Affinisation

The process of untwisted affinisation is laid out in Kac. We can consider this to define an “affinisation” process on a Cartan matrix of finite indecomposable type, which turns it into a Cartan matrix of affine type. What we do is take the simple roots \alpha_1, \ldots, \alpha_r, and add a new “fake” simple root \alpha_{r + 1} = -\tilde{\alpha} where \tilde{\alpha} is the highest root. We correspondingly add the dual root \alpha^\vee_{r + 1} = - \tilde{\alpha}^\vee, the negative of the corresponding coroot (alternatively, the highest short coroot). The matrix recording the pairing \innprod{\alpha_i^\vee, \alpha_j}_{i = 1}^{r + 1} is the untwisted affine Cartan matrix corresponding to A.

Wheras the original Cartan matrix was invertible, the affine matrix will have corank 1. Write a_1, \ldots, a_n for the coefficients \widetilde{\alpha} = \sum_{i = 1}^r a_i \alpha_i of the highest root, then since a_1 \alpha_1 + \cdots + a_r \alpha_r + \alpha_{r + 1} = 0 the vector (a_1, \ldots, a_r, 1) gives a linear dependence among the columns of the affine Cartan matrix. Similarly, write \widetilde{\alpha}^\vee = \sum_{i = 1}^r a_i^\vee \alpha_i^\vee for the coefficients of the highest short coroot. Then the vector (a_1^\vee, \ldots, a_r^\vee, 1) gives a linear dependence among the rows of the affine matrix. In the untwisted case, we define a_{r + 1} = 1 and a_{r + 1}^\vee = 1.

The numbers h = \sum_{i = 1}^{r + 1} a_i and h^\vee = \sum_{i = 1}^{r + 1} a_i^\vee are called the Coxeter number and dual Coxeter number associated to the affine diagram (or if the affine diagram is untwisted, associated to the underlying finite type root system).

IMPORTANT NOTE: h(R^\vee) \neq h^\vee(R), or in other words the Coxeter number of the dual root system is not the dual Coxeter number of the original system. In the most simple terms, this is because the coefficients of the highest root, highest short root, highest coroot, and highest short coroot may all be different: a_i^\vee(R) is not equal to a_i(R^\vee). The slogan here is that affinisation and duals do not commute. As a concrete example, starting with a root system of type C_r and affinising gives what Kac calls C_r^{(1)}, however the dual of this (the transpose of the affine Cartan matrix) is what Kac calls D_{r + 1}^{(2)}, an affine algebra that only arises from the twisted construction. As another example, both G_2 and F_4 are self-dual, but their affine algebras are not self-dual.

Now, in any realisation of the affine system, we can define canonically the null root \delta = \sum_{i = 1}^{r + 1} a_i \alpha_i, which lies in the \bbR-span of the simple roots. Since a_{r+1} = 1 in the untwisted case, the affine root can be expressed as \alpha_{r + 1} = \delta - \sum_{i = 1}^r a_i \alpha_i = \delta - \widetilde{\alpha}. The null root is not seen by any simple coroot (not even the affine simple coroot), meaning that \innprod{\alpha_i^\vee, \delta} = 0, and hence it is fixed by all reflections: s_i(\delta) = \delta for i = 1, \ldots, r+1. Furthermore, the finite reflections send the finite simple roots to other finite (perhaps non-simple) roots. If the simple affine roots are linearly independent, the finite reflections look like

s_i = \begin{pmatrix} s_i |_{\Phi_f} & 0 \\ 0 & 1 \end{pmatrix}

however the affine reflection s_{r + 1} acts on the finite root \alpha_i as \begin{aligned} s_{r+1}(\alpha_i) &= \alpha_i - \innprod{\alpha_{r+1}^\vee, \alpha_i} \alpha_{r+1} \\ &= \alpha_i - \innprod{\alpha_{r+1}^\vee, \alpha_i}(\delta - \widetilde{\alpha}) \end{aligned}

Character rings for reductive groups

Each split reductive group G over \bbZ boils down to the combinatorial data of its root datum: a pair (X^\vee, X) of free \bbZ-modules of finite rank, in some perfect pairing \innprod{-, -} \colon X^\vee \times X \to \bbZ, along with choices of simple coroots \alpha_i^\vee \in X^\vee and simple roots \alpha_i \in X. The Cartan matrix A = [a_{ij}] = [\innprod{\alpha_i^\vee, \alpha_j}]_{ij} records the ‘core’ of this data, the isogeny class of the group.

A morphism of root data is a pair (f^\vee, f) \colon (X^\vee, X) \to (Y^\vee, Y) of (necessarily adjoint) maps f^\vee \colon X^\vee \to Y^\vee and f \colon Y \to X such that the simple roots and coroots of one root datum are sent to the simple roots and coroots of another. Thus the root data of the same Cartan type form a category. This category has initial and final objects, called the simply connected and adjoint root data respectively. The simply-connected datum has X^\vee_{sc} the free \bbZ-module with basis \alpha_i^\vee, and X_{sc} the free \bbZ-module with basis \varpi_i, with \innprod{\alpha_i^\vee, \varpi_i} = \delta_{ij}. The simple roots are the unique solutions to \innprod{\alpha_i^\vee, \alpha_j} = a_{ij}.

Returning to an arbitrary root datum (X^\vee, X), characters of the torus live in \bbZ[X], while characters of representations of the group live in the subring \bbZ[X]^W. We introduce the notation:

• e(\lambda) is the basis of \bbZ[X] as a group algebra.
• m(\lambda) = \sum_{\mu \in W \cdot \lambda} e(\mu) is the sum over the orbit of \lambda under W.
• \chi(\lambda) is the Weyl character associated to \lambda.

The elements \set{m(\lambda) \mid \lambda \in X_+} and \set{\chi(\lambda) \mid \lambda \in X_+} form bases of \bbZ[X]^W. A key step is being able to calculate the Kostka numbers, the coefficients which expand \chi(\lambda) in terms of the m(\mu):

\chi(\lambda) = \sum_{\mu \in X_+} K_{\lambda \mu} m(\mu) = \sum_{\mu \in X} K_{\lambda \mu} e(\mu).

A classic (but impractical) way of calculating these numbers is by the Weyl character formula. A much more practical method to calculate them for dominant \lambda is by Freudenthal’s formula, which recurses down from the highest weight. For non-dominant weights, the relation \chi(w \bullet \lambda) = \det(w) \chi(\lambda) may be used, where w \bullet \lambda = w(\lambda + \rho) - \rho is the reflection around -\rho. If \lambda has a stabiliser under the \bullet-action (i.e. \innprod{\alpha_i^\vee, \lambda} = -1 for some i) then \chi(\lambda) = 0.

The generalised tensor identity implies that if \sum_\mu a_\mu e(\mu) is symmetric under the W-action, then \chi(\lambda) \sum_{\mu} a_\mu e(\mu) = \sum_\mu a_\mu \chi(\lambda + \mu), which can be used to calculate tensor product multiplicities.

Implementing Freudenthal’s formula for an arbitrary root datum is not difficult, but things can be made far more efficient if it is only implemented for semisimple Lie algebras. It can be connected to an arbitrary root datum in the following way. Define X_0 = \set{\lambda \in X \mid \innprod{\alpha_i^\vee, \lambda} = 0 \text{ for all } i}: this is the “central” part of the root datum which plays almost no role in the Lie theory (aside from just shifting things around). It is the kernel of the map f \colon X \to X_{sc}.

SL2 and quantum numbers

The simple reductive group \SL_2 has well-understood representation theory, both in characteristic zero and in characteristic p > 0. The weight lattice of \SL_2 is identified with the integers, with the positive root \alpha = 2 and the fundamental weight \varpi = 1. The characters of \SL_2-modules are then Laurent polynomials in a single variable.

For \lambda \geq 0, the induced module \nabla_\lambda has character \ch \nabla_\lambda = q^{\lambda} + q^{\lambda - 2} + \cdots + q^{2 - \lambda} + q^{-\lambda} = \frac{q^{\lambda + 1} - q^{-(\lambda + 1)}}{q - q^{-1}} = [\lambda + 1]_q, where [n]_q denotes the quantum integer n. In characteristic zero, the induced modules are irreuducible and so the quantum integers describe all the characters of \SL_2.

In positive characteristic however, things are more interesting. Suppose we are in characteristic p > 0 for some prime p, then

1. By general theory, the module \nabla_\lambda for \lambda in the fundamental alcove \overline{C} = \set{\lambda \mid 0 \leq \innprod{\alpha^\vee, \lambda} < p \text{ for all } \alpha \in \Phi^+} remain irreducible, and
2. By Steinberg’s tensor product theorem, it is enough to know simple representations in the region X_1 = \set{\lambda \mid 0 \leq \innprod{\alpha^\vee, \lambda + \rho} \leq p \text{ for all } \alpha \in S}.

Usually the set of weights \overline{C} is a strict subset of X_1 and therefore more work needs to be done to find all the simple characters. But in the special case of \SL_2, all the dominant weights inside X_1 are also inside C and these weights are \set{0, 1, \ldots, p - 1}. So whenever 0 \leq \lambda < p, we may write \ch L(\lambda) = \ch \nabla_\lambda = [\lambda + 1]_q.

The Steinberg tensor product theorem says the following: write a dominant weight \lambda as a sum \lambda = \sum_{i \geq 0} p^i \lambda_i with each \lambda_i \in X_1. Then L(\lambda) = L(\lambda_0) \otimes L(\lambda_1)^{[1]} \otimes L(\lambda_2)^{[2]} \otimes \cdots, where V^{[k]} is the p^k-twist of the module V. We do not need to know much about the functor (-)^{[k]} aside from the fact that it dialates characters by p^k. If we consider a character \ch V to be a Laurent polynomial f(q) \in \bbZ[q^{\pm 1}], then the character \ch V^{[1]} is f(q^p).

Hence we can immediately write down the characters of \SL_2 for p = 3, say:

\begin{aligned} 0 = (0)_3 :&& \ch L(0) &=& &1 \\ 1 = (1)_3 :&& \ch L(1) &=& q^{-1} &+ q \\ 2 = (2)_3 :&& \ch L(2) &=& q^{-2} + &1 + q^{2} \\ 3 = (10)_3 :&& \ch L(3) &= \ch L(1)^{[1]} =& q^{-3} &+ q^{3} \\ 4 = (11)_3 :&& \ch L(4) &= \ch L(1)^{[1]} \otimes L(1)^{[0]} =& q^{-4} + q^{-2} &+ q^{2} + q^{4} \\ 5 = (12)_3 :&& \ch L(5) &= \ch L(1)^{[1]} \otimes L(2)^{[0]} =& q^{-5} + q^{-3} + q^{-1} &+ q + q^3 + q^5 \\ 6 = (20)_3 :&& \ch L(6) &= \ch L(2)^{[1]} =& q^{-6} + &1 + q^{6} \end{aligned}

Using this, it can be seen that the (n - 2i) weight space in L(n) is nonzero iff \binom{n}{i} \not\equiv 0 \pmod{p}. The diagram below shows the pattern of nonzero binomial coefficients modulo p, where p is any natural number (of course, only p = 0 and p prime correspond to representation theory). The curious case of p = 1 is because I have defined \binom{p}{0} = \binom{p}{p} = 1 regardless of characteristic, with the other coefficients determined by the usual Pascal’s triangle recurrence relation.

Computing in Lie theory

I’ll go over here in brief how I am computing all the basic combinatorial objects in Lie theory: root systems and statistics on them, Weyl groups, characters of irreps, etc. I hope to expand this into something more illuminating, but for now it’s basically a set of notes-to-self.

The real question we’re answering here is: starting from a Cartan matrix, how do you produce everything else?