Applied Representation Theory

Notation and Basic Definitions

We will collect any definitions and notational quirks and questions that folks have from the lectures here. It may start out a bit disorganized, but will eventually take reasonable shape as the questions roll in. We’ll also provide some links to external context and ideas. Since the course rides the line between physics and mathematics, no doubt there will be questions. it’s totally normal and encouraged!

Basic Set Theory

A set is a collection of objects subject to a few axioms that will typically not concern us. Finite sets are straight forward. The axioms are primarily involved with infinite sets. The Zermelo-Fraenkel axioms, extended with the axiom of choice is the de facto standard for most applications in mathematics, although work in this area is ongoing. (See also these two popular accounts).

When being explicit about sets, we usually write them with braces. For example, you might consider the set

$$A = \left\{a,b,c\right\}\quad \mathrm{or} \quad B = \left\{1,2,3\right\}.$$

The cardinality of a set is the number of elements it has. For finite sets, this is sometimes called the order of a set. For infinite sets, the idea of cardinality gets pretty wonky and we have to sometimes have to start worrying about things like the axiom of choice.

Sets are often defined in a prescriptive manner. Notationally this is written as

$$S = \left\{ a \;\Big|\; \mathrm{some\,condition\,on\,} a\right\},$$

which essentially reads:

“the set S equals the set of all $a$ such that $a$ is subject to some condition.”

For example, the set $X$ of even numbers less than 50 would be written:

$$X = \left\{ x \in \mathbb{N} \;\Big|\; x < 50\; \mathrm{and}\; x\;\mathrm{even}\right\}.$$

Here we’ve added a bit of implicit grammar, moving part of the “predicate” to the “subject” by saying

“The set $X$ is the set of all $x$ in the natural numbers such that x is less than 50 and also even.”

This is normal. It’s equivalent to saying

$$X = \left\{ x\;\Big|\; x \in \mathbb{N} \; \mathrm{and} x < 50\; \mathrm{and}\; x\;\mathrm{even}\right\}.$$

That weird fork symbol, $\in$ means “in” or “belongs to” or “is a member of”. As in $x$ is in the set $\mathbb{N}$.

We’ve also implicitly used the definition of $\mathbb{N}$ - the natural numbers - which are all the positive integers that are not negative. In particular, zero is in $\mathbb{N}$. There are other symbols we’ll use a lot too. $\mathbb{Z}$ is the set of integers. The rational numbers $\mathbb{Q}$ are given by

$$\mathbb{Q} = \left\{ \frac{a}{b} \;\Big|\; a,b \in \mathbb{Z}\right\}.$$

We also have the real numbers $\mathbb{R}$ and the complex numbers $\mathbb{C}$. Defining those as sets is where all that trouble with infinity starts to arise.

A subset is exactly what you think it is, a restriction of some set to a “smaller” collection of elements. If $V$ is a subset of $W$ we write,

$$V\subset W.$$

Mathematicans are pedantic about what smaller means and as a result we have the entire field of set theory. Note that sometimes they’re so pedantic that they’ll allow a set to be a subset of itself.

There’s also the empty set, which has no elements at all, $\emptyset$

A function is a map between sets that takes objects of one set to objects of another. We write

$$f : A\rightarrow B,$$

for a function that takes items of a set $A$ to items of a set $B$. $A$ is typically called the domain of f, and $B$ the range of f, in this context. Category theorists are obsessed with duality and will sometimes call the range the codomain. A function is said to be injective or “one to one” if this mapping uniquely pairs items in $A$ with items in $B$.

It’s typically assumed that every item in $A$ is acted upon by $f$. Meaning for each $a$ in $A$, there is an $f(a)$ in $B$. The converse of that statement is not typically assumed. For example, if $A$ is $\mathbb{N}$ and $f(x) = 2x$, $f$ will only map $\mathbb{N}$ to even numbers. To distinguish between items in $B$ that have a corresponding element in $a$, we define the subset $f[A]\subset B$,

$$f[A] = \left\{ b \in B \;\Big|\: f(a) = b,\;\mathrm{for\,some\,}a\in A\right\}.$$

This subset is called the image of $A$ under $f$. Sometimes it makes sense to talk about the dual notion of a preimage,

$$f^{-1}[B] = \left\{ a \in A \;\Big|\: f(a) = b,\;\mathrm{for\,some\,}b\in B\right\}.$$

This might be a weirder concept to grasp, but you might be interested in understanding, for example, how the preimage emerges as a subset of $A$ from of a subset of the range $B$.

If the image of a function $f$ happens to be the entire range,

$$f[A] = B,$$

then we say that $f$ is surjective. Any function that is both injective and surjective is said to be bijective. A bijective map means that two sets are effectively identical - at least as far as the theory and practice of sets is concerned. If a bijection - a bijective function - exists between two sets, we also call those two sets bijective to each other.

For example, if $A = \{a,b,c\}$ and $B = \{1,2,3\}$, the function

$$f(a) =1,\quad f(b) = 2,\quad, f(c) = 3,$$

is a bijection and these two sets are bijective. We don’t care about what’s inside, all that matters is we can make one to look like the other.

Sets are the skeleton upon which much of modern mathematics is built. Fancier things like groups or algebras are defined to be sets with additional properties. The notion of a bijection typically extends to these newer, fancier things, by including those other properties.

To make sure you understand the basics of set theory, you might consider proving the following proposition:

Proposition: Every finite is bijective with a subset of $\mathbb{N}$.