Intereting Posts

Probability of opening all piggy banks
$A \subseteq \mathbb R^n $ s.t. for every continuous function $f : A \to \mathbb R$ , $f(A)$ is closed in $\mathbb R$ , is $A$ closed $\mathbb R^n$?
How to evaluate $\int \cot^2(x) \;\mathrm dx$?
Easy proof for existence of Lebesgue-premeasure
How to find the probability of truth?
A trigonometric inequality: $\cos(\theta) + \sin(\theta) > 0$
process of finding eigenvalues and eigenvectors
Possibly rotated parabola from three points
Proving Binomial Identity without calculus
Exercise review: perpendicular-to-plane line
Why does $\sum a_i \exp(b_i)$ always have root?
Atiyah and Macdonald, Proposition 2.9
Paradox as to Measure of Countable Dense Subsets?
Do all polynomials of even degree start by decreasing as you plot from $-\infty$ upward?
totient function series diverges?

Let $S_n$ be the group of permutations of $\{1, 2, \ldots, n\}$. A “character” for $S_n$ is a function $\chi\colon S_n \to \mathbb{C} \setminus \{0\}$ with $\chi(ab) = \chi(a)\chi(b)$ for all $a, b \in S_n$ [NOTE: this is not 100% accurate – see comments and answers below (OP)]. What is an *irreducible* character? I have searched the Web, and all the references I found were inadequate or told me more than I wanted to know.

- Sorting numbers in a matrix by moving an empty entry through other entries is not always possible .
- For which numbers there is only one simple group of that order?
- Can we ascertain that there exists an epimorphism $G\rightarrow H$?
- Transitive subgroup of symmetric group
- Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.
- Unique subgroup of index 2 in a finite abelian group.
- Nilpotent groups are solvable
- Question on groups of order $pq$
- Why is $S_5$ generated by any combination of a transposition and a 5-cycle?
- Finite group with unique subgroup of each order.

The homomorphisms $G\to \Bbb C^\times$ are actually not the *whole* story of character theory, but are a very tidy chapter in it. If $V$ is a vector space (over $\Bbb C$) and $G$ finite, the homomorphisms $G\to GL(V)$ from $G$ into the general linear group of invertible linear maps are called *representations*, which are essentially the ways to equip $V$ with a linear $G$ action. If $\rho$ is a representation, then the map given by $\chi_\rho:G\to \Bbb C:g\mapsto\mathrm{tr}\,\rho(g)$ (the trace of the linear map associated to $g$, which is independent of basis or coordinate choice for $V$) is called a *character* of $G$.

If $V$ is one-dimensional (in which case we call $\rho$ and $\chi_\rho$ one-dimensional as well) then $\rho=\chi_\rho$ and the characters are multiplicative. Note that $\mathrm{tr}\,\rho(e_G)=\dim\,V$ shows the dimension can be directly computed from the character, so there is no ambiguity with respect to what dimension a character may have. With a distinguished basis we have $V\cong \Bbb C^n$ in an obvious way, and so we can write $GL(V)$ as $GL_n(\Bbb C)$, in which case we are working with *matrix* representations specifically.

If there is a proper nontrivial subspace $W\subseteq V$ that is invariant under the linear $G$-action, that is if we have $\rho(g)W\subseteq W$ for each $g\in G$, then the restriction of $G$’s linear action to $W$ then forms a *subrepresentation* of $V$. If a representation $\rho:G\to GL(V)$ has *no* subrepresentations, it is called an *irreducible* representation, in which case $\chi_\rho$ is also called irreducible. This is well-defined because (over any field of characteristic zero) representations are uniquely determined by their characters, which again prevents any sort of ambiguity or conflicting descriptions from arising.

A one-dimensional space has no proper nontrivial subspaces, so one-dimensional representations and characters (the ones you are talking about) are all automatically irreducible. Using the very nifty Schur’s lemma, one can see that representations of abelian groups decompose into a direct sum of one-dimensional representations and thus are irreducible precisely when they are one-dimensional (you may encounter Schur’s and direct sums later).

I can recommend this excellent note on the representation theory of the symmetric groups and Young tableaux, where the irreducible representations are obtained through Specht modules. To understand this fully you will need relatively good familiarity with representation theory, algebra, combinatorics and permutations, so maybe come back to it later. (Also for other readers.)

Representation theory doesn’t need to be done in $\Bbb C$, we can work in arbitrary fields. However, as I hinted at a moment ago, if the characteristic divides the order of the finite group $G$ then things get complicated and we are working in modular representation theory. Things also get complicated when $G$ is a topological group, in which case we need theory from abstract harmonic analysis.

There are two somewhat conflicting definitions of characters, and the one you cited is not the relevant one.

- A character $\chi$ of a linear representation $\pi$ of a group $G$ is its composition with trace $\operatorname{tr}\circ \pi$.
- A character of a group is a character of any its representation.
- An irreducible character is the character of an irreducible representation.

What you cited is a multiplicative character. (Over $\bf C$, not sure how it is in general), multiplicative characters are exactly the characters of one-dimensional representations, and they are irreducible.

So a multiplicative character is an irreducible character, but not the other way around: any $S_n$ has a natural irreducible $n-1$-dimensional representation.

Worth noting, iirc, the only multiplicative characters of a symmetric group are the trivial one and the sign, so they are rather irrelevant in this case.

On the other hand, for abelian groups, irreducible characters are exactly the one-dimensional (so multiplicative) characters (that’s why in the context of abelian groups, commutative Fourier analysis etc., the definition you cited is usually used).

- proving of Integral $\int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}dx = \ln\left(\frac{a}{b}\right)$
- Reinventing The Wheel – Part 1: The Riemann Integral
- Difference between complete and closed set
- A 10-digit number whose $n$th digit gives the number of $(n-1)$s in it
- Covergence test of $\sum_{n\geq 1}{\frac{|\sin n|}{n}}$
- Existence and value of $\lim_{n\to\infty} (\ln\frac{x}{n}+\sum_{k=1}^n \frac{1}{k+x})$ for $x>0$
- Surjective function into Hartogs number of a set
- Prove: $a_n \leq b_n \implies \limsup a_n \leq \limsup b_n$
- Brownian bridge expression for a Brownian motion
- Proof of $\sqrt{n^2-4}, n\ge 3$ being irrational
- Cross product of two vectors, given magnitudes and angle
- A Ramanujan infinite series
- Boundedness of functions in $W_0^{1,p}(\Omega)$
- Converting triangles to isosceles, equilateral or right???
- Characterisation of the squares of the symmetric group