Representation theory facts for kids

Representation theory helps us understand how math objects "act" on other things. For example, it shows how the symmetries of a hexagon (like flipping or turning it) change the hexagon itself.

Representation theory is a part of mathematics that helps us study complex math structures, like groups or algebras, in a simpler way. It does this by "representing" their elements as linear transformations, which are like special kinds of changes or movements, in a vector space. Think of it like giving these abstract math objects a "role" to play or a "picture" to show how they behave.

Imagine you have a group of numbers that you can multiply together. Representation theory lets you turn these numbers into matrices (which are grids of numbers). When you multiply the original numbers, it's like multiplying their corresponding matrices. This is helpful because matrices and their operations are much easier to work with and understand.

This field of math is very useful because it turns hard problems in abstract algebra into easier problems in linear algebra. Linear algebra is a part of math that deals with vectors, spaces, and transformations, and it's very well understood.

Representation theory is used in many areas of mathematics and physics. For example, in physics, it helps describe how the symmetries of a physical system affect how that system behaves. It also helps with Fourier analysis (breaking down complex waves into simpler ones) and connects to geometry and number theory.

There are many ways to study representation theory, using tools from different math fields like algebraic geometry and topology.

What is a Representation?
Different Kinds of Representation Theory
Beyond Basic Representations
See also

What is a Representation?

Let's imagine a space called V, which is a vector space. You can think of it as a grid or a space where you can move things around, like the 3D space we live in. In representation theory, we often use spaces like Rⁿ or Cⁿ, which are standard spaces made of column vectors (lists of numbers).

The main idea is to turn abstract math objects into n × n matrices (square grids of numbers).

There are three main types of math objects that representation theory focuses on:

Groups: A group is a set of things (like numbers or movements) that you can combine (like adding or multiplying) and always get another thing in the set. The set of all invertible n × n matrices forms a group when you multiply them. So, the representation theory of groups shows how to describe group elements using these matrices.
Associative Algebras: These are like groups but with more operations, similar to how numbers can be added and multiplied. All n × n matrices form an associative algebra with matrix addition and multiplication.
Lie Algebras: These are special kinds of algebras where the multiplication rule is a bit different, often related to the "commutator" (like MN − NM for matrices).

This idea works for any vector space V and any field (a set of numbers where you can add, subtract, multiply, and divide). Instead of matrices, we use linear maps, which are functions that transform vectors in a linear way.

How Representations Work

There are two main ways to think about a representation:

Action: This way focuses on how the elements of a group or algebra "act" on the vectors in the space V. Imagine a group G and a vector space V. A representation is a rule that tells you how each element g from G moves or changes any vector v in V. It's like a special kind of movement where:

* The movement is always "linear" (it keeps lines straight and doesn't bend the space). * If you apply two group elements one after another, it's the same as applying their combined group operation.

Mapping: This way focuses on a function that sends each element of the group or algebra to a specific linear map. So, for a group G, a representation is a function that takes an element g from G and gives you a linear map that transforms vectors in V. This map must follow the rules of the group's operations.

Important Terms

Representation Space: This is the vector space V where the representation "lives" or "acts." Its dimension (how many independent directions it has) is also called the dimension of the representation.
Faithful Representation: This is a special kind of representation where every different element of the group or algebra gets a different linear map. It means the representation truly shows all the unique properties of the original object.

Equivalent Representations

Imagine you have two different representations, but they essentially give you the same information about the group. These are called isomorphic representations or equivalent representations. It's like having two different maps of the same city; they look different, but they show the same streets and buildings. In representation theory, we often try to classify representations "up to isomorphism," meaning we group together all the representations that are essentially the same.

Building Blocks of Representations

Subrepresentations: Sometimes, a smaller part of the representation space V (a subspace) is also preserved by the group's actions. This smaller part is called a subrepresentation.
Irreducible Representations: These are the simplest possible representations. They cannot be broken down into smaller, non-trivial subrepresentations. Think of them as the "atoms" or "building blocks" of all other representations.
Reducible Representations: If a representation can be broken down into smaller subrepresentations, it's called reducible.

For many groups, especially finite groups, all representations can be built by combining irreducible representations. This is like saying all complex molecules are made of simpler atoms.

Direct Sums: If you have two representations, you can combine them into a larger one called a direct sum. This new representation doesn't give you more information than the two original ones.
Indecomposable Representations: These are representations that cannot be broken down into a direct sum of smaller, non-trivial representations.

Combining Representations

You can also combine representations using something called a tensor product. If you have two representations of a group, you can create a new, larger representation on a "tensor product" space. This is a more advanced way of combining them. For example, in the representation theory of the group SU(2) (important in quantum mechanics), combining two representations with labels like l₁ and l₂ results in a new representation that can be broken down into a sum of simpler representations.

Different Kinds of Representation Theory

Representation theory has many different branches, depending on what kind of math object you're studying and what kind of space you're using.

Finite Groups: These are groups with a limited number of elements. Their representations are very important for understanding their structure. For example, for finite groups, representations can often be understood using "character theory," which assigns a special number (a character) to each group element.

* A cool fact: For finite groups, if you're working with numbers like real or complex numbers, any representation can be made to preserve an "inner product," which is like a way to measure distances and angles in the vector space. This makes them "unitary" representations.

Modular Representations: These are representations of finite groups over special fields of numbers where some of the usual math rules (like dividing by the group's size) don't apply. This area is more complex but very important in advanced group theory.

Unitary Representations: These are representations where the linear maps are "unitary operators," meaning they preserve distances and angles in the vector space. They are extremely important in quantum mechanics because they describe how symmetries affect quantum systems.

Lie Groups: These are groups that are also "smooth manifolds," meaning they look like smooth surfaces or curves. Many groups important in physics and chemistry are Lie groups. Studying their representations helps us understand things like the symmetries of particles.

Lie Algebras: These are closely related to Lie groups and can be thought of as the "infinitesimal symmetries" of Lie groups. Their representation theory is also very important in physics and geometry.

Invariant Theory: This branch studies how groups act on geometric objects and looks for things that don't change, or are "invariant," under these actions. For example, it might look for polynomial functions that stay the same even after a transformation.

Automorphic Forms: These are special functions that have certain symmetry properties. They are very advanced and connect representation theory to number theory, especially in the Langlands program, which tries to find deep connections between different areas of mathematics.

Associative Algebras: Representations of these algebras are closely linked to the idea of "modules" in algebra. Many results in representation theory can be seen as special cases of how modules behave.

Beyond Basic Representations

Representation theory can be generalized even further:

Set-Theoretic Representations: Instead of acting on a vector space, a group can act on a simple set of objects, moving them around. This is also called a "group action" or "permutation representation."

Representations in Other Categories: In advanced math, a "category" is a collection of objects and the ways they relate to each other. Representation theory can be extended to show how groups "act" within other types of categories, not just vector spaces or sets.

Representations of Categories: You can even study representations of more general "categories" themselves, not just groups. A simple example is the representation theory of "quivers," which are like directed graphs. This helps solve complex problems by turning them into simpler ones.