Last update: 12 August 2013
Homework Problem 1.16 Study for large groups (the algebra of functions on Go find out about compact groups and representation theory.
If is a (finite dimensional) of dimension then there is an associated representation, which we shall also denote by such that
Definition 1.17 Let be an algebra, and and be The direct sum is the of all pairs where Addition is defined componentwise; the action of is
We can make a parallel definition for representations.
Definition 1.18 Let be an algebra, and and representations of Then the direct sum is the representation given by
Definition 1.19 An is completely decomposable if it is isomorphic (as an to a direct sum of simple
Problem 1.20 Let be an algebra.
|1.||What are the irreducible representations of|
|2.||Are the other representations of completely decomposable?|
Definitions 1.21 Given an algebra A trace on is a linear functional such that for all The center of is the subalgebra An idempotent is an element such that and A central idempotent is an idempotent such that An ideal of is an ideal in the ring theory sense.
We will use the words “trace” and “character” interchangeably; some would have us call our traces “virtual characters”.
Definition 1.22 Let be a representation. Then the character of the representation is given by where denotes the vanilla trace of high school matrix theory. Any such character is a trace in the sense defined above.
The following propositions can be nicely summarized as follows:
Theorem 1.23 There exists a unique nonzero
Proposition 1.24 There is a unique (up to scalar multiples) nonzero trace on
Let be a trace, and let
Thus any such is a scalar multiple of the ordinary trace functional.
Proposition 1.25 There is a unique nonzero element in (up to scalar multiples).
Let Now and implies that for some and the coefficient Fix these and
Let Since is central, On the other hand,
Thus if then Thus central implies that is a diagonal matrix.
To show that central implies is scalar, fix and consider For any Since this holds for all it follows that for all
Proposition 1.26 There is a unique nonzero ideal in
Proposition 1.27 There is a unique central idempotent in
Let with central implies that Now thus is a complex number satisfying so either or This is one place where these developments will not go through so nicely in other characteristics.
Proposition 1.28 There is a unique (nonzero) irreducible representation of (up to equivalence).
First we shall prove that such a representation exists.
Let be the space of all column vectors of length acts on by matrix multiplication.
Claim: is irreducible.
Let be the standard basis for Let be a subrepresentation, with Choose with and fix an such that
Claim: (whence proving the previous claim.)
Thus for every Thus
Continued in next lecture.
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.