My OperaCharacters, it is
Thursday, November 17, 2005 10:29:55 PM
As you can see [ here ] representation theory is applied in chemistry. Personally I am only interested in the pure mathematics aspects of the subject. If you look at the character table of D4 you can see that there are five representations: the trivial rep, a faithfull rep as rotations and reflections: in fact the Dihedral Group, and three other reps of dimension 1 which I wouldn't have find immediately. I would like to know if there is an algorithm I could use to calculate Character Tables.
Character X.
X: G -> C: X(g) = Tr(p(g))
Now in words.
The character is a map from a group to the complex numbers. G->C
p is a representation of the group which maps each element to a matrix. p(g)
A trace is the sum of the matrix elements on the diagonal. Tr(p(g)).
The character is the map group element to matrix trace. X(g) = Tr(p(g)).
Types of characters:
- linear: has degree 1
- trivial: is character of the trivial representation and is 1
- permutation character pi(g) = | fix(g) |
Properties
Equivalent representations have equal characters.
Equal characters represent equivalent or isomorphic CG modules.
X(1) is the dimension of the CG module, also called the degree of the character.
Group elements which are conjugate have the same character value.
Kernel of a character X is the set of all group elements which have the same value as X(1).
The sum of two characters is itself a character.
( More, more, more ... )
Sofar about characters for today. A pleasant surprise was that the permutation character pi(g) = | fix(g) |
Example S3 has three conjugacy classes with reps
() leaves 3 points fixed
( 12 ) leaves 1 point fixed
( 123 ) leaves 0 point fixed
Adding the reps of the S3 character table to a 3 dimension character gives
X( () ) = 3
X( (12) )= 1
X( (123) )= 0
or fix(g). Cool.
None of this comes from The Book. :-(, This does however. "... that the characters of the irreducible representations form an orthonormal basis for the space of class functions on G ...". Sounds interesting to me!
Character X.
X: G -> C: X(g) = Tr(p(g))
Now in words.
The character is a map from a group to the complex numbers. G->C
p is a representation of the group which maps each element to a matrix. p(g)
A trace is the sum of the matrix elements on the diagonal. Tr(p(g)).
The character is the map group element to matrix trace. X(g) = Tr(p(g)).
Types of characters:
- linear: has degree 1
- trivial: is character of the trivial representation and is 1
- permutation character pi(g) = | fix(g) |
Properties
Equivalent representations have equal characters.
Equal characters represent equivalent or isomorphic CG modules.
X(1) is the dimension of the CG module, also called the degree of the character.
Group elements which are conjugate have the same character value.
Kernel of a character X is the set of all group elements which have the same value as X(1).
The sum of two characters is itself a character.
( More, more, more ... )
Sofar about characters for today. A pleasant surprise was that the permutation character pi(g) = | fix(g) |
Example S3 has three conjugacy classes with reps
() leaves 3 points fixed
( 12 ) leaves 1 point fixed
( 123 ) leaves 0 point fixed
Adding the reps of the S3 character table to a 3 dimension character gives
X( () ) = 3
X( (12) )= 1
X( (123) )= 0
or fix(g). Cool.
None of this comes from The Book. :-(, This does however. "... that the characters of the irreducible representations form an orthonormal basis for the space of class functions on G ...". Sounds interesting to me!
