# Introduction

This chapter gives functionality for direct sums of *highest weight representations* (or modules). This is an important class of representations of (almost) semisimple Lie algebras (Chapter [Lie Algebras](../LieAlgebras/index-lie-algebras.md#chapalglie)) and connected reductive algebraic groups (Chapter [Groups of Lie Type](../ChevalleyGroups/index-chevalley-groups.md#chapgrplie)). This class includes all finite dimensional representations if the base field is the complex field.

The representations we are considering are in bijection with sets of dominant weights with multiplicities. Such sets are called *decomposition multisets*. Many interesting computations in representation theory can be done combinatorially with weight multisets, without the need to construct the module itself. Examples of the things we can compute include: module dimension, the multiset of all the weights, and decomposition multisets for symmetric powers, alternating powers, and tensor products. We can also restrict a decomposition multiset to a subgroup or induce it to a supergroup.

The code for such combinatorial computations is based on the LiE software package [[van Leeuwen *et al.*, 1992](../../references.md#cite-leeuwen-cohen-lisser)]. The algorithms for computing the actual representations are from [[de Graaf, 2001](../../references.md#cite-wdg-lie-hwrep)] in the Lie algebra case, and from [[Cohen *et al.*, 2004](../../references.md#cite-cohenmurraytaylor)] in the group case.

## Highest Weight Modules

This introduction is inspired by the LiE manual [[van Leeuwen *et al.*, 1992](../../references.md#cite-leeuwen-cohen-lisser)].

First consider connected reductive Lie groups over the complex field. If $G$ is a connected reductive complex Lie group, then it is a homomorphic image $G=\xi(G')$, where $\xi$ is a Lie-group homomorphism with finite kernel and $G'$ is the direct product of a simply connected group and a torus. Recall that a simply connected group is a direct product of *simple* simply connected groups. In particular, such groups are determined by their Cartan name and the dimension of the torus. For example, we denote the direct product of the group of type $A_4 C_3 B_2$ with a two dimensional torus by $A_4 C_3 B_2 T_2$. Most of the code ported from LiE works only for groups of this form. Similar terminology is used for the root datum corresponding to a group.

Connected reductive complex Lie groups have a very pleasing representation theory:

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Every module decomposes as a direct sum of irreducible representations.

**–**
The (finite dimensional) irreducible representations correspond to dominant weights.

It follows that representations correspond to finite sets of dominant weights with multiplicity. These multisets are called *decomposition multisets*. We can use this classification to do useful computations about representations, without having to explicitly construct them.

Multisets of weights can be used for other purposes as well: The multiset of all weights occurring in a module $M$ is called the *character multiset*. Since the Weyl group permutes the weights occurring in the character of $M$, it suffices to consider only the dominant weights with their multiplicities. This is called the *dominant character multiset*. In the LiE system, multisets of weights are represented by polynomials: for example, the decomposition multiset is called a decomposition character.

When using the functions in this section, it is important to keep track of which kind of multiset you are using. For example, if you input a decomposition multiset to a function that expects a dominant character multiset, the output is meaningless.

We often abbreviate decomposition multiset to *decomposition*, and similarly for character multisets. Write $R_D$ for the root datum of the group of the decomposition $D$. Denote the irreducible module for the group with root datum $R$ with highest weight $v$ by $V^R_v$, or to $V_v$ if $R$ is clear from the context.

It is often useful to define consider *virtual* multisets, which allow weights to have negative multiplicities. We call a virtual multiset *proper* if its weights all have nonnegative multiplicities. A decomposition corresponds to an actual module if and only if it is proper.

## Toral Elements

Many functions use a special syntax for finite-order elements of the torus of a Lie group $G$ (we are rarely interested in infinite-order elements). Recall that a weight is in fact a mapping from the torus $T$ to $C^\star$, and thus a weight $\lambda$ can be evaluated at an element $t \in T$. The resulting element is written $t^\lambda$. A set of fundamental weights $\omega_1, \ldots, \omega_r$ has the property that any element $t \in T$ is uniquely determined by the values $t^{\omega_1}, \ldots, t^{\omega_r}$. Therefore, we may represent $t$ as a vector $(a_1, \ldots, a_r, n)$, with the property that $t^{\omega_i} = e^{2 \pi i a_i / n} = \zeta_n^{a_i}$, where $\zeta_n = e^{2 \pi i / n}$ is the canonical $n$-th root of unity. An example of a function which uses this syntax for toral elements is [`Spectrum`](representation-properties.md#function-spectrumrvt). This function also provides a means to convert toral elements into a more natural form: see Example [Example: Spectrum](representation-properties.md#example-ex-66b7b0).

## Other Highest Weight Representations

Magma can also construct highest weight representations for:

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(Almost) reductive Lie algebras (Chapter [Lie Algebras](../LieAlgebras/index-lie-algebras.md#chapalglie)); and

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Split groups of Lie type (Chapter [Groups of Lie Type](../ChevalleyGroups/index-chevalley-groups.md#chapgrplie)).

If the base field has positive characteristic, highest weight representations are indecomposable, but not necessarily irreducible. In some cases there are irreducible representations which are not highest weight representations.

For groups of Lie type, we consider projective representations (i.e., homomorphisms to a projective general linear group). Suppose $G$ is a split group of Lie type defined over the field $k$ and $r$ is the least common multiple of the nonzero abelian-group invariants of the coisogeny group of $G$ (see Section [Isogeny of Split Reduced Root Data](../RootData/introduction.md#subsectrdisogeny)). Let $K$ be an extension of $k$ containing at least one $r$th root of each element of $k$ (i.e., $K$ contains a Kummer extension). Then highest weight representations are projective representations defined over $K$, and are constructed using polynomial functions and $r$th roots.

If $k$ already contains all $r$th roots, then no extension is needed and the representation will be linear rather than projective. This happens when $r=1$, i.e., the coisogeny group is torsion free. This includes direct products of a simply connected group and a torus. The general linear group also has this property. It also happens when $k$ is the complex field or field of algebraic numbers, when $k$ is the real field and $r$ is odd, and when $k$ is finite and $|k|-1$ is coprime to $r$.

The functions give a warning when the representation is not linear, but this can be avoided using the optional parameter `NoWarning`. Note that an appropriate extension $K$ can be constructed for all fields other than rational function fields, fields of Laurent series, and local fields. In these cases, as well as for nonfields, the representations can only be computed when $r=1$.