Last update: 12 August 2013
Having shown that is an irreducible irreducible representation of we proceed to prove uniqueness of up to equivalence.
Let be any simple module of and choose Then
and so for some The set is a nonzero left submodule of and hence by the simplicity of
We claim that as left The map defined by
realizes this isomorphism, where the nonzero entries on the right appear in the column (the proof is left as an exercise).
We may then define a linear transformation
by This map is a module homomorphism, for if we choose an arbitrary matrix unit then
and the module property follows by linearity.
Since is a submodule and the simplicity of forces Similarly, is a nonzero submodule of the simple module hence The map is therefore an isomorphism between and
Next, let us fix some notation. As usual, let be the complex numbers, the reals, the integers. We depart from standard notation to write for the set of nonnegative integers and let denote the set of positive integers. We will use interval notation to denote the set of integers between two endpoints (inclusive).
Definition 1.29 A simple algebra is an algebra with no nonzero proper two-sided ideals. For this course, a simple algebra is any algebra such that for some
Definition 1.30 We will say a finite dimensional algebra is semisimple provided where is some finite index set, and
We will often write for the simple component of indexed by Thus a typical element is of the form
where each which we regard as a matrix.
Every finite dimensional semisimple algebra has a basis of matrix units which correspond to the usual matrix units in That is, the structure constants are given by
Define a trace by where That is is the trace of the block of
Proposition 1.31 Let be a trace on and let Then
By 1.23, tr is the unique trace on hence restricted to is the unique trace on Thus, for each there exists such that for all In particular,
The proposition follows by linearity of
Definition 1.32 Let be a trace on The vector weight vector of
The previous proposition shows that a trace is completely determined by its weight vector. Conversely, one may define a trace by specifying its weight vector. Hence the trace functionals on are in one to one correspondence with the vectors in which explains the notation
After classifying the traces on we turn to the ideal structure. From the general theory of semisimple algebras, we know that possesses a finite collection of minimal two-sided ideals (i.e. nonzero ideals of containing no proper nonzero two-sided ideals). The algebra is the direct sum of these minimal ideals and, furthermore, every ideal of is a direct sum of some subset of these ideals.
Since is simple and evidently the minimal two-sided ideals of are the simple components
Definitions 1.33 Two idempotents and are said to be orthogonal provided An idempotent is said to be minimal (or principal) provided cannot be written as a sum of orthogonal idempotents.
From the general theory, every minimal two-sided ideal is generated by a central idempotent i.e. Since must be the identity of necessarily Note that the are orthogonal as they belong to different minimal ideals of
However, the minimality of the two-sided ideals does not quite translate into minimality of the in the sense of the above definition, since the are orthogonal idempotents which sum to Yet is minimal in the sense that it may not be written as the sum of central orthogonal idempotents:
Proposition 1.34 The minimal central idempotents of are In particular, this set forms a basis for the center
One may verify by using the basis of matrix units that Since is a direct sum of matrix algebras, the second statement of the proposition follows from the definition of the
Suppose is any central idempotent of If we express in terms of the new basis for the center, we obtain
using the orthogonality of the By uniqueness of expression, we have and hence Thus, any central idempotent of is a sum of one or more The result then follows from the independence of the
Homework Problem 1.35 Show that every element of the form
where each is an idempotent which generates a minimal left ideal of Conclude that these elements are minimal idempotents of
Similarly, show that the idempotents
generate minimal right ideals of and hence are minimal idempotents.
Finally, we turn to the irreducible representations and simple modules of Let be defined by where is the of the element
Proposition 1.36 The representations are the complete set of irreducible representations of up to equivalence.
The restriction is the unique irreducible representation of Any of is also a of hence is zero on However, Hence, on and is an irreducible representation of
If is any irreducible representation of then is an ideal of and hence is the direct sum of one or more of the If is in the complement of then is a summand of Irreducibility of then implies that is equivalent to exactly one of the
Turning to simple modules, denote the space by and let act on by The proof of the following proposition is left as an exercise.
Proposition 1.37 The set form a complete set of simple modules of up to isomorphism.
Observe also that the irreducible representation defined by a has character defined by
Hence the irreducible characters of are the traces
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.