PHY 745 Group Theory 11 11 50 AM

  • Slides: 18
Download presentation
PHY 745 Group Theory 11 -11: 50 AM MWF Olin 102 Plan for Lecture

PHY 745 Group Theory 11 -11: 50 AM MWF Olin 102 Plan for Lecture 5: Representations, characters, and the “great” orthogonality theorem Reading: Chapter 3 in DDJ 1. Finish proof of “Great Orthogonality Theorem” 2. Character of a representation 3. Great orthogonality theorem for characters. PHY 745 Spring 2017 -- Lecture 5 1/23/2017 1

1/23/2017 PHY 745 Spring 2017 -- Lecture 5 2

1/23/2017 PHY 745 Spring 2017 -- Lecture 5 2

The great orthogonality theorem on unitary irreducible representations 1/23/2017 PHY 745 Spring 2017 --

The great orthogonality theorem on unitary irreducible representations 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 3

Proof of the great orthogonality theorem • Prove that all representations can be unitary

Proof of the great orthogonality theorem • Prove that all representations can be unitary matrices • Prove Schur’s lemma part 1 – any matrix which commutes with all matrices of an irreducible representation must be a constant matrix • Prove Schur’s lemma part 2 • Put all parts together 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 4

Proof of the great orthogonality theorem • Prove that all representations can be unitary

Proof of the great orthogonality theorem • Prove that all representations can be unitary matrices • Prove Schur’s lemma part 1 – any matrix which commutes with all matrices of an irreducible representation must be a constant matrix • Prove Schur’s lemma part 2 • Put all parts together 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 5

1/23/2017 PHY 745 Spring 2017 -- Lecture 5 6

1/23/2017 PHY 745 Spring 2017 -- Lecture 5 6

1/23/2017 PHY 745 Spring 2017 -- Lecture 5 7

1/23/2017 PHY 745 Spring 2017 -- Lecture 5 7

Proof continued: 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 8

Proof continued: 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 8

1/23/2017 PHY 745 Spring 2017 -- Lecture 5 9

1/23/2017 PHY 745 Spring 2017 -- Lecture 5 9

Characters 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 10

Characters 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 10

Great orthogonality theorem for characters 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 11

Great orthogonality theorem for characters 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 11

Example – P(3): 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 12

Example – P(3): 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 12

Character table for P(3): 1/23/2017 C 1 3 C 2 2 C 3 c

Character table for P(3): 1/23/2017 C 1 3 C 2 2 C 3 c 1 1 c 2 1 -1 1 c 3 2 0 -1 PHY 745 Spring 2017 -- Lecture 5 13

Check orthogonality: C 1 3 C 2 2 C 3 c 1 1 c

Check orthogonality: C 1 3 C 2 2 C 3 c 1 1 c 2 1 -1 1 c 3 2 0 -1 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 14

Some further conclusions: The characters ci behave as a vector space with the dimension

Some further conclusions: The characters ci behave as a vector space with the dimension equal to the number of classes. The number of characters=the number of classes 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 15

The regular representation is composed of h x h matrices constructed as follows, shown

The regular representation is composed of h x h matrices constructed as follows, shown with the P(3) example: 100000 reg Greg (E)= 0 1 0 0 G (A)= E-1 A-1 B-1 C-1 D-1 F-1 EABCDF AEDFBC BFEDCA CDFEAB FBCAED DCABFE Greg 1/23/2017 (D)= 001000 000100 000010 000001 Greg (B)= 00001000 00010000 000001 100000 001000 000010 1000001 010000 000100 Greg (F)= Greg (C)= 00000100 010000 00100000 000010 PHY 745 Spring 2017 -- Lecture 5 0100000 0000010 0001000 000100 0000010 100000 0010000 16

Regular representation continued – Note that the regular representation matrices satisfy the multiplication table

Regular representation continued – Note that the regular representation matrices satisfy the multiplication table of the group and have the same class structure as the group. 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 17

Example for P(3): 1/23/2017 C 1 3 C 2 2 C 3 c 1

Example for P(3): 1/23/2017 C 1 3 C 2 2 C 3 c 1 1 c 2 1 -1 1 c 3 2 0 -1 PHY 745 Spring 2017 -- Lecture 5 18