These notes are all old and probably slightly incorrect, and are more just notes-to-self for while I’m programming.

# 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

- 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
- 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.

## Quantum integers

Define the *quantum integers* [n] \in \bbZ[v^{\pm 1}] for n \geq 0 as follows:

\begin{aligned} [0] &= 0 \\ [1] &= 1 \\ [2] &= v^{-1} + v^{1} \\ [3] &= v^{-2} + 1 + v^{2} \\ [4] &= v^{-3} + v^{-1} + v^{1} + v^{3}, \end{aligned}

and further define [-n] = -[n]. This can be compactly summarised in the geometric series formula [n] := \frac{v^n - v^{-n}}{v - v^{-1}}.

The quantum integers are fixed under the ring homomorphism v \mapsto v^{-1}, which we write as [n]_{v^{-1}} = [n]. We also have the *quantum addition formula*

[n] +_v [m] := v^{-m}[n] + v^n[m] = v^{m}[n] + v^{-n}[m]

and the *quantum multiplication formula*

[n] \times_v [m] := [n]_v [m]_{v^n}.

It is straightforward to show that [n] +_v [m] = [n + m] and [n] \times_v [m] = [nm].

We may also define the *quantum factorial* for n \geq 0:

[n]^{!} := [n][n - 1] \cdots [2][1],

and the *quantum binomial coefficient* for n \in \bbZ and r \in \bbN

\qbinom{n}{r} := \frac{[n]^!}{[r]^![n-r]^!} = \frac{[n][n-1] \cdots [n-1+r]}{[r][r-1] \cdots [1]}.

The quantum binomaial coefficients satisfy a Pascal-style recurrence relation

\qbinom{n}{k} = v^k \qbinom{n - 1}{k} + v^{n - k} \qbinom{n - 1}{k - 1}

which together with \qbinom{n}{0} = \qbinom{n}{n} = 1 shows that they are Laurent polynomials, again symmetric over v \mapsto v^{-1}.

### Classic quantum integers

A more common definition of the quantum integers is by taking [n]_q := \frac{q^{n} - 1}{q - 1} = 1 + q + \cdots + q^{n - 1}. We will differentiate between these quantum integers and the others by using q vs v as the variable. They are related by v^{-n + 1}[n]_{q \mapsto v^2} = [n]_v, as we can see in the formula: v^{-n + 1}[n]_{q \mapsto v^2} = \frac{v^{-n}}{v^{-1}} \cdot \frac{v^{2n} - 1}{v^2 - 1} = \frac{v^n - v^{-n}}{v - v^{-1}} = [n]_v. In the opposite direction, we have [n]_q = (v^{n - 1}[n]_v)_{v \mapsto q^{\frac{1}{2}}}

# 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?

## The goals

Some of the goals are as follows:

- Compute irreducible characters and tensor product multiplicities etc for arbitrary finite type root data.
- Compute various other statistics and interesting numbers on their associated root systems (eg heading towards
*the poster*). - Be able to compute in Weyl groups, both in a compact enumerated form (eg for computing KL polynomials), and in a more sparse non-enumerated form (for occasional computations, or computations where we quotient by a large parabolic).

In order to go towards goal 1, we can use the fact that every reductive group G admits a central isotypy G_{sc} \to G from the simply-connected group in its isogeny class, which we can use to pull back representations of G to representations of G_{sc}. Irreps should stay irreducible (why?), and then we are always working within the basis of fundamental weights where computations simplify somewhat.

This means that (modulo some mappings between root data) solving 2 well will also give most of the tools to solving 1.

## Building root systems

Let A = [a_{ij}] be an n \times n Cartan matrix of finite type, i.e. a GCM with nonzero (and therefore positive, integer) determinant. It defines a root system, root lattice, and weight lattice, and dually a coroot system, coroot lattice, and coweight lattice. Theoretically, these are all sitting inside some finite-dimensional vector space V of dimension n over the reals, but we never work explicitly with this space; instead whenever we build roots we will just build their root and weight coordinates without worrying about trying to embed them into anything, instead studying their relationship to each other, and how the pairing works on them.

To begin with, the simple roots are coordinate vectors in the simple root basis, and the columns of the Cartan matrix in the fundamental weight basis. Dually, the simple coroots are coordinate vectors in the simple coroot basis, and the rows of the Cartan matrix in the fundamental coweight basis.
\alpha_i = \sum_j \delta_{ij} \alpha_j = \sum_j a_{ji} \varpi_j, \quad
\alpha_i^\vee = \sum_j \delta_{ij} \alpha_j^\vee = \sum_j a_{ij} \varpi_j^\vee, \quad
For brevity, from now on we call these the *root, weight, coroot*, and *coweight* bases. The pairings on these bases are:
\innprod{\alpha_i^\vee, \varpi_j} = \delta_{ij}, \quad
\innprod{\alpha_i^\vee, \alpha_j} = a_{ij}, \quad
\innprod{\varpi_i^\vee, \alpha_j} = \delta_{ij}, \quad
\innprod{\varpi_i^\vee, \varpi_j} = a_{ij}^{-1},
where a_{ij}^{-1} denotes the (i, j) entry of A^{-1}. Consequently the pairing between the weight and coweight bases is not integral, and we prefer to make use of the others.

When building the root systems, for each root and coroot we record it in both the weight and root (or coweight and coroot) bases simultaneously. There are two “games” that can be played to construct the root systems: the first is the *root reflection* game, and the second is the *root addition* game. The reflection game is important for us because if we play the games in the same order on both the root and coroot sides, then it constructs the root-coroot bijection. The addition game is important because it records data about how far each root is along each i-string, and constructs the root poset.

### The root reflection game

This is the “Weyl group centric” method of building roots, and could really be done for any finite Coxeter group (the difficulty for other finite Coxeter groups being that the Cartan matrix defining the reflections is not integral, and so becomes more difficult to work with). The fact that we have a crystallographic root system makes things very nice here.

Let W be the Weyl group: it is generated by the simple reflections r_i, where r_i(v) = v - \innprod{\alpha_i^\vee, v} \alpha_i for any v \in V. Dually, we have r_i^\vee(f) = f - \innprod{f, \alpha_i} \alpha_i^\vee for any f \in V^*. From now on we call both these reflections by the name r_i and hope that which kind of vector we are reflecting is clear from context. We can write down how these reflections behave on the root and weight basis elements: r_i(\alpha_j) = \alpha_j - a_{ij} \alpha_i, \quad r_i(\varpi_j) = \varpi_j - \delta_{ij} \alpha_i. On the root basis, the simple reflection r_i only modifies the ith coordinate: if \lambda = \sum_j x_j \alpha_j in the root basis then x = \begin{pmatrix} x_1 \\ \vdots \\ x_i \\ \vdots \\ x_n \end{pmatrix} \xto{r_i} \begin{pmatrix} x_1 \\ \vdots \\ x_i - x \cdot A_{i, -} \\ \vdots \\ x_n \end{pmatrix}, where x \cdot A_{i, -} denotes the dot product of the vector x with the ith row of the Cartan matrix (which is the simple coroot \alpha_i^\vee in the coweight basis). (TODO: this has a lovely interpretation in terms of Dynkin diagrams). In the weight basis on the other hand, we have if \lambda = \sum_j y_j \varpi_j then y = \begin{pmatrix} y_1 \\ \vdots \\ y_i \\ \vdots \\ y_n \end{pmatrix} \xto{r_i} y - y_i A_{-, i}, where A_{-, i} this time represents the ith column of the Cartan matrix (which is the simple root \alpha_i in the weight basis).

Armed with the knowledge of how to reflect column vectors, we start off the game having only the simple roots. To each of these starting points, we try to apply the n possible reflections. Three possiblities occur:

- We produce a new vector we have not yet seen before. This is a new root of the root system, and we record it.
- We produce a vector that we have seen before. This is again a root of the root system, but since it’s already recorded we can ignore it.
- We produce a negative root (a root whose coordinates in the root basis are all \leq 0). This only happens for r_i applied to \alpha_i, in fact. We ignore these.

Playing this game, building roots layer by layer, produces the whole positive root system. If we play the game in the same order on the dual system, we produce the bijection \alpha \mapsto \alpha^\vee between roots and coroots. The layer at which we uncover a new root is called the *depth* of that root: the minimal number of simple reflections needed to transform the root into a negative root, all the simple roots having depth 1.

### The root addition game

This is the “Lie algebra centric” method of building roots. We know that the root system (along with a Cartan subalgebra of dimension n) is the adjoint representation of a semisimple Lie algebra, and so in particular we can look at this Lie algebra’s restriction to the copy of \mathfrak{sl}_2 corresponding to each simple root i, henceforth called the i-subalgebra. This is a finite-dimensional representation of \mathfrak{sl}_2 and so breaks up into i-strings.

Each root in the root system is a member of the i-string (perhaps the i-string has length zero, in which case it is a member of a boring string). We will label the lengths of i strings by their highest weight, which records the number of edges in the string. For each root \alpha and each i, we need to know:

- How long is the i-string passing through \alpha, and
- What position is \alpha along this string?

We will record a root’s position along its string by the \mathfrak{sl}_2-weight, so for example if \alpha is in weight 2 on a 1-string then

(DIAGRAM HERE)

If \alpha is not at the top of its i-string, then \alpha + \alpha_i is also a root, and we can use this fact to determine the “legal moves” which will produce more roots. In addition to keeping track of each root in the root and weight basis (the ith coordinate in the weight basis gives us the position along the i-string), we need to keep track of the highest weight of the i-string for each root: another vector of length n to store with each root.

- \alpha_i is at the top of its i-string, since \alpha_i + \alpha_i is not a root. Also, the i-string passing through \alpha_i has length 3 (since -\alpha_i is a root, but -2\alpha_i is not).
- If j \neq i, then \alpha_i is at the bottom of its j-string, since \alpha_i - \alpha_j is not a root (all roots are either positive or negative).

We then proceed layer by layer to build the whole positive root system. Each root \alpha can be “promoted” to the next layer up for each i such that \alpha is not at the top of its i-string. Once promoted, we know that \alpha + \alpha_i lies on the same i-string as \alpha which has the same length: we record this length. There may be another way to get to \alpha + \alpha_i from the same layer, say \beta + \alpha_j, in which case we also fill in the j-string length coming from \beta.

After filling in a layer, there may be string lengths which are not recorded since they came from nothing below. For example if \beta is on the new layer, and the i-string length for \beta is unrecorded, then we may assume that \beta is at the bottom of its i-string, and fill in the string length data from the i-weight of \beta.

After this game finishes, the whole positive root system is constructed. If the positive root system is indecomposable, then the resulting poset has a unique maximal element, which is the *highest root*.

### Norms of roots

The last piece of data we need is the *norm* (\alpha, \alpha) of a root. We know that if the root system is indecomposable, then there exists a unique W-invariant inner product on V. Hence we can define a canonical inner product on V by declaring that (\alpha, \alpha) = 2 for all of the short simple roots.

In finite type, there are at most 2 lengths of root in each indecomposable system. If we can figure out what the norms of the simple roots should be we will be done: the norms of all the other roots can be calculated directly from the pairing matrix, or transferred upwards from the simple roots while playing the reflection game.

The W-invariant inner product should satify \innprod{\alpha_i^\vee, -} = 2 \frac{(\alpha_i, -)}{(\alpha_i, \alpha_i)} for each i. Playing around with this identity leads to \frac{(\alpha_i, \alpha_i)}{(\alpha_j, \alpha_j)} = \frac{\innprod{\alpha_j^\vee, \alpha_i}}{\innprod{\alpha_i^\vee, \alpha_j}} = \frac{a_{ji}}{a_{ij}}, which gives the ratio of lengths for any two simple roots which are adjacent in the Dynkin diagram. Since we have fixed the smallest roots to have square length 2, the lengths of the other roots can be extrapolated from this ratio; for example by classifying the Dynkin diagram into simply-laced components with an arrow between them.

Let D be the diagonal matrix with entries (\alpha_i, \alpha_i)/2, then DA is a symmetric matrix representing the inner product just defined. Almost anything we do with this inner product will be scaling-invariant, so we won’t worry about fixing notation too much with it until later on: this is the “gist” of how to create a W-invariant inner product.

## The Weyl group

- Formula for the order of the Weyl group in terms of highest root
- How to adapt the formula for decomposable systems, and parabolic subgroups.
- Longest word
- Enumerated Coxeter group
- Word Coxeter group

## Affinisation

- Untwisted affine Cartan matrix
- Affine reflection matrices for the affine Weyl group
- Coordinates for the affine Weyl group
- Plotting the affine Weyl group (how to go from a positive-definite Gram matrix to some vectors in \bbR^n?)

## Dimensions and multiplicities of irreps

- Weyl dimension formula (not much to say)
- Dominant characters via Freudenthal’s formula
- Tensor product multiplicities via the generalised tensor identity

## Working with integral matrices

- Hermite normal form
- Inverses and adjugates
- Smith normal form