Group and Simple Representations Essay

Submitted By cisco1823
Words: 589
Pages: 3
Open Document
Course 424
Group Representations
Dr Timothy Murphy
GMB ??

Friday, ?? 2005

??:00–??:00

Attempt 7 questions. (If you attempt more, only the best 7 will be counted.) All questions carry the same number of marks.
Unless otherwise stated, all groups are compact (or finite),, and all representations are of finite degree over C.
1. Define a group representation. What is meant by saying that 2 representations α, β are equivalent? Find all representations of S3 of degree
2 (up to equivalence).
What is meant by saying that a representation α is simple? Find all simple representations of D4 from first principles.
2. What is meant by saying that a representation α is semisimple?
Define a measure on a compact space. State carefully, and outline the main steps in the proof of, Haar’s Theorem on the existence of an invariant measure on a compact group.
Prove that every representation of a compact group is semisimple.
3. Define the character χα of a representation α, and show that it is a class function (ie it is constant on conjugacy classes).

1

Define the intertwining number I(α, β) of 2 representations α, β of a group G, and show that if G is compact then
I(α, β) =

χα (g)χβ (g) dg.
G

Prove that a representation α is simple if and only if I(α, α) = 1.
4. Draw up the character table for S4 .
Determine also the representation-ring for this group, ie express the product αβ of each pair of simple representation as a sum of simple representations. 5. Show that the number of simple representations of a finite group G is equal to the number s of conjugacy classes in G.
Show also that if these representations are σ1 , . . . , σs then dim2 σ1 + · · · + dim2 σs = |G|.
Determine the dimensions of the simple representations of S5 , stating clearly any results you assume.
6. Determine the conjugacy classes in SU(2); and prove that this group has just one simple representation of each dimension.
Find the character of the representation D(j) of dimensions 2j + 1
(where j = 0, 12 , 1, 23 , . . . ).
Express each product D(i)D(j) as a sum of simple representations
D(k).
7. Define the exponential eX of a
Show More

Related Documents: Group and Simple Representations Essay

Group Representation Research Paper

Course 3413 — Group Representations Sample Paper II Dr Timothy Murphy December 2010 Attempt 6 questions. (If you attempt more, only the best 6 will be counted.) All questions carry the same number of marks. Unless otherwise stated, all groups are compact (or finite), and all representations are of finite degree over C. 1. What is a group representation? What is meant by saying that a representation is simple? What is meant by saying that a representation is semisimple? Prove that every finite-dimensional…

Words 384 - Pages 2

Essay on Representation Theory and Simple Representations

Course 424 Group Representations Dr Timothy Murphy G.M.B. Friday, 22 May 1995 14:00–16:00 Answer as many questions as you can; all carry the same number of marks. Unless otherwise stated, all representations are finite-dimensional over C. 1. Define a group representation. What is meant by saying that 2 representations α, β are equivalent? What is meant by saying that the representation α is simple? Determine all simple representations of D4 (the symmetry group of a square) up to equivalence, from…

Words 597 - Pages 3

Exam99 1a Essay

Course 424 Group Representations Ia Dr Timothy Murphy Seminar Room Thursday, 17 June 1999 16:30–18:30 Attempt 6 questions. (If you attempt more, only the best 6 will be counted.) All questions carry the same number of marks. Unless otherwise stated, all groups are finite, and all representations are of finite degree over C. 1. Define a group representation. What is meant by saying that 2 representations α, β are equivalent? What is meant by saying that the representation α is simple? Determine…

Words 351 - Pages 2

Exam89 1 Essay

Course 424 Group Representations I Dr Timothy Murphy Arts Block A2039 Friday, 20 January 1989 15.45–17.45 Answer as many questions as you can; all carry the same number of marks. Unless otherwise stated, all groups are finite, and all representations are finite-dimensional over C. 1. Define a group representation. What is meant by saying that 2 representations α, β are equivalent? Determine all 2-dimensional representations of S3 up to equivalence, from first principles. 2. What is meant by saying…

Words 365 - Pages 2

Essay about Exam91 3

Course 424 Group Representations III Dr Timothy Murphy Sam Beckett Theatre Wednesday, 10 June 1991 14:00–16:00 Answer as many questions as you can; all carry the same number of marks. Unless otherwise stated, all Lie algebras are over R, and all representations are finite-dimensional over C. 1. Define the exponential eX of a square matrix X. Determine eX in each of the following cases: X= 1 0 0 2 , X= 0 1 1 0 , X= 0 1 -1 0 , X= 1 0 0 -1 Which of these 5 matrices X are themselves expressible…

Words 340 - Pages 2

Exam03 3 Essay

Course 424 Group Representations III Dr Timothy Murphy Joly Theatre Friday, 25 April 2003 15:15–16:45 Attempt 5 questions. (If you attempt more, only the best 5 will be counted.) All questions carry the same number of marks. In this exam, ‘Lie algebra’ means Lie algebra over R, and ‘representation’ means finite-dimensional representation over C. 1. Define the exponential eX of a square matrix X, and show that eX is invertible. Determine eX in each of the following cases: X= X= 0 0 1 0 1 , X=…

Words 373 - Pages 2

Mahzarin Banaji Stereotypes

exist the duality of the conscious and unconscious mind. Author, Mahzarin Banaji, analyzes one of societies most commonly used forms of representations and focuses on the idea that their construction is completely unconscious. Through the characterization of stereotypes as implicit social representations, Banaji is able to propose the idea that that these representations are formed without our conscious consent and we are able to measure them through specific tests. In this paper I will analyze Banaji’s…

Words 1418 - Pages 6

The Two Sides In The Connecticut Compromise

What is your topic? My topic is The Connecticut Compromise. What sides/groups/organizations were involved in the disagreement? The two groups that disagreed under my topic were the small states and the larger states. What were the sides in the conflict most concerned about? They were most concerned about how their state would be represented during an election. How can this compromise prevent future conflict? It might prevent the large states or the small states from trying to argue with the government…

Words 725 - Pages 3

Evolution Of Racial Stereotypes In The Media

Although media and the representation of racial stereotypes have evolved over many decades, we cannot simple say we live in a colour-blind society. Despite denial, most, if not all individuals carry around some level of racial bias whether it stems from the media, globalization, technology or the way we have been raised. Throughout the movie “Racial Stereotypes in the Media”, one can notice the great evolution of the representation of race. Examining a few decades previous to today, the media specifically…

Words 289 - Pages 2

Essay on Assess The Arguments For The Retention

the retention of the FPTP for the UK general elections FPTP is a simple plurality system which means that the winner is the candidate who gains the most votes. Some may argue that it should be abolished in favour of Proportional representation because it is will provide the government with more legitimacy. However, I am of the opinion that it should be retained because it produces a strong government. Proportional representation would produce a weak and ineffective government because it will create…

Words 1102 - Pages 5