS 90 o a b 80 o 70 o
51 DIHEDRAL GROUPS 90 o a b ab 80 o 70 o
52 DIHEDRAL GROUPS 90 o a b ab aba 80 o 70 o
53 DIHEDRAL GROUPS 90 o a b ab aba 80 o 70 o abab
54 DIHEDRAL GROUPS 90 o a b ab aba 80 o 70 o abab ababa
55 DIHEDRAL GROUPS 90 o a b ab aba 80 o 70 o abab ababa ababab
56 DIHEDRAL GROUPS 90 o a b ab aba 80 o 70 o abab ababa ababab e
57 DIHEDRAL GROUPS 90 o a b ab aba 80 o 70 o abab ababa ababab e a = e,
58 DIHEDRAL GROUPS 90 o a b ab aba 80 o 70 o abab ababa ababab e a = e, b = e,
59 DIHEDRAL GROUPS 90 o a b ab aba 80 o 70 o abab ababa ababab e a = e, b = e, (ab) 4 = e
60 DIHEDRAL GROUPS 90 o a b ab aba 80 o 70 o abab ababa ababab e I (4) =ha, b a = e, b = e, (ab) 4 = ei
61 DIHEDRAL GROUPS
62 DIHEDRAL GROUPS 7 o 44 o a b ab aba (ab) 6 o 88 o (ab) a (ab) (ab) a (ab) 4 e
63 DIHEDRAL GROUPS 7 o 44 o a b ab aba (ab) 6 o 88 o (ab) a (ab) (ab) a (ab) 4 e I (5) =ha, b a,b, (ab) 5 i
64 DIHEDRAL GROUPS 7 o 44 o a b ab aba (ab) 6 o 88 o (ab) a (ab) (ab) a (ab) 4 e I (m) =ha, b a,b, (ab) m i
65 MATRIX REPRESENTATION
66 MATRIX REPRESENTATION v v
67 MATRIX REPRESENTATION v v
68 MATRIX REPRESENTATION v v v v
69 MATRIX REPRESENTATION v v v v apple 0 0 a
70 MATRIX REPRESENTATION v v apple 0 0 a
71 MATRIX REPRESENTATION v v apple 0 0 a
72 MATRIX REPRESENTATION v v v v apple 0 0 a
73 MATRIX REPRESENTATION v v v v apple 0 apple 0 0 a 0 - b
74 MATRIX REPRESENTATION v v apple 0 apple 0 0 a 0 - b
75 MATRIX REPRESENTATION v v apple 0 apple 0 0 a 0 - b
76 MATRIX REPRESENTATION v v v v apple 0 apple 0 0 a 0 - b
77 MATRIX REPRESENTATION v v v v apple 0 apple 0 apple a b ab
78 MATRIX REPRESENTATION apple 0 apple 0 apple 0 - apple a b ab aba apple - 0 apple 0 - apple 0 apple abab ababa ababab e
79 REPRESENTATIONS
80 REPRESENTATIONS V an n-dimensional C-vector space.
81 REPRESENTATIONS V an n-dimensional C-vector space. GL(V) = GL n (C) the group of all invertible linear transformations f : V! V.
82 REPRESENTATIONS V an n-dimensional C-vector space. GL(V) = GL n (C) the group of all invertible linear transformations f : V! V. A representation of a group (G,?) is a map : G! GL(V) such that for all a, b G we have (a? b) = (a) (b)
83 CHARACTERS Let : G! GL n (C) be a representation of a group (G,?). The function : G! C defined by (a) =Tr( (a)) is called the character of.
84 CHARACTERS Let : G! GL n (C) be a representation of a group (G,?). The function : G! C defined by (a) =Tr( (a)) is called the character of. e a b ab aba abab ababa ababab apple 0 0 apple 0 0 apple apple 0-0 apple apple apple apple
85 CONJUGACY
86 CONJUGACY For any matrices A, B GL n (C) recall that Tr(ABA - )=Tr(B)
87 CONJUGACY For any matrices A, B GL n (C) recall that Tr(ABA - )=Tr(B) Hence for any two elements a, b G and any character : G! C we have (a? b? a - )= (b)
88 CONJUGACY For any matrices A, B GL n (C) recall that Tr(ABA - )=Tr(B) Hence for any two elements a, b G and any character : G! C we have (a? b? a - )= (b) We say a, b G are conjugate if there exists an element x G such that. x? a? x - = b
89 CONJUGACY For any matrices A, B GL n (C) recall that Tr(ABA - )=Tr(B) Hence for any two elements a, b G and any character : G! C we have (a? b? a - )= (b) We say a, b G are conjugate if there exists an element x G such that. x? a? x - = b This defines an equivalence relation on G. The resulting equivalence classes are called conjugacy classes.
90 CONJUGACY a ababa b aba 90 o 70 o 80 o ab ababab abab e
91 IRREDUCIBLE CHARACTERS
92 IRREDUCIBLE CHARACTERS A representation : G! GL(V) is irreducible if there is no proper subspace W V which is invariant under G. By this we mean that for all g G we have (g)w W.
93 IRREDUCIBLE CHARACTERS A representation : G! GL(V) is irreducible if there is no proper subspace W V which is invariant under G. By this we mean that for all g G we have (g)w W. We have : G! GL(V) is irreducible if and only if G X gg (g) (g) =
94 IRREDUCIBLE CHARACTERS A representation : G! GL(V) is irreducible if there is no proper subspace W V which is invariant under G. By this we mean that for all g G we have (g)w W. We have : G! GL(V) is irreducible if and only if G X gg (g) (g) = A character with this property is also called irreducible.
95 IRREDUCIBLE CHARACTERS a b
96 IRREDUCIBLE CHARACTERS a b 8 ( (-) ) =
97 CHARACTER TABLES Theorem The number of distinct irreducible characters of a finite group is equal to the number of conjugacy classes.
98 CHARACTER TABLES Theorem The number of distinct irreducible characters of a finite group is equal to the number of conjugacy classes. Let g,...,g n G be representatives for the conjugacy classes and let,..., n be the irreducible characters of G. The square matrix ( i (g j )) 6i,j6n is called the character table of G.
99 CHARACTER TABLES I (4) e a b ab (ab)
100 CHARACTER TABLES I (m) e a b (ab) r (ab) m - (-) r (-) m - (-) r (-) m j 0 0 " jr + " -jr (-) j
101 CHARACTER TABLES I (m) e a b (ab) r (ab) m - (-) r (-) m - (-) r (-) m j 0 0 " jr + " -jr (-) j 6 j, r 6 m - " = e i/m
102 SYMMETRIC GROUPS
103 SYMMETRIC GROUPS S n is the group of all bijective functions f : {,..., n}! {,..., n}.
104 SYMMETRIC GROUPS S n is the group of all bijective functions f : {,..., n}! {,..., n}. S n We call the symmetric group on n points.
105 SYMMETRIC GROUPS S n is the group of all bijective functions f : {,..., n}! {,..., n}. S n We call the symmetric group on n points. S n = n! which can be very large even for small n. For example S 0 =
106 SYMMETRIC GROUPS
107 SYMMETRIC GROUPS Example ( n = )
108 SYMMETRIC GROUPS Example ( ) n =
109 SYMMETRIC GROUPS = Example ( ) n =
110 SYMMETRIC GROUPS = = Example ( ) n =
111 SYMMETRIC GROUPS A function f S n is called a cycle of length k if there exists a subset X = {x,...,x k } {,..., n} such that f(i) =i for any integer i 6 X and f acts on the elements of X in the following way f x f f x k x f x k- x f f x 4 f
112 SYMMETRIC GROUPS Lemma Every element of S n is a product of disjoint cycles.
113 SYMMETRIC GROUPS Lemma Every element of S n is a product of disjoint cycles. =
114 SYMMETRIC GROUPS
115 SYMMETRIC GROUPS A partition of n is a sequence µ =(µ,...,µ k ) of integers such that µ > > µ k > and µ + + µ k = n.
116 SYMMETRIC GROUPS A partition of n is a sequence µ =(µ,...,µ k ) of integers such that µ > > µ k > and µ + + µ k = n. For example the partitions of 5 are (5), (4, ), (, ), (,, ), (,, ), (,,, ), (,,,, )
117 SYMMETRIC GROUPS A partition of n is a sequence µ =(µ,...,µ k ) of integers such that µ > > µ k > and µ + + µ k = n. For example the partitions of 5 are (5), (4, ), (, ), (,, ), (,, ), (,,, ), (,,,, ) Given f S n let f f k be a decomposition of f into a product of disjoint cycles. If µ i denotes the length of the cycle f i then the sequence µ(f) =(µ,...,µ k ) is a partition of n, after possibly reordering the entries. We call µ(f) the cycle type of f.
118 SYMMETRIC GROUPS Theorem Two elements of the symmetric group are conjugate if and only if they have the same cycle type.
119 SYMMETRIC GROUPS Theorem Two elements of the symmetric group are conjugate if and only if they have the same cycle type. We will write P(n) for the set of all partitions of n.
120 SYMMETRIC GROUPS Theorem Two elements of the symmetric group are conjugate if and only if they have the same cycle type. We will write P(n) for the set of all partitions of n. For any partition P(n) we denote by an irreducible character of. S n
121 SYMMETRIC GROUPS Theorem Two elements of the symmetric group are conjugate if and only if they have the same cycle type. We will write P(n) for the set of all partitions of n. For any partition P(n) we denote by an irreducible character of. S n P(0) = 67
122 HOOKS OF PARTITIONS Consider the partition µ =(5, 5, 4,,, ) P(9).
123 HOOKS OF PARTITIONS Consider the partition µ =(5, 5, 4,,, ) P(9).
124 HOOKS OF PARTITIONS Consider the partition µ =(5, 5, 4,,, ) P(9).
125 HOOKS OF PARTITIONS Consider the partition µ =(5, 5, 4,,, ) P(9).
126 HOOKS OF PARTITIONS Consider the partition µ =(5, 5, 4,,, ) P(9). H
127 HOOKS OF PARTITIONS Consider the partition µ =(5, 5, 4,,, ) P(9). H h = 6
128 HOOKS OF PARTITIONS Consider the partition µ =(5, 5, 4,,, ) P(9). H h = 6 l = 6
129 HOOKS OF PARTITIONS Consider the partition µ =(5, 5, 4,,, ) P(9). h = 6 l = 6
130 HOOKS OF PARTITIONS Consider the partition µ =(5, 5, 4,,, ) P(9). R h = 6 l = 6
131 HOOKS OF PARTITIONS Consider the partition µ =(5, 5, 4,,, ) P(9). R µ \ R h = 6 l = 6
132 CHARACTER TABLE Theorem (Murnaghan Nakayama Formula) Write f S n as a product f f k of disjoint cycles. Assume is a cycle of length m then the element f k g = f f k- is contained in the symmetric group S n-m. For any partition P(n) we have (f) = X (-) l ij \R ij (g) h ij =m
133 CHARACTER TABLE S
Sign elements in symmetric groups
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