Essay on Representation Theory and Simple Representations

Submitted By cisco1823
Words: 597
Pages: 3
Open Document
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 first principles.
2. What is meant by saying that the representation α is semisimple?
Prove that every finite-dimensional representation α of a finite group over C is semisimple.
Define the character χα of a representation α.
Define the intertwining number I(α, β) of 2 representations α, β. State without proof a formula expressing I(α, β) in terms of χα , χβ .
Show that the simple parts of a semisimple representation are unique up to order.
3. Draw up the character table of S4 .
Determine also the representation ring of S4 , ie express each product of simple representations of S4 as a sum of simple representations.
4. 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.
5. Explain the division of simple representations of a finite group G over
C into real, essentially complex and quaternionic. Give an example of each (justifying your answers).
Show that if α is a simple representation with character χ then the value of χ(g 2 ) g∈G determines which of these 3 types α falls into.
6. Define a measure on a compact space. State carefully, but without proof, Haar’s Theorem on the existence of an invariant measure on a compact group. To what extent is such a measure unique?
Prove that every representation of a compact group is semisimple.
Which of the following groups are (a) compact, (b) connected:
O(n), SO(n), U(n), SU(n), GL(n, R), SL(n, R)?
(Justify your answer in each
Show More

Related Documents: Essay on Representation Theory and Simple Representations

Essay about Response: Drawing and Stick Figures

Response Paper 2 This article is very much similar to Mitchells who quotes, “Representation is always of something or someone by something or someone, to someone. Representation isn’t that simple, it’s more complex than just “a dab of paint”, representing that the stone represents someone” This simply means that everything we read represents something. Belsey’s article goes in detail of “how” this representation works. This is quite interesting since before this article is read, no one really…

Words 386 - Pages 2

Questions On Psychology

QUIZ 2 – List of terms 1. Theory of mind- the capacity to reflect on one’s own thoughts and to distinguish them from other peoples thoughts and ideas. 2. Joint attention- paying attention to something that another person is referencing. 3. False belief- children who haven’t developed theory of mind yet, they attribute their thoughts onto someone else. Eg. Crayon box full of crackers- next person will think crackers when only having seen the crayon box. 4. Personification- the attribution…

Words 598 - Pages 3

EC141 Artice Review Week 5 Essay example

and it’s behavior. We may presume that individuals and global society have the same needs and wants. The economist theories touch on economic perspective, behavior, principle, budget, laws and benefits, except are the theories a true representation of what economy means to individuals and global markets? One such representation of economist theories that is acclaimed by society, individuals and global markets is that of The Ponzi Scheme. Kaushik Basu’s article (2014) “The Ponzi…

Words 701 - Pages 3

Masculine Representations in The Magic Toyshop Essay

Jessica Kazmin 1430215 Modes of Reading Joseph Shafer Examining "the Man": Masculine Representations in Angela Carter's The Magic Toyshop As a well-established champion of feminist prose, Angela Carter was published and known before the freethinkers of the 1970s began production of essays on feminist theory. The radical movers and shakers of the 1960s were inescapable. In Angela Carter's novel Love, Carter described herself as one of "the children of Nescafe and the Welfare State... The pure…

Words 3435 - Pages 14

Essay on Semiotics and Signs

Semiotics What are the fundamental tenets of semiotic theory, and what is its significance for twentieth century and/or contemporary art? Summarize its basic precepts and implications, and critically evaluate its strengths and weaknesses for interpreting at least one twentieth century artist or artistic movement. ----------------------------------------------------------------- Semiotics, a well established discipline of signs and their use in human and…

Words 1762 - Pages 8

Participant Observation In Qualitative Research

different realities that are subject to the interpretation of social actors, i.e. the social representation of meanings. The process of Document Analysis described relates to an interpretative view of the reality from the viewpoint of previous information gathered. This process has predominated in qualitative research, either by criteria of social representations theory or theory of action. Such theories seek to understand the reality of the information from a perspective declared by it. Such analysis…

Words 1500 - Pages 6

GRAPH ALGORITHMS Essay

Pratyusha Sampathirao 11/19/2014 Agenda        Graph Graph Theory Categories Applications Graph Properties and Measurements References Graph   A graph is a representation of a set of objects where some pairs of objects are connected by links.  The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges.  Set of dots : Vertices(Nodes or points)  Joining lines or curves : Edges(Arcs or…

Words 794 - Pages 4

Attention and Feature Integration Theory Essay

Computational psychology – too many ambiguity to be considered, so we developed ways to resolve these ambiguity. Rabbit? (mom points to an object and said rabbit, how does tommy know which thing mom was pointing at?) We go for natural level of representations, in this case the whole object. Dis Office: 6578 Neurons. Signal is fed into dendrite. -> information sent thru neurons. inside of neurons is more negative than outside. -> when signal is send to dendrite, depolarization occurs and if enough…

Words 3311 - Pages 14

Tradition Definition Essay

artworks are characterized by their possession of, respectively, representational, expressive, and formal properties. It is not difficult to find fault with these simple definitions. For example, possessing representational, expressive, and formal properties cannot be sufficient conditions, since, obviously, instructional manuals are representations, but not typically artworks, human faces and gestures have expressive properties without being works of art, and both natural objects and artifacts produced…

Words 521 - Pages 3

Implicit Attitude Paper

Could one’s implicit attitudes about mathematics affect their attitudes towards other disciplines? Implicit attitudes and stereotypes about mathematics do not have to be available to conscious introspection in order to affects one’s performance or involvement in a given field. But could one’s implicit attitudes towards one field affect their attitude towards another? In this paper, we will examine whether there is an association between gender and math and whether it divides the sciences as well…

Words 1439 - Pages 6