Representations of Rank 2 Reductive Groups

A set of interactive visualisations of concepts encountered in the representation theory of algebraic groups. This page has all the visualisations I’ve been working on in one spot, with minimal explanations. Everything here is still a work in progress – any and all comments or suggestions are welcome. The exact content on this page may change in the future as I reorganise things, but I’ll leave a link to wherever its new home is.

I think that interactivity is one of the best ways of building intuition about a problem or a mathematical concept. For example, being able to tweak parameters in a graph and see how the graph changes in real time is a fantastic way of building a mental model about how a class of graphs looks. However, this kind of interactivity has been lacking for other mathematical concepts such as representation theory – most of the software tools I know of have a very long feedback loop involving typing a question into a console, and even after that the user has to take a lot of steps to render that answer in any visual way.

This page contains my attempts to change this situation, by showing a lot of concepts one comes across in representation theory in a visual and interactive way. To make visualisation as consistent as possible, only rank 2 groups are allowed – in other words the groups on this list: T_2, A_1 \times A_1, \GL_2, A_2, B_2, G_2. All of the visualisations on this page run completely in the browser.

Orthogonal reflections

A Euclidean space is a finite-dimensional real vector space V equipped with an inner product (-, -) \colon V \times V \to \bbR (a symmetric positive-definite bilinear form). Given any hyperplane H \subseteq V and vector v, the inner product defines an orthogonal reflection of v over H, by travelling from v to H along the shortest path, and then travelling that distance again. The inner product can also define the hyperplane H as the kernel of the map v \mapsto (\alpha, v) for some fixed vector \alpha \in V, in which case we use the notation H_\alpha for the hyperplane.

The following diagram shows the situation in two dimensions, with the vector \alpha defining the hyperplane H_\alpha, and s_\alpha \colon V \to V the reflection-over-H_\alpha operator. Try moving \alpha and v and see how the diagram changes.

As illustrated in the diagram, the formula for the reflection s_\alpha is s_\alpha(v) = v - 2 \frac{(v, \alpha)}{(\alpha, \alpha)}.

The rank 1 and 2 root systems

A Euclidean root system is a finite set \Phi of nonzero vectors inside a Euclidean space (V, (-, -)), which satisfy the two additional properties:

1. (Reflection invariance) We have s_\alpha(\Phi) = \Phi for all \alpha \in \Phi, and
2. (Crystallographic) For every pair \alpha, \beta \in \Phi, the number 2 \frac{(\alpha, \beta)}{\alpha, \alpha} is an integer.

The vectors in \Phi are called roots, while \dim (\span_\bbR \Phi) is called the rank of the root system.

There are exactly two rank-1 root systems. The A_1 root system consists of \{\pm \alpha\} for any nonzero vector \alpha \in V, and the BC_1 root system consists of \{\pm \alpha, \pm 2 \alpha\}.

A simple application of the crystallographic property shows that if \alpha is a root, then the only multiples of \alpha which could also be roots are \pm \frac{1}{2} \alpha, \pm \alpha, or \pm 2 \alpha, and so indeed BC_1 is as complicated as a rank 1 system could get. We call a root \alpha \in \Phi divisible if \frac{1}{2} \alpha is also a root, and indivisible otherwise. The root system A_1 is called reduced, meaning that every root is indivisible, while the root system BC_1 is non-reduced.

Rank 2 root systems are where the geometry starts getting interesting. There are seven kinds of rank 2 root systems, four of which are reduced. All of these except for A_2 and G_2 are constructed in the Euclidean space \bbR^2, while A_2 and G_2 are constructed in the plane Z \subseteq \bbR^3 defined by x_1 + x_2 + x_3 = 0. Move your mouse over the roots to see their coordinates.

The root systems A_1 \times A_1, A_1 \times BC_1, and BC_1 \times BC_1 are called reducible because we can decompose the ambient space into two orthogonal subspaces, each containing a part of the root system. The other root systems A_2, B_2, G_2 and BC_2 are called irreducible since they cannot be decomposed in this way. (Be careful not to confuse “reduced” with “irreducible”). The rank 2 root systems which are both reduced and irreducible are A_2, B_2, and G_2.

Building a root system

It is not difficult to prove that the only rank 2 root systems are those shown above. In order to give some intuition for this problem, here is a game where the goal is to build a rank 2 root system.

Positive and simple systems

Planned for this section:

1. Show and explain gridlines, root lattice, weight lattice.
2. Positive and simple systems
3. Coxeter and Cartan matrices

Weyl characters

Fix a pinning T \subseteq B \subseteq G of a reductive group G so that we can talk about the weight lattice X(T) = \Hom(T, \bbG_m). For each weight \lambda \in X(T) there is a natural way of constructing a G-module via induction: let k be the base field and denote by k_\lambda the one-dimensional B-representation with weight \lambda, then we say that H^0(\lambda) := \ind_B^G(k_\lambda) is the induced module for weight \lambda. (Ignore the strange notation H^0 for a moment). It turns out that the induced module is nonzero precisely when \lambda is a dominant weight, and in this case we define the Weyl module V(\lambda) = H^0(- w_0 \lambda)^* and the Weyl character \chi(\lambda) = \ch H^0(\lambda) = \ch V(\lambda).

There is a way to extend the definition of \chi(\lambda) to non-dominant \lambda. Since the induction functor \ind_B^G \colon B\dashmod \to G\dashmod is left exact, we can form its right derived functors H^i(\lambda) := R^i \ind_B^G(k_\lambda). Note that H^0(\lambda) = \ind_B^G(k_\lambda) is the induced module, explaining the strange notation choice from the previous paragraph. We then may define \chi(\lambda) for any \lambda \in X(T) by taking an alternating sum across all of the derived functors: \chi(\lambda) = \sum_{i \geq 0} (-1)^i \ch H^i(\lambda). This agrees with the previous definition by Kempf’s vanishing theorem, which states that H^i(\lambda) = 0 for i > 0 and dominant \lambda.

The following visualisation shows the Weyl character of the selected weight. The blue dots show positive contributions, while the orange dots show negative contributions. The larger the dot, the larger the magnitude of the contribution. You can pan by clicking and dragging, and zoom by holding shift and scrolling. You can click once to freeze a weight in place, and double-click anywhere else to unfreeze.

The computation of each Weyl character is done using the Demazure character formula (which works without modification for non-dominant weights). For each simple root \alpha \in S, the isobaric Demazure operator \pi_\alpha is a linear endomorphism on the space \bbZ[X(T)] of characters, defined as the following geometric series: \pi_\alpha(e^\lambda) = \frac{e^\lambda - e^{s_\alpha \lambda - \alpha}}{1 - e^{-\alpha}}. There are essentially three cases in the expansion, depending on whether \innprod{\lambda, \alpha^\vee} is greater than, less than, or equal to -1. You can look at the effect of the \pi_\alpha operator by switching the visualisation above to show Demazure characters. It might be handy to also turn on the “Show Weyl orbit” option…

Orbits under the (Affine) Weyl group

The Weyl group W associated to a root system \Phi consists of all reflections over the hyperplanes H_\alpha = \ker \innprod{-, \alpha^\vee} where \alpha is a root. Another group which often arises is the p-dialated affine Weyl group, consisting of all reflections over the hyperplanes H_{\alpha, p} = \ker(\innprod{-, \alpha^\vee} - p). There are two different actions of the (affine) Weyl group of interest to us: the usual action just stated, and the \rho-shifted action w \bullet \lambda = w(\lambda + \rho) - \rho, where \rho is half the sum of the positive roots. Note that the \bullet action depends on a choice of simple system.

In the visualisation, the groups acting are all of the form W_p = p \bbZ \Phi \rtimes W, where \bbZ \Phi is the root lattice and W is the Weyl group of the root system. Note that W_0 is the usual Weyl group, and W_1 is the affine Weyl group for the dual root system (using the Bourbaki definition of the affine Weyl group).

The thin grey lines mark the hyperplanes defined by those weights \mu such that \innprod{\mu, \alpha^\vee} \in \mathbb{Z} for all coroots \alpha^\vee. The thick black lines mark the weights where \innprod{\mu, \alpha^\vee} = 0, and the blue lines mark the weights where \innprod{\mu, \alpha^\vee} \in p \bbZ. When the \rho-shifted action is selected, the blue lines instead mark the weights \mu where \innprod{\mu + \rho, \alpha^\vee} \in p \bbZ. Therefore when the shifted action is selected, the small blue chambers are the facets of the W_p \bullet action.

Move your mouse over the diagram to select a weight \lambda (shown in green). The red points will be the orbit of the green point under the group W_p. Clicking a weight will freeze the selection, and double-clicking anywhere else will un-freeze the selection.

The Jantzen Filtration

The Jantzen filtration is a certain filtration of a Weyl module V(\lambda) into G-modules. There is a sum which overcounts the character of the first proper submodule occuring in this filtration, and by “getting lucky” with this sum (for example, when the sum just happens to be multiplicity-free), we can compute some simple representations in positive characteristic.

Fix a dominant weight \lambda \in X(T)_+. Then there exists a filtration V(\lambda) = V(\lambda)^0 \supseteq V(\lambda)^1 \supseteq V(\lambda)^2 \supseteq \cdots of the Weyl module V(\lambda), such that V(\lambda)^0 / V(\lambda)^1 \cong L(\lambda), and furthermore such that the sum of the characters of the filtered pieces is \sum_{i > 0} \ch V(\lambda)^i = \sum_{\alpha \in R^+} \sum_{0 < mp < \innprod{\lambda + \rho, \alpha^\vee}} \nu_p(mp) \chi(s_{\alpha, mp} \bullet \lambda). The notation here is:

1. The characteristic of the underlying field is p > 0.
2. L(\lambda) is the simple module with highest weight \lambda.
3. \chi(\lambda) is the Weyl character for the weight \lambda. Even though the \lambda appearing in the sum are not dominant, we can use \chi(w \bullet \lambda) = \det(w) \chi(\lambda) to restate the sum in terms of dominant \lambda – see the reflect to dominant option in the visualisation.
4. \nu_p(a) is the largest n such that p^n \mid a.
5. s_{\alpha, mp} is the affine reflection in the hyperplane \innprod{-, \alpha^\vee} = mp, or alternatively \lambda \mapsto s_\alpha(\lambda) + mp \alpha. (Note that this appears in the formula with the \bullet action, not as just written).

The visualisation below shows the various terms \chi(\mu) appearing in the sum \sum_{i > 0} \ch V(\lambda)^i. You can pan by clicking and dragging, and zoom by holding shift and scrolling. You can click once to freeze a weight in place, and double-click anywhere else to unfreeze.

Since V(\lambda) / V(\lambda)^1 \cong L(\lambda) and \ch V(\lambda) is known, a formula for the character of the first proper submodule V^1(\lambda) would give a formula for the character of L(\lambda). Unfortunately, the sum above overcounts: \begin{aligned} \ch V(\lambda)^1 &= \ch V^1 / V^2 + \ch V^2 / V^3 + \ch V^3 / V^4 + \cdots \\ \sum_{i > 0} \ch V(\lambda)^i &= \ch V^1 / V^2 + 2 \ch V^2 / V^3 + 3 \ch V^3 / V^4 + \cdots \end{aligned} However, if (for example) the sum \sum_{i > 0} \ch V(\lambda)^i happened to be multiplicity-free when decomposed into sums characters of simple G-modules, then we would know that everything is concentrated in the first term \ch V^1 / V^2, and hence multiplicity-free sums give the actual character of V^1(\lambda). This works to determine all simple characters in types A_2 and B_2, and most in type G_2, as is shown in the next section. (See the section on sticking points for some examples of where I can’t yet pull enough information out of the Jantzen filtration to determine characters.

Characters of simple modules

Let T \subseteq B \subseteq G be a pinned reductive group defined over a field k of characteristic p > 0, and \lambda \in X(T)_+ a dominant weight. A major open problem is to determine a character formula for the simple module L(\lambda). For the groups of type A_2 and B_2, all simple characters can be computed using standard facts and the Jantzen filtration, and the same tactic almost works for G_2. The following visualisation shows various ways of looking at the simple characters for these groups.

Selecting simples in terms of Weyls will show the coefficients of the expansion L(\lambda) = \sum_{\mu} c_\mu \chi(\mu), which is the most performant as there is the least amount of work to do computationally. Selecting Weyls in terms of simples will show the coefficients the other way around, of the expansion \chi(\lambda) = \sum_{\mu} b_\mu \ch L(\mu). This has some nasty behaviour as the weights get large (inverting the upper triangular matrix requires incredibly large numbers), and so this mode peters out after a while. Finally, selecting simples in the standard basis will show the honest character L(\lambda) = \sum_{\mu} a_\mu e^\mu inside the group algebra \bbZ[X(T)]. The performance of this mode is dependent on the number of terms in the “simples in terms of Weyls” expansion.

You can pan by clicking and dragging, and zoom by holding shift and scrolling. You can click once to freeze a weight in place, and double-click anywhere else to unfreeze. If you select an incredibly large weight while in a “slow” mode, you might have to close your browser tab and re-open it. Sorry!

I’ve checked my program’s output on p-restricted weights for A_2, B_2, and G_2 against tables published by Frank Luebeck, available here. Any weight which is not p-restricted computed via the Steinberg tensor product theorem, and hopefully I have not made an error while programming that part. Any suggestions for how to check my results there would be very welcome.

Future work

This is a todo list for myself in rough order of priority, for when I have time.

1. Add a more in-depth section on how the code works (also as a reference for myself). This would start of well paired with the positive and simple systems section, since we can explicitly examine a whole bunch of stuff to do with embeddings, Cartan matrices, Dynkin diagrams, etc.
2. Clean up the visualisation code and the algebra code so that it’s ready to be shown to people.
3. Add better support for groups like GL_2 whose rank is not equal to its semisimple rank. At the moment the code responsible for the Steinberg tensor product gives up on GL_2, because it doesn’t know how decompose a weight into a p-restricted one.
4. The simple “multiplicity-free” tactic with the Jantzen filtration gives all the simple characters for A_1, A_2, A_3, and B_2, however it is not enough to get all the simple characters of G_2. Is there some other tactic that leverages the Jantzen filtration for all weights of G_2?
5. Are tilting characters easy to implement?

Sticking points

• Computing the decomposition of \chi(1, 4) in G_2 with p = 5. The Jantzen sum evaluates as L(0, 5) + 2 L(1, 3) + L(2, 0) and hence the multiplicity-free tactic does not apply. However, the formal character \chi(1, 4) - (\ch L(0, 5) + 2 L(1, 3) + L(2, 0)) has some negative terms, and so we know that actually the simple L(1, 3) should only occur once, rather than twice, as a composition factor of the Weyl module. This strategy is now programmed in, and gets a few more G2 weights than previously.
• Computing the decomposition of \chi(1, 1) in G_2 with p = 3. The Jantzen sum is \chi(0, 0) + \chi(0, 1) + \chi(1, 0) = L(0, 0) + 2 L(0, 1) + L(1, 0), but unfortunately the previous strategy can’t proceed further here..
• Similarly, Computing the decomposition of \chi(1, 0, 1) in A_3 with p = 2. In this case, the Jantzen sum gives 2 \chi(0, 0, 0), and so we have two possibilities: the Weyl module V(1, 0, 1) has either one or two trivial composition factors (the answer should be 1, since the simple L(1, 0, 1) has a two-dimensional zero weight space according to Luebeck’s tables).