Symmetric Group To Prove: Conjugation preserves the cycle structure - - PowerPoint PPT Presentation

Symmetric Group

To Prove: Conjugation preserves the cycle structure

Cayley Theorem proof

 For a finite group G={g1 = e, g2, g3, ….gn }  Define a map to symmetry Πg defined as  Show that Πg is isomorphic to G and hence G is

a subgroup of the symmetry group of degree n.

Semi-direct products

Generators of alternating groups

Show that the alternating group is generated by the set of all the three-cycles on n letters.

Bilateral axis

 If the principal axis is in the plane of mirror

symmetry, then it is called bilateral axis

 Principal axis is bilateral if we have U2 two-fold

axis perpendicular to principal axis

 Point group with the bilateral n-fold axis

requires Ck

n to be conjugate to Cn-k n

 Using this, write the conjugate classes of

Dihedral Dn and Cnv groups

Dihedral groups

Tetrahedral, cubic groups

Principal axis is bilateral if we have Try to write the

group elements of the proper group symmetry of the cube called Octohedral symmetry and denoted as O

Similarly, proper group symmetries of the

regular Tetrahedron is denoted as T

Then including improper reflection planes, we

have Oh, Td – try to write the conjugacy classes

Tetrahedral molecule

C2 Three C2 C3 Four 3-fold axes Pure Rotations give group T

Including reflections

σ is the mirror (reflection) plane six mirror reflection planes (6σ) S4 is a rotation by 90° followed by a mirror reflection a tetrahedral Structure has total 24 symmetry operations

Representation of

What is reducible and irreducible representation?

Examples

 Write two-dimensional matrix representation for the

group C3v

 Write 3-dimensional matrix representation for

symmetric group of degree 3

 What is the difference  Can we a matrix S such that the above 3-dimensional

matrices can be brought to block diagonal form?

Reducible Representation

 If we can find a S which diagonalises or block

diagonalises n * n matrix representations of the group elements of the group

 Is there such S ? That may become tedious if matrix

size is large.

 Further, for a given group how many non-trivial

representations which cannot be further broken into diagonal or block diagonal form? Such representations are called irreducible representations (irreps).

Reducible and Irreducible

Bring it to block-diagonal form if not diagonal

Each block-diagonal component is called irreducible representation We discussed in class, (i) two-dimensional rep of C3v (ii) another three-dimensional rep of C3v Claim : 2d rep is irreducible but 3d rep is reducible.

Notations



Characters χ(trace of matrices)do not change under similarity

• transformations. Characters will be same for all the group elements within

the same class.



For abelian groups, number of classes=h (order of G)



Aim is to find the number of irreducible representations Гα (g), their dimensions ℓα for every group G and the characters χα(g)

 Гred(g)=∑ aα Гα (g) where aα gives # of times irrep α appears in the

reducible representation

 Postulates: (i)

Number of Гα (g) is equal to number of classes (p)

(ii)

∑ (ℓα )2 = h

Great Orthogonality theorem

number of elements in class i

Character Tables

 Using the orthogonality properties of the great

• rthogonality theorem, let us work out the character

table C2v and for symmetric group of degree 3

 Write character table for C4 and C4v

Mulliken Symbol Notation

 One-dimensional irreps are denoted by A if character

for Cn is 1

 One dimensional irreps are denoted by B if character

for Cn is -1

 Two dimensional irreps are denoted by E  Three dimensional irreps are denoted by F or T  We introduce subscript g(gerade) or u(ungerade)

depending on σh character is +1 or -1 respectively

Character table

Symmetric group and irrep diagrams

 Recall number of classes is equal to number of irreps.

So, the same Young diagrams used for cycle structure can be used to denote irreps.

 Symmetrizer – horizontal row of boxes-trivial

representation is denoted by this diagram

 Antisymmetrizer- vertical column of boxes-A2 of S3 is

denoted by vertical column of 3 boxes

 Other diagrams are called mixed representations.

Mixed representation diagrams

 How do we determine the dimensions of any Young

• diagram. Symmetrizer and antisymmetrizer diagrams

are 1-dimensional irreps. To find dimensions of mixed rep of ς(n), use hook formula:



Count the number of boxes the hook for every box traverses. I have indicated two hooks- one traversing 1box and the other traversing 4 boxes. Do for all boxes. Dimension of irrep diagram is n! / hook number

Resolve reducible into irreps

 For reducible rep :

Гred(g)=∑ aαГα (g) where aα gives # of times irrep α appears in the reducible representation

Irreps in reducible representation

Using the orthogonality property, aα for reducible representation Гred is

Typically, a system with symmetry group G when acted by an external perturbation will no longer have G as symmetry. In fact, it will be subgroup K C G. Then the irreps of G will be reducible representation for K

Regular representation-definition

 Take the multiplication table of as rows and columns of

matrix representation

 Write the 4 * 4 matrix elements using

where δ(e)= 1 and zero otherwise

Are they diagonal matrices? Are they reducible representation?



Write the characters of the regular representations (trace of the matrix Г ) denoted as χreg(g) and obtain aα

 Please work this out and we will discuss in the next class

Hamiltonian and symmetry

 For system with symmetry group G, Hamiltonian H

commutes with the elements of G : [H,g]=0

 Implies, Ψ(x) and g Ψ(x) will have same energy

eigenvalue

 If Ψ(x) is a non-degenerate eigenfunction of H, then g

Ψ(x) = c Ψ(x)

 If Ψ(x) is one of the degenerate eigenfunction of H,

then g Ψ(x)= linear combination of the set of degenerate eigenfunctions

Character table and degeneracy

 Character table gives the characters and dimensions of the

irreducible representations α’s

 Гα(g) irrep for group G will act on ℓα dimensional basis states

ξi where i takes 1,2 , …. ℓα.

 Гα(g) ξi = will be linear combinations of ξi’s 

Both Гα(g) ξi and ξi have same energy if H is invariant under the group symmetry G- hence dimensionality ℓα> 1

• f irreps indicate degenerate eigenfunctions

 Relook at particle in a 2-d square box and its group

symmetry which allows 2-fold degenerate wavefunctions

Character table and basis states

g z= (+1) z is eigenvalue eqn with eigenvalue +1 for all g which are characters of irrep A1. Similarly acting on axial vectors and x,y coordinates will shown the eigenvalues matching with the other irreps as we discussed in class.