PHY 745 Group Theory 11 11 50 AM
- Slides: 18
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
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 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 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 7
Proof continued: 1/23/2017 PHY 745 Spring 2017 -- Lecture 5 8
1/23/2017 PHY 745 Spring 2017 -- Lecture 5 9
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
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 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 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 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 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 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 1 c 2 1 -1 1 c 3 2 0 -1 PHY 745 Spring 2017 -- Lecture 5 18
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