Hello and welcome! I’m Sam, a Ph.D. candidate studying modular representation theory - in particular, equivalences between representation spaces. If you don’t know what those words mean, fear not. This series of blog posts will attempt to introduce my field and more! My end goal is to build up to painting a picture of the question which guides my dissertation, getting into some, but not too many, technical details. The intended audience is an undergraduate student who’s taken a course on group and ring theory, with basic linear algebra assumed as well. Though if you’re not there yet, don’t worry! You still might get something out of these posts, and I might write an appendix-like primer for everything you need from those courses to get what I’m doing here.

Today we’re going to discuss a few terms and features of modular representation theory, some of which will be necessary for a greater understanding of what comes later, and others which I’m partial to and think it would be neat to talk about. Some of these features can be seen as invariants of indecomposable modules, which can help us understand them better. Things will be a bit vaguer today, since much of the theory to understand the concepts we’ll be discussing is beyond the scope of these posts. However, I’ll still do my best to paint a picture of what’s going on!

# Blocks

As usual, $G$ will denote a finite group, $p$ a prime which divides $\mid G \mid$, and now, $k$ will denote a field of characteristic $p$. Whenever we want to talk about the classical case, we’ll just refer to $\mathbb{C}G$ for the time being.

To motivate the notion of blocks, we’ll begin by stating a version of a landmark structural result of Wedderburn and Artin, which in greatest generality, describes the structure of any semisimple ring. Here, we state the corollary of the result for the group algebra $\mathbb{C}G$, i.e. the classical case.

Theorem: (Artin, Wedderburn) Let $\chi_i: G \to GL_{n_i}(\mathbb{C})$ be a representative of the $i$th irreducible representation of $G$ over $\mathbb{C}$, for $i \in \{1,\dots, k\}.$ Then, the following is an isomorphism of $\mathbb{C}$-algebras:

\begin{align*} \Phi: \mathbb{C}G &\to \text{Mat}_{n_1}(\mathbb{C})\times \cdots \times \text{Mat}_{n_k}(\mathbb{C}),\\ \sum_{g\in G}a_g G &\mapsto \left(\sum_{g\in G}a_g \chi_1(g), \dots, \sum_{g\in G}a_g \chi_k(g) \right) \end{align*}

Here, $\text{Mat}_n(\mathbb{C})$ is the ring of $n\times n$ matrices with the usual addition and multiplication. So in short, $\mathbb{C}G$ is a direct product of matrix rings over $\mathbb{C}$, and the isomorphism is determined by the irreducible representations of $G$ over $\mathbb{C}$! But there’s more - this also allows us to compute the primitive central idempotents of $\mathbb{C}G$.

Definition: Recall an idempotent of a ring $R$ is an element $i \in R$ which satisfies $i^2 =i$.

• A central idempotent of a ring $R$ is an idempotent $i \in Z(R)$.
• A primitive idempotent of a ring $R$ is an idempotent $i \in R$ which cannot be expressed as $i = i_1 + i_2$ for two idempotents $i_1, i_2$ satisfying $i_1i_2 = i_2i_1 = 0$ (these are orthogonal idempotents).

It isn’t too difficult to show that the only central idempotent of $\text{Mat}_{n}(\mathbb{C})$ is the identity matrix $I\_n$, so it follows that in $\text{Mat}\_{n\_1}(\mathbb{C})\times \cdots \times \text{Mat}\_{n\_k}(\mathbb{C})$, the primitive central idempotents are precisely of the form

$(0, \dots, 0, I_{n_i}, 0,\dots, 0) \in \text{Mat}_{n_1}(\mathbb{C})\times \cdots \times \text{Mat}_{n_k}(\mathbb{C}).$

Taking the preimage via $\Phi^{-1}$ gives us the central idempotents $e_i \in \mathbb{C}G$. These $e_i$ satisfy $e_1 + \cdots + e_k = 1_{\mathbb{C}G}$ and $e_ie_j = 0$ when $i \neq j$. Furthermore, using $\Phi$, it’s straightforward to note that the $\mathbb{C}$-algebra $\mathbb{C}G e_i$ (i.e. elements of the form $m e_i$ for $m \in \mathbb{C}G$) is isomorphic to the matrix algebra $\text{Mat}_{n_i}(\mathbb{C})$. We say $e_i$ is the central idempotent associated to $\chi_i$.

Furthermore, one can show given the above properties of the central idempotents that for any $\mathbb{C}G$-module $M$, it can be partitioned up into a direct sum by those idempotents:

$M = e_1M \oplus \cdots \oplus e_k M,$

and each $e_iM$ can be equivalently considered a $\mathbb{C}Ge_i$-module. Moreover, if $M_j$ is the corresponding simple module to $\chi_j$, and $m_j \in M_j$ then

$e_im_j = \begin{cases} m_j & i = j \\\ 0 & i \neq j \end{cases}.$

In other words, $e_iM_i = M_i$ and each $e_j$ annihilates $M_i$ when $i \neq j$. To make a bit of a handwavey statement, these idempotents go hand in hand (in fact correspond bijectively) with simple modules, and can be used to break down any $\mathbb{C}G$-module to obtain the copies of the corresponding simple module that lives inside. These idempotents “break up” the representation space into a bunch of individual chunks, and after hitting a module with $e_i$, whatever remains “belongs to” $e_i$ in some sense! In this case, what ends up belonging is a direct sum of copies of the simple module $M_i$.

Let’s move back now to $kG$, which is not semisimple by Maschke’s theorem, so the Artin-Wedderburn theorem does not apply. However, $kG$ still has primitive central idempotents! Without bothering to compute what they are, let’s call these $e_1,\dots, e_l\in Z(kG)$ as before. What happens if we try hitting modules, or even $kG$, with those idempotents, as in the classical case? Well, it turns out that a lot of things are the same! As before,

$kG = kGe_1 \times \cdots \times kGe_l.$

Alternatively, we can consider $kG$ as a $(kG,kG)$-bimodule to phrase this in terms of direct sums.

Definition: A $(R,S)$-bimodule $M$ is both a left $R$-module and a right $R$-module for which the actions commute, i.e. $r(ms) = (rm)s$ for all $r\in R, s\in S, m\in M$.

In this case, structurally $kG$ as a $k$-algebra and $kG$ as a bimodule are fundamentally the same (though note that the notions of homomorphisms are not the same, since module homomorphisms are $kG$-linear, but $k$-algebra homomorphisms commute with multiplication - try explicitly writing down the definitions of each type of homomorphism to see what I mean). In fact, we could have done this in the $\mathbb{C}G$ case as well, but the Artin-Wedderburn theorem is stated explicitly in terms of algebras. Now identifying $kG$ as a bimodule, internal direct sums make sense, so we can write:

$kG = kGe_1 \oplus \cdots \oplus kGe_l,$

this follows since $1_{kG} = e_1+ \dots + e_l$ is the unique decomposition of $1_{kG} \in Z(kG)$ into primitive idempotents. This is the unique decomposition of $kG$ as a bimodule into indecomposable direct summands!

Here, we have a name for these idempotents and resulting $k$-subalgebras. We say the central idempotents $e_i$ are the blocks of $kG$ and the $kGe_i$ are the block algebras (or just blocks again for short). It turns out that for any $kG$-module, the same decomposition via the central idempotents applies:

$M = e_1M \oplus \cdots \oplus e_l M$

and each $e_iM$ is a $kGe_i$-module as well. This time though, these blocks do not correspond bijectively to the simple $kG$-modules. However, from the above decomposition, it follows that for every indecomposable $kG$-module $M$, there exists a unique block $e_i$ for which $e_im = m$ for all $m \in M$ and $e_jM = 0$ for $j \neq i$. In this case, we say $M$ belongs to the block $e_i$ if $e_iM = M$, and can consider $M$ a $kGe_i$-module as well.

So the blocks give a way of partitioning down the set of indecomposible (and simple) modules further - and since all $kG$-modules are direct sums of indecomposable modules, it lets us break up the problem of describing the space of $kG$-modules in general to describing the space of $kGe_i$-modules for all $i \in \{1,\dots, l\}$ (this is me hinting at what will come later…). And blocks truly do partition up the space - they determine what composition factors a module can have, maps between modules, and much more. For example, if a principal indecomposable module belongs to $e_i$, its corresponding simple module will belong to $e_i$ as well.

There are many more reasons blocks are fascinating, some of which I may be able to touch upon at the end of this post, but unfortunately most other results about blocks I will probably not be able to reach given the scope of these posts. (I do really wish that I could discuss how blocks affect the Cartan and decomposition matrices, so if you want to read further, I suggest you look up what those two matrices are). However, one final fact I’d like to discuss - recall earlier how I said that we don’t just do representation theory over $kG$, we have a $p$-modular system $(K,\mathcal{O},k)$, and that representation theory over $KG$ is the same as $\mathbb{C}G$.

Compute the central idempotents of $KG$ in the same way as I described for $\mathbb{C}G$, and call the resulting idempotents $e_\chi$, where $\chi$ is a simple $KG$-module. Then, lift the blocks of $kG$ to obtain the blocks of $\mathcal{O}G$. We have in fact an equality:

$e_i = \sum_{ \chi(e_i) \neq 0} e_\chi.$

Another way of stating this in words is: the block idempotent $e_i \in \mathcal{O}G$ is the sum of all the central idempotents of $KG$ corresponding to simple $KG$-modules which belong to $e_i$ (or alternatively, are not annihilated by $e_i$). And the only thing keeping the block idempotent from “breaking down further” is that $\mathcal{O}$ simply isn’t large enough - however in $K$, it can!

# Vertices, Sources, and Defect Groups

To begin, let’s ask a simple question: suppose $H \leq G$ are groups. How can we lift a representation of $H$ to a representation of $G$? Or semi-equivalently, how can we turn a $kH$-module into a $kG$-module in a canonical way? The primary answer to this question is via induction.

Definition: Given a left $kH$-module $M$, we define a left $kG$-module $\text{Ind}^G_H M$ as follows:

$\text{Ind}^G_H M = kG \otimes_{kH} M.$

If you haven’t seen a tensor product before, I’m going to skip all the technicalities of how this works. The important thing to know is that there is a canonical (in fact, functorial, whatever that means) way of turning $kH$-modules into $kG$-modules (I really hope this doesn’t come back to bite me later, but I think I can get away with not explaining all the technicalities here). Induction and restriction (take a module, but treat it as a $kH$-module rather than a $kG$-module) go hand in hand (and are in fact adjoint functors), and there are a number of results that describe how the two interact, such as Frobenius reciprocity and the celebrated Mackey formula.

Now that we’ve “defined” induction, we can introduce a generalized notion of projectivity. Recall that a $R$-module was projective if it was a direct summand of a free module, (where a free module is a direct sum of copies of $R$) - given any ring $R$, it’s always an important question to classify its projective modules. For example in $\mathbb{C}G$, every nonzero module is projective! First, let’s introduce some notation. Given two $R$-modules $M,N$ we write $M \mid N$, literally $M$ “divides” $N$, if $M$ is a direct summand of $N$, up to isomorphism.

But wait! Is this notion well-defined? What happens if there are two direct sum decompositions of $N$ where one has a submodule isomorphic to $M$ and the other does not? Good question, I’m glad you asked. The Krull-Schmidt theorem prevents any of those shenanigans from occurring, at least while working over group algebras.

Theorem: (Krull-Schmidt for $kG$-modules) If $M$ is a $kG$-module, then if $M = M_1 \oplus \cdots \oplus M_k = N_1 \oplus \cdots \oplus N_l,$ then up to isomorphism and reordering, $k = l$ and these direct sum decompositions are the same.

Great, so when we say $M$ “divides” $N$, we truly mean that it doesn’t depend on choice of direct sum decomposition, and even that there is a divisor and dividend here! Now, let’s define a generalized version of projectivity.

Definition: Let $H \leq G$. Say a $kG$-module $M$ is relatively $H$-projective if $M \mid \text{Ind}^G_H(N)$ for some $kH$-module $N$.

Why is this a generalized version of projectivity? Well at least when $k$ is a field (remember, in $p$-modular systems, $\mathcal{O}$ is not a field), then relative 1-projectivity is equivalent to projectivity! In some sense, this measures how “close” to being projective a $kG$-module is, with smaller groups being “more” projective. There are a number of theorems which apply as well to relatively $H$-projective modules that sort of mirror projectivity theorems, but that are formulated in terms of $H$.

Using properties of induction, one can show that if $M$ is $H$-projective, then it is ${}^gH$ projective for any $g \in G$, and if $H \leq H’$, then $M$ is also $H’$-projective. It follows that for any module, there is a conjugacy class of minimal subgroups $H \leq G$ for which $M$ is $H$-projective. This leads us to our next definitions, which give us more invariants of indecomposable $kG$-modules.

Definition: Suppose $M$ is an indecomposable $kG$-module. A subgroup $H$ for which $M$ is minimally $H$-projective is called a vertex of $M$. An indecomposable $kH$-module $N$ for which $M \mid \text{Ind}^G_H N$ is called a source of $M$ with respect to $H$. $(H,N)$ is a vertex-source pair of $M$.

One can show that in fact, if $H$ is a vertex of any indecomposable $kG$-module, then $H$ is a $p$-group, i.e. $\mid H\mid = p^k$ for some $k$ - I haven’t been mentioning it much, but all the invariants and decompositions I’ve been talking about are extremely dependent on $p$, and the structure of $kG$ can vary greatly for different characteristics of $k$ (remember - if $p$ doesn’t divide the order of $G$, the situation is the same as $\mathbb{C}G$)!

For example, the indecomposable modules which have $1$ as a vertex are exactly the principal indecomposable modules - they are the direct summands of $\text{Ind}^G_1(k) \cong kG$. So every indecomposable module has invariants such as the block they belong to, and their vertex-source pairs. There are theorems that describe how some of these invariants can lead to bijections between sets of indecomposable modules, such as the Green correspondence (which I will not be getting into), and plenty more conjectures (some of which I will be getting into, but not today).

There are two more things I must discuss before we wrap up for today. You may be asking what some properties having a certain vertex or a certain source may tell us about a module. We’ve already noted that modules with trivial vertex are projective. What about modules with trivial source? Well, these trivial source modules can be shown to in fact be precisely the modules which are indecomposable direct summands of permutation modules, or $p$-permutation modules as we call them! These modules will be quite important later - one of their special features is that in the interplay between $\mathcal{O}G$-modules and $kG$-modules, most modules do not lift uniquely from $kG$ to $\mathcal{O}G$, but $p$-permutation modules do.

The other thing I want to mention is an invariant not of an indecomposable module, but of a block! Recall that a block $kGe_i$ can equivalently be considered an indecomposable $(kG, kG)$-bimodule. This ends up being equivalent to considering it as a $k[G\times G]$-module, under the identification

$(g,h)\cdot m := g\cdot m \cdot h^{-1}$

which means that it has a vertex and a source! A defect group of a block $kGe_i$ is a vertex of $kGe_i$ when considered as a $k[G\times G]$-bimodule. This is one definition, but as it turns out, there are many other equivalent ones, some using minimality conditions, others using maximality, some using the module structure of a block, and others using its ring structure! Ultimately, I’m not able to go into much further detail there without taking some pretty significant detours.

The defect group of a block is a bit of a nebulous creature. One conjecture, the Donovan conjecture, claims that only finitely many blocks (up to an equivalence condition, which I may detail later) can have a $p$-subgroup $P$ as a defect group. Blocks which share defect groups tend to be linked as well. One result of Brauer, which unfortunately I cannot state in its entirety, gives a bijection between blocks of $kG$ with defect group $P$ and blocks of $kN_G(P)$ with defect group $P$ (or in fact, any subgroup $N_G(P) \leq H \leq G$). There is one other conjecture which we will save for next time, which arises when two blocks arising from $kG$ and $kN_G(P)$ share an abelian defect group $P$ - we will build up to this conjecture in the next post!

### Bibliography

• Course notes
• Linkelmann, “The Block Theory of Finite Groups”
• Nagao, Tsushima, “Representations of Finite Groups”