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Previous parts: - Part 1 (Simple Modules)

There are a few results that come up when dealing with simple modules that are important but not necessarily important in my story, so I wanted to leave them here. Most of these are directly analogous to the version with G-actions on vector spaces, really only with one exception at the end.

Since this is an addendum, I also won't provide proofs for everything, not that I've been giving anything more than proof sketches for any of the results in the previous parts anyways.

Recall our conventions: - G is a finite group - k is a field of characteristic p (which can be zero) - kG-modules are finite dimensional over k - A is a k-algebra that is also finite dimensional over k.

Furthermore, recall that from last time kG is semisimple if and only if p does not divide the order of G, and if kG is semisimple, it is isomorphic to a direct sum of matrix algebras over k.

Prop 3.1 If S is a simple A-module, it is isomorphic to A/m, where m is a maximal submodule of A. Cor 3.2 If A is semisimple, then every simple module of A is a direct summand of A.

Both of these statements are extremely classical, and the proof of the proposition will be left to the reader. The corollary follows from the proposition using techniques in the previous part: since A is semisimple, we can decompose A as A = ⨁ S_i where S_i are the simple summands of A. Now, for any simple A-module S, the natural map A → S decomposes into maps S_i → S, each of which is nonzero if and only if S_i is isomorphic to S. Since the map A → S is nonzero, at least one of the maps S_i → S is nonzero, proving the corollary.

Prop 3.3 If A is semisimple, S is a simple A-module which is of dimension d as a k-vector space, then S shows up in the decomposition of A exactly d times.

While this proposition takes some work, it still follows fairly easily from Wedderburn-Artin, and inspection of the decomposition of A into matrix algebras leads to this result very quickly. (Hint: for a matrix algebra M_n(k), show the subset of matrices that have zero entries everywhere except for the jth column is a simple module for any 1 ≤ j ≤ n).

Readers familiar with representation theory will see that these results are restatements of results about G-representations, in particular stating that the regular representation is exactly the direct sum of all the irreducible representations, where each irreducible representation shows up a number of times equal to its dimension. Another important result that is still missing gives us a equivalence between the number of conjugacy classes of G and the number of irreducible representations of G. Phrased with simple modules, we have:

Thm 3.4 If kG is semisimple, then the number of simple kG-modules (up to isomorphism) is exactly the number of conjugacy classes of G.

Note that this requires that kG is semisimple, which happens exactly when p does not divide |G|. It turns out that we do have a more general result:

Thm 3.5 The number of simple kG-modules (up to isomorphism) is exactly the number of conjugacy classes of G of elements with order not divisible by p.

The proof for the case where kG is semisimple is pretty much exactly the same as in representation theory: when we consider the center of kG, since kG is a direct sum of matrix algebras A_i, we find that Z(kG) = Z(A_1 + ... + A_r) = Z(A_1) + ... + Z(A_r)

However, the center of a matrix algebra is the scalar matrices, so for e_i the identity matrix in A_i for each i, we find that Z(kG) = ke_1 + ... + ke_r which is an r-dimensional space.

On the other hand, for C_1, ... , C_s the conjugacy classes of G, we can show that the center is exactly the submodule generated by the elements ∑g_i where we sum over all the elements g_i in the conjugacy class C_i. Since there are s such linearly independent elements, the submodule generated by these elements has dimension s, and so performing the computations shows s = r.

Unfortunately, this proof can't easily be extended to the case where kG is not semisimple, and since that proof is much more complicated, I won't sketch it here.

It occurred to me that a tumblr post would probably not be a bad way to collect my thoughts on a topic if I wanted to give a talk on this topic in the future, so with that in mind, here are some assorted notes on representation theory.

For this post, we will assume the following conventions:

G is a finite group, k is an algebraically closed field with characteristic p, A is a k-algebra that is finite dimensional over k, p is either a prime or zero.

Most results will still work in characteristic 0 when sensible. Since Sylow p-subgroups show up later, in that case p is a positive prime, but otherwise p = 0 works everywhere else.

Background

The most basic story of representation theory is as follows: let k = C be the complex numbers, V a finite dimensional vector space over k. Then, a representation of a group G is a map G → GL(V), though usually it will be denoted simply by V when the context is clear. In other words, we get an action of G on V that is compatible with the vector space structure. A subspace W of V is a subrepresentation if W is G-invariant, so for every g in G, gW ⊆W. Maps between representations can be defined in a fairly natural way, and as you might expect, for any two representations U and V of a group G, we can make the direct sum U⊕V, the tensor product U⊗V, and the dual U* into representations of G in fairly natural ways.

From this framework, a couple of classical results follow, the most important of which is Maschke's theorem. If we have a G representation U with a subrepresentation V, then there exists a complementary subrepresentation W such that U = V⊕W as an internal direct sum. One consequence of this is that we can study an arbitrary representation by dividing it up until you can no longer find a nontrivial subrepresentation. This leads to the following definition: a G representation V is irreducible if every subrepresentation of V is either V itself, or the trivial subspace 0. An immediate corollary of Maschke's theorem, then, is that every G-representation can be decomposed as a direct sum of irreducible representations. Thus, just as we use various forms of prime factorization to study other mathematical objects, here we can study arbitrary representations simply by studying irreducible representations. Indeed, much is known about irreducible representations, of which one of the most interesting results is that the number of distinct irreducible representations up to isomorphism of a group G is exactly the number of conjugacy classes of G.

The ability to find an irreducible decomposition of a G representation is extremely convenient, but the conditions to find such a convenient result are fairly stringent. Recall that the above results are currently only stated for k = C and G a finite group; if we were to consider an arbitrary field and an arbitrary group, Maschke's theorem certainly would not hold. While there are several ways to proceed from here, the approach we take will involve a rephrasing of representation theory from a new perspective.

Group algebras and semisimplicity

One of the strengths of representation theory is that in the study of a group G, by studying the G representations we can use the powerful tools of linear algebra. In particular, since we consider vector spaces over an algebraically closed field, we in particular have the powerful tools of Jordan canonical form and spectral theory to throw at our problems.

Taking this one step further, then, instead of considering a group action of G on V, sharp-eyed readers will notice that this action gives V the structure of a kG-module, where kG is the group algebra. This shift in perspective will let us throw the tools of ring theory at our problems, giving us an even more extensive repertoire of results at our disposal.

Def 1.1 Let A be as above, and for the sake of convention, we consider A-modules to be those that are finite dimensional as a k-vector space. 1. An A-module M is simple if any submodule N of M is either trivial, so N = 0, or M itself, so N = M. 2. An A-module M is semisimple if it is a direct sum of simple modules. 3. A is semisimple as a ring/algebra if A is semisimple as a left A-module. Equivalently, A is semisimple as a ring/algebra if every A-module is semisimple. 4. An A-module M is indecomposable if whenever M is written as a direct sum M = S⊕T, either S = 0 or T = 0.

Immediately, there are two notable points. Firstly, Maschke's theorem from above can be restated as the following: if k = C, then kG is semisimple. Secondly, in the case where k = C, simple modules and indecomposable modules coincide. This is not true in general, we will show later that indecomposable kG-modules that are not simple exist. Similarly, for general k, we find that kG is not always semisimple. The relevant result for this is as follows:

Thm 1.2 The group algebra kG is semisimple if and only if the characteristic of k does not divide the order of G.

Proving this is simple enough. In the backwards direction, it suffices to show that for any kG-module U and any submodule V, we can find a direct sum decomposition of U into V⊕W. Since k does not divide |G|, the standard proof for the characteristic 0 case still works. Take a projection ρ of U onto V, the associated projection ρ' = 1/|G| * ∑ gρg^(-1), where we take the sum over all g∈G, is well defined, and for W the kernel of ρ', we get that this V and W give us a direct sum decomposition. On the other hand, if k does divide the order of G, then this proof clearly fails due to the fact that |G| = 0 in k.

To properly prove the reverse direction, we introduce our first tool from ring theory.

Def/Prop 1.3 The following are equivalent: 1. The set of elements J of A that annihilate every semisimple module. 2. The smallest submodule J of A such that A/J is a semisimple A-module. 3. The intersection of all the maximal submodules of A. 4. The largest nilpotent ideal of A. The ideal J that satisfies any of the equivalent conditions above is the radical of A, denoted rad(A).

Note that this is the Jacobson radical of A, and while A is in general not commutative, meaning we do not have powerful tools such as Nakayama lemma at our disposal, there are still some properties we can use. Furthermore, note that an ideal J is nilpotent if J^n = 0 for some n, which is not in general the same as saying J is an ideal of nilpotent elements. When G is a finite group, however, the nilradical of kG is a nilpotent ideal since kG is Noetherian.

Similarly, for modules we have:

Def/Prop 1.4 Let U be an A-module. Then, the following are equivalent: 1. (rad(A))U 2. The smallest submodule V of U such that U/V is semisimple. 3. The intersection of all maximal submodules of U. The submodule V that satisfies any of the equivalent conditions above is the radical of U, denoted rad(U).

Proofs of these equivalencies is left as an exercise to the reader. With these tools, however, we can easily prove the other direction of Theorem 1.2.

Let G be a group such that the characteristic of k divides |G|. Consider the element s = ∑ g where we take the sum over all g∈G. Then, on one hand for any h∈G we have that hs = ∑ hg = s, but on the other hand we have that s^2 = ∑ hs = |G|s = 0. Thus, s is a nonzero nilpotent element, and thus rad(kG) is nonzero, so kG is not semisimple.

This proves a more general version of Maschke's theorem, and note that the case k = C is covered since char k = 0, and no group G has order divisible by 0.

We're not quite done, however, and there is one important result I want to leave in this blog post.

Wedderburn-Artin Theorem

As it turns out, when we have a semisimple algebra, its structure is extremely rigid, and the proof of this result is an excellent demonstration of how all our structure lets us prove difficult problems.

Lemma 2.1 Let A be as above, S a simple A-module. Then, End(S)≅k.

(God is the isomorphism symbol ugly on web tumblr)

This is simple (no pun intended): since k is algebraically closed, for every ρ∈End(S), we know that ρ has at least one eigenvalue λ. This means that ρ - λ1 has nontrivial kernel, and so the image (ρ - λ1)S is a proper submodule of S. Since S is simple, (ρ - λ1)S = 0, and so ρ = λ.

Lemma 2.2 For S, T nonisomorphic simple A-modules, Hom(S, T) = 0.

Again, if we consider any φ∈Hom(S, T), if we suppose φ is not the zero map, then the image is a nonzero submodule of T, and so is equal to T. Similarly, the kernel is a proper submodule of S, and so is 0. Thus, φ is an isomorphism of S and T, which contradicts our assumption.

Lemma 2.3 For A as above, End(A) is isomorphic to A with opposite multiplication.

This is a standard result in ring theory: every endomorphism of A is characterized by where it sends 1. The only reason we need to involve opposite multiplication is that function composition reverses the order of multiplication.

Putting these two lemmas together, we get the theorem:

Theorem 2.4 We call a k-algebra A simple if it has no ideals beside 0 and A itself. If A is simple, then A is a matrix algebra over k.

The proof of this follows very elegantly from the above: consider A as a left module over itself, let S be a simple submodule of A, and U the sum of all submodules of A isomorphic with S. Then, for every ρ∈End(A), ρ(U) is contained in U by Lemma 2.2. Thus, U is a nonzero ideal in A, and so must be equal to A itself. Note this proves that if A is a simple algebra, A is a semisimple module over itself.

Now, suppose that U is the direct sum of n copies of S. Then, End(A) = End(U) ≅M_n(k) the algebra of n×n matrices with entries in k.

This last isomorphism deserves some explanation. Since U is isomorphic to a direct sum of n copies of S, say S_1, ... , S_n, every endomorphism of U can be characterized by all the induced maps S_i → S_j. Since Hom(S_i, S_j) is just k by Lemma 2.1, this gives End(U) the structure of a matrix algebra with the ij entry corresponding to Hom(S_i, S_j). To finish off the proof, by Lemma 2.3, we have that A is isomorphic to M_n(k) up to matrix transpose.

Cor 2.5 (Wedderburn-Artin Theorem) If A is a semisimple algebra, then A is a direct sum of matrix algebras.

Again, we decompose A as a module into a direct sum U_1 + ... + U_r where each U_i is a direct sum of all the copies of some simple module in A. Again, by the same argument as above, every endomorphism of A is characterized by all the induced maps Hom(U_i, U_j). However, if U_i and U_j are direct sums of nonisomorphic simple groups, by Lemma 2.2 again we get Hom(U_i, U_j) = 0. Thus, End(A) is given by the set of block diagonal matrices, with each block corresponding to Hom(U_i, U_i), which is a matrix algebra by Theorem 2.4.