Group Theory and Representation

A group is the mathematical encoding of symmetry — the set of invertible transformations of an object closed under composition. The concept arose in Évariste Galois’s 1832 night-before-his-duel manuscript on the solvability of polynomial equations and crystallised through the nineteenth and early twentieth centuries into the central organising idea of modern algebra. Today groups are the bedrock of every place symmetry matters: Galois theory of field extensions, the classification of crystals, the gauge groups of particle physics, the symmetries of differential equations, robotics kinematics, and the algorithms that detect equivalent molecules, knots, and graphs. Representation theory — the study of how groups act linearly on vector spaces — turns abstract symmetry into the concrete language of matrices, characters, and harmonic analysis, and was the engine of Élie Cartan’s classification of Lie algebras, Hermann Weyl’s reformulation of quantum mechanics, and the contemporary geometric Langlands program. This note covers the group-theoretic foundations through the classification of finite simple groups, the structure theory of solvable, nilpotent, and abelian groups, free groups and presentations, group cohomology, and the representation theory of finite and compact Lie groups including the character theory of $S_n$, the Peter-Weyl theorem, and applications to crystallography, particle physics, and Galois theory.

See also

• linear-algebra-essentials — representations are linear actions; matrix groups; characters live in ${\mathbb{C}}^n$.
• algebraic-geometry-foundations — algebraic groups, homogeneous spaces, geometric Langlands.
• lie-groups-so3-se3 — Lie groups used in robotics and physics; the present note covers the abstract structure and reps.
• lie-group-catalog — catalog of specific Lie groups and their representations.
• number-theory — Galois groups and the inverse Galois problem; class field theory.
• graph-theory — Cayley graphs, expander families, automorphism groups.
• combinatorial-optimization — group-theoretic methods in counting (Burnside, Pólya).
• complex-analysis — modular group and modular forms.

1. Groups and subgroups

1.1 Definition

A group $(G,\cdot )$ is a set $G$ with a binary operation $\cdot :G\times G\to G$ such that:

1. Associativity: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
2. Identity: there is $e\in G$ with $e\cdot a=a\cdot e=a$ for all $a$.
3. Inverses: for every $a$ there is $a^{-1}$ with $a\cdot a^{-1}=a^{-1}\cdot a=e$.

If additionally $a\cdot b=b\cdot a$ for all $a,b$, the group is abelian (commutative).

1.2 Examples

• $(\mathbb{Z},+),(\mathbb{Q},+),(\mathbb{R},+),(\mathbb{C},+)$ — additive abelian groups.
• $({\mathbb{Q}}^{\times },\cdot ),({\mathbb{R}}^{\times },\cdot )$ — multiplicative.
• $(\mathbb{Z}/n\mathbb{Z},+)$ — cyclic group of order $n$, written $C_n$ or $\mathbb{Z}/n$.
• $S_n$ — symmetric group on $n$ letters; order $n!$.
• $A_n$ — alternating group of even permutations; order $n!/2$.
• $D_n$ — dihedral group of order $2n$, symmetries of a regular $n$-gon.
• ${\text{GL}}_n(k),{\text{SL}}_n(k),O(n),U(n),{\text{Sp}}_{2n}(k)$ — matrix Lie groups.
• $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$ — quaternion group of order 8 (non-abelian, order-2 centre).
• Free group $F_n$ on $n$ generators.

1.3 Order

The order $|G|$ is the cardinality of $G$; finite or infinite. The order of an element $a\in G$ is the smallest $n\ge 1$ with $a^n=e$, or $\infty$ if no such $n$.

1.4 Subgroups

A subset $H\subseteq G$ is a subgroup ($H\le G$) if it is closed under product and inverses (equivalently, $h_1{h}_2^{-1}\in H$). Always contains $e$.

Cyclic subgroup: $⟨a⟩=\{a^n:n\in \mathbb{Z}\}$ generated by $a$; isomorphic to $\mathbb{Z}$ if $a$ has infinite order, to $\mathbb{Z}/n$ if $a$ has order $n$.

1.5 Cosets and Lagrange’s theorem

For $H\le G$ and $g\in G$, the left coset $gH=\{gh:h\in H\}$. Cosets partition $G$ and all have the same cardinality $|H|$. The index $[G:H]$ is the number of cosets.

Lagrange’s theorem (Joseph-Louis Lagrange 1771): for finite $G$, $|G|=[G:H]\cdot |H|.$ Hence $|H|$ divides $|G|$, and the order of any element divides $|G|$.

Corollaries: groups of prime order are cyclic. Fermat’s little theorem is the special case in $(\mathbb{Z}/p\mathbb{Z}{)}^{\times }$.

2. Normal subgroups and quotients

2.1 Normal subgroups

$N\le G$ is normal ($N⊴G$) if $gN{g}^{-1}=N$ for all $g\in G$, equivalently $gN=Ng$ as sets.

Examples:

• $\{e\}$ and $G$ are always normal.
• All subgroups of abelian groups are normal.
• $A_n⊴S_n$ (index 2).
• ${\text{SL}}_n(k)⊴{\text{GL}}_n(k)$ (kernel of $\det$).
• The centre $Z(G)=\{a:ag=ga\text{ for all }g\}$ is normal.

2.2 Quotient groups

For $N⊴G$, the quotient $G/N$ has elements the cosets $gN$ and operation $(g_{1}N)(g_{2}N)=(g_1{g}_2)N$. Well-defined precisely because $N$ is normal.

Examples:

• $\mathbb{Z}/n\mathbb{Z}$ from $n\mathbb{Z}⊴\mathbb{Z}$.
• $S_n/A_n\cong \mathbb{Z}/2$.
• ${\text{GL}}_n(k)/{\text{SL}}_n(k)\cong k^{\times }$.
• $G/Z(G)=\text{Inn}(G)$ — inner automorphism group.

2.3 Simple groups

A group is simple if its only normal subgroups are $\{e\}$ and itself. The atoms of group theory under quotients.

Examples:

• Cyclic groups of prime order $C_p$.
• Alternating groups $A_n$ for $n\ge 5$.
• ${\text{PSL}}_n({\mathbb{F}}_q)$ for $n\ge 2$ (except ${\text{PSL}}_2({\mathbb{F}}_2),{\text{PSL}}_2({\mathbb{F}}_3)$).
• Sporadic simple groups (see §6.2).

3. Homomorphisms and isomorphism theorems

3.1 Group homomorphisms

A homomorphism $\phi :G\to H$ satisfies $\phi (ab)=\phi (a)\phi (b)$. Necessarily $\phi (e_G)=e_H$ and $\phi (a^{-1})=\phi (a{)}^{-1}$.

• Kernel: $\ker \phi =\{a\in G:\phi (a)=e_H\}$; a normal subgroup of $G$.
• Image: $\text{im}\phi$; a subgroup of $H$.
• Isomorphism: bijective homomorphism. Two groups are isomorphic ($G\cong H$) if there is an isomorphism between them.
• Automorphism: isomorphism $G\to G$. The set $\text{Aut}(G)$ is itself a group under composition.

3.2 First isomorphism theorem

$\phi :G\to H$ a homomorphism. Then $G/\ker \phi \cong \text{im}\phi .$ The fundamental construction reducing any homomorphism to a quotient followed by an injection.

3.3 Second isomorphism theorem

For $H\le G$ and $N⊴G$: $H/(H\cap N)\cong HN/N$.

3.4 Third isomorphism theorem

For $N⊴K⊴G$ with $N⊴G$: $(G/N)/(K/N)\cong G/K$.

3.5 Correspondence theorem

For $N⊴G$, subgroups of $G/N$ correspond bijectively to subgroups of $G$ containing $N$, preserving normality and indices.

4. Group actions

4.1 Definition

A left action of $G$ on a set $X$ is a map $G\times X\to X$ with $e\cdot x=x$ and $g\cdot (h\cdot x)=(gh)\cdot x$. Equivalently a homomorphism $G\to \text{Sym}(X)$.

4.2 Orbits and stabilisers

For $x\in X$:

• Orbit: $G\cdot x=\{g\cdot x:g\in G\}$.
• Stabiliser: $G_x=\{g:g\cdot x=x\}$, a subgroup of $G$.

Orbits partition $X$. The action is transitive if there is a single orbit; faithful if ${\bigcap }_x{G}_x=\{e\}$; free if all $G_x=\{e\}$.

4.3 Orbit-stabiliser theorem

$|G\cdot x|=[G:G_x].$ For finite $G$, $|G\cdot x|\cdot |G_x|=|G|$.

4.4 Burnside’s lemma (Cauchy-Frobenius)

For finite $G$ acting on finite $X$, the number of orbits is $|X/G|=\frac{1}{|G|}{\sum }_{g\in G}|X^g|$ where $X^g=\{x:g\cdot x=x\}$ is the fixed-point set.

Standard application: counting necklaces, colourings, isomorphism classes of small structures.

4.5 Pólya enumeration

George Pólya 1937 (Acta Mathematica 68): refines Burnside to count by weight/colour. The cycle index polynomial of a permutation group $G\le S_n$ is $Z_G(x_1,\dots ,x_n)=\frac{1}{|G|}{\sum }_{g\in G}{\prod }_k{x}_k^{c_k(g)}$ where $c_k(g)$ counts $k$-cycles of $g$. Substituting $x_k={\sum }_j{a}_j^k$ gives the generating function for $G$-orbits of colourings.

4.6 Class equation

For finite $G$ acting on itself by conjugation: $|G|=|Z(G)|+{\sum }_i[G:C_G(g_i)]$ where $C_G(g_i)=\{h:h{g}_i{h}^{-1}=g_i\}$ is the centraliser and the sum runs over conjugacy classes outside the centre. Used to prove the centre of a $p$-group is non-trivial.

5. Sylow theorems

Peter Sylow 1872, in Mathematische Annalen. For a finite group $G$ of order $p^{a}m$ with $\gcd (p,m)=1$:

5.1 First Sylow theorem

$G$ has a subgroup of order $p^a$ — a Sylow $p$-subgroup.

5.2 Second Sylow theorem

All Sylow $p$-subgroups are conjugate to each other.

5.3 Third Sylow theorem

The number $n_p$ of Sylow $p$-subgroups satisfies $n_p\equiv 1(\mathrm{mod}p)$ and $n_p\mid m$.

5.4 Applications

Sylow theorems are the workhorse for classifying groups of small order:

• Groups of order $p$: cyclic.
• Groups of order $p^2$: abelian, either $C_{p^2}$ or $C_p\times C_p$.
• Groups of order $pq$ ($p<q$, $p\nmid q-1$): cyclic. If $p\mid q-1$: also a non-abelian semidirect product $C_q⋊C_p$.
• Groups of order $\le 60$ classified explicitly. $A_5$ is the smallest non-abelian simple group ($|A_5|=60$).

5.5 $p$-groups

A $p$-group has order $p^k$ for some $k\ge 0$. Properties:

• Centre $Z(G)\ne \{e\}$ for any $p$-group with $|G|>1$.
• $p$-groups are nilpotent (see §7.3) and hence solvable.
• Every maximal subgroup is normal of index $p$.

6. Classification of finite simple groups

6.1 Statement

The Classification of Finite Simple Groups (CFSG) — one of the largest collaborative theorems in mathematics, completed in stages between 1955 and 2004 with corrections of “quasi-thin” cases by Aschbacher-Smith. Every finite simple group is one of:

1. Cyclic $C_p$ of prime order.
2. Alternating $A_n$ for $n\ge 5$.
3. Groups of Lie type: classical (e.g., ${\text{PSL}}_n,{\text{PSU}}_n,{\text{PSp}}_{2n},\text{P}{\Omega }_n^{ϵ}$) and exceptional ($G_2,F_4,E_6,E_7,E_8$, and twisted analogues), each over a finite field ${\mathbb{F}}_q$.
4. 26 sporadic groups, including the Monster $M$ of order $\approx 8\times 10^{53}$.

Total length: ~10,000-15,000 pages across many papers. A “second-generation” simplified proof is in progress (Gorenstein-Lyons-Solomon, ongoing; ~10 volumes).

6.2 The sporadic groups

The 26 sporadic groups split into the happy family (20 groups that are quotients of subgroups of the Monster) and 6 pariahs. The Monster has order $|M|=2^{46}\cdot 3^{20}\cdot 5^9\cdot 7^6\cdot 11^2\cdot 13^3\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot 59\cdot 71=808017424794512875886459904961710757005754368000000000$.

Monstrous moonshine (John Conway and Simon Norton 1979, conjecture; Richard Borcherds 1992, proof and Fields Medal 1998): the $j$-invariant Fourier coefficients are dimensions of natural Monster representations. The proof uses vertex operator algebras and ties the Monster to string theory.

6.3 Groups of Lie type

Built as $G({\mathbb{F}}_q)/Z$ for $G$ a simple algebraic group. Their representation theory (Deligne-Lusztig 1976 Annals of Math 103) is a paragon of geometric representation theory and was a key input to the Langlands program.

7. Solvable and nilpotent groups

7.1 Commutator subgroup

The commutator $[a,b]=ab{a}^{-1}{b}^{-1}$. The derived subgroup $[G,G]$ (or $G^{\prime }$) is the subgroup generated by all commutators. It is the smallest normal subgroup such that $G/[G,G]$ is abelian. The quotient $G^{ab}=G/[G,G]$ is the abelianisation.

7.2 Derived series and solvability

The derived series: $G^{(0)}=G$, $G^{(n+1)}=[G^{(n)},G^{(n)}]$. $G$ is solvable if $G^{(k)}=\{e\}$ for some $k$.

Equivalent: $G$ has a chain $\{e\}=G_0⊴G_1⊴\cdots ⊴G_n=G$ with each $G_{i+1}/G_i$ abelian.

Examples:

• All abelian groups (solvable with $k=1$).
• $S_3,S_4$ solvable.
• $S_n$ for $n\ge 5$ — not solvable.
• All $p$-groups (in fact nilpotent).
• Galois groups of cyclotomic extensions and abelian extensions.

Application: Galois’s theorem (1832): a polynomial equation is solvable by radicals iff its Galois group is solvable. The general quintic has Galois group $S_5$ (non-solvable), hence no general radical formula.

7.3 Lower central series and nilpotency

The lower central series: ${\gamma }_1(G)=G$, ${\gamma }_{n+1}(G)=[G,{\gamma }_n(G)]$. $G$ is nilpotent if ${\gamma }_k(G)=\{e\}$ for some $k$. Nilpotency class is the smallest such $k-1$.

Equivalent for finite $G$: nilpotent iff direct product of Sylow subgroups iff every Sylow subgroup is normal.

Nilpotent implies solvable (strict for $S_3$ etc.). $p$-groups are nilpotent. Abelian iff nilpotent of class $\le 1$.

7.4 Heisenberg group

A canonical example of nilpotent non-abelian group: $3\times 3$ upper-triangular matrices with $1$ on the diagonal over a ring $R$: $H=\left\{\left(\begin{array}{ccc}1& a& c\\ 0& 1& b\\ 0& 0& 1\end{array}\right):a,b,c\in R\right\}.$ Nilpotent of class 2 (centre is the $c$ entry). Appears throughout quantum mechanics (canonical commutation relations) and harmonic analysis on ${\mathbb{R}}^{2n+1}$.

8. Free groups and presentations

8.1 Free groups

The free group $F_S$ on a set $S$ is the set of finite reduced words in $S\cup S^{-1}$ under concatenation with cancellation. Universal property: every map $S\to G$ to a group extends uniquely to $F_S\to G$.

Subgroups of free groups are free (Nielsen-Schreier theorem 1921/1927). $F_2$ contains $F_n$ for every $n$ — and even $F_{\infty }$.

8.2 Presentations

Every group $G$ is a quotient $F_S/N$ for some set $S$ and normal subgroup $N$. Writing $N$ as the normal closure of a set $R$ of relators gives a presentation $G=⟨S\mid R⟩.$

Examples:

• $C_n=⟨a\mid a^n⟩$.
• $D_n=⟨r,s\mid r^n,s^2,(sr{)}^2⟩$.
• $S_n=⟨{\sigma }_1,\dots ,{\sigma }_{n-1}\mid {\sigma }_i^2,({\sigma }_i{\sigma }_{i+1}{)}^3,({\sigma }_i{\sigma }_j{)}^2\text{ for }|i-j|>1⟩$ (Coxeter presentation).
• Braid group $B_n=⟨{\sigma }_1,\dots ,{\sigma }_{n-1}\mid {\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1},({\sigma }_i{\sigma }_j)=({\sigma }_j{\sigma }_i)\text{ for }|i-j|>1⟩$ (Artin).

8.3 Word and isomorphism problems

Max Dehn 1911 posed three decision problems for finitely presented groups:

1. Word problem: given a word $w$ in generators, is $w=e$?
2. Conjugacy problem: are two words conjugate?
3. Isomorphism problem: are two presentations of the same group?

Pyotr Novikov 1955 / William Boone 1958: the word problem is undecidable in general (there exist finitely presented groups with no algorithm to decide $w=e$). All three problems are undecidable. Decidable for important classes: hyperbolic groups (Gromov 1987), automatic groups, residually finite groups (with algorithm).

8.4 Cayley graphs

The Cayley graph $\Gamma (G,S)$ for a generating set $S$ has vertex set $G$ and edges $\{g,gs\}$ for $s\in S$. $G$ acts on $\Gamma$ by left translation, freely and transitively on vertices.

Cayley graphs of ${\text{SL}}_2({\mathbb{F}}_p)$ with generators of small word length give expander graphs — sparse but highly connected. Critical in computer science (extractors, derandomisation) and group-theoretic property (T) (Kazhdan 1967).

9. Abelian groups

9.1 Fundamental theorem of finitely generated abelian groups

Every finitely generated abelian group $A$ is isomorphic to $A\cong {\mathbb{Z}}^r\oplus \mathbb{Z}/d_1\oplus \mathbb{Z}/d_2\oplus \cdots \oplus \mathbb{Z}/d_k$ with $d_1\mid d_2\mid \cdots \mid d_k$ — the invariant factor decomposition — or equivalently $A\cong {\mathbb{Z}}^r\oplus {⨁}_{p,i}\mathbb{Z}/p^{e_{p,i}}$ — the primary decomposition. The integer $r$ is the rank, and the orders are unique.

Special cases: finite abelian groups are direct sums of cyclic groups of prime-power order.

9.2 Smith normal form

Algorithmic. For any $m\times n$ matrix $M$ over a PID (e.g., $\mathbb{Z}$), there are invertible $P,Q$ with $PMQ$ diagonal with $d_1\mid d_2\mid \cdots$. Used to compute the structure of finitely generated abelian groups presented as ${\mathbb{Z}}^n/M{\mathbb{Z}}^m$.

9.3 Divisible groups

An abelian group $A$ is divisible if for every $a\in A$ and $n\ge 1$, the equation $nx=a$ has a solution. Examples: $\mathbb{Q},\mathbb{R},\mathbb{Q}/\mathbb{Z},{\mathbb{C}}^{\times }$. Structure: every divisible abelian group is a direct sum of copies of $\mathbb{Q}$ and Prüfer groups $\mathbb{Z}(p^{\infty })$.

10. Semidirect products and group extensions

10.1 Direct product

$G_1\times G_2$ with componentwise operation. Every element $(g_1,g_2)$ decomposes uniquely; both factors are normal subgroups; the factors commute.

10.2 Semidirect product

For $N$ a group, $H$ a group, and $\phi :H\to \text{Aut}(N)$ a homomorphism, the semidirect product $N{⋊}_{\phi }H$ has underlying set $N\times H$ and operation $(n_1,h_1)(n_2,h_2)=(n_1\cdot \phi (h_1)(n_2),h_1{h}_2).$

In a semidirect product, $N$ is normal, $H$ is a (not necessarily normal) subgroup, and $N\cap H=\{e\}$. Direct product is the case $\phi$ trivial.

Examples:

• $D_n=C_n⋊C_2$ where $C_2$ acts by inversion.
• $S_n=A_n⋊C_2$ (for $n\ge 2$).
• Affine group $\text{Aff}(n)={\mathbb{R}}^n⋊{\text{GL}}_n(\mathbb{R})$.
• Euclidean group $E(n)={\mathbb{R}}^n⋊O(n)$, motion group of ${\mathbb{R}}^n$. Crystallographic groups are discrete subgroups.

10.3 Group extensions and cohomology

A group extension is a short exact sequence $1\to N\to G\to Q\to 1$. Equivalence classes of central extensions (where $N\subseteq Z(G)$) are classified by $H^2(Q,N)$, the second group cohomology. Non-trivial extensions encode “obstructions” to splitting.

$H^1(G,A)$ classifies derivations modulo principal derivations, equivalently splittings of $A⋊G$. Foundational in Galois cohomology and class field theory.

10.4 Group cohomology primer

For $G$ a group and $A$ a $G$-module:

• $H^0(G,A)=A^G$ (invariants).
• $H^1(G,A)$ = derivations / inner derivations.
• $H^2(G,A)$ = extensions of $G$ by $A$.

Computational tools: bar resolution, group cohomology via classifying space $BG$, spectral sequences. Hochschild-Serre spectral sequence for $N⊴G$. See standard texts (Brown Cohomology of Groups, Weibel Introduction to Homological Algebra).

11. Representation theory: foundations

11.1 Linear representations

A (finite-dimensional, complex) representation of $G$ is a homomorphism $\rho :G\to \text{GL}(V)$ for a finite-dim complex vector space $V$. Equivalently, $V$ is a left $\mathbb{C}[G]$-module (where $\mathbb{C}[G]$ is the group algebra).

A morphism of representations is a $\mathbb{C}$-linear map $T:V\to W$ with $T{\rho }_V(g)={\rho }_W(g)T$ for all $g$.

11.2 Irreducible representations

A representation is irreducible if $V$ has no proper non-zero $G$-invariant subspace. Indecomposable if not a direct sum. For finite $G$ over $\mathbb{C}$, irreducible = indecomposable.

11.3 Schur’s lemma

Issai Schur 1905. For irreducible representations $V,W$ over $\mathbb{C}$: ${\dim }_{\mathbb{C}}{\text{Hom}}_G(V,W)=\{\begin{array}{ll}1& V\cong W,\\ 0& V\ncong W.\end{array}$ Consequence: the only $G$-equivariant endomorphisms of an irreducible $V$ are scalars.

11.4 Maschke’s theorem

Heinrich Maschke 1898. Let $G$ be a finite group and $k$ a field with $\text{char}(k)\nmid |G|$. Every finite-dim representation of $G$ over $k$ is a direct sum of irreducibles.

Proof: Given a subrepresentation $W\subseteq V$, average a projection over $G$ to get a $G$-equivariant projection onto $W$. The averaging requires $|G|$ to be invertible in $k$.

For $\text{char}(k)\mid |G|$, modular representations are not semisimple; modular representation theory is much harder.

11.5 Group algebra structure

For finite $G$, the group algebra $\mathbb{C}[G]$ decomposes as a direct product of matrix algebras: $\mathbb{C}[G]\cong {\prod }_i{M}_{d_i}(\mathbb{C})$ where $d_i$ are the dimensions of the irreducible representations. Hence $|G|={\sum }_i{d}_i^2.$

The number of irreducible representations equals the number of conjugacy classes.

12. Character theory

12.1 Characters

The character of a representation $\rho$ is ${\chi }_{\rho }(g)=\text{tr}(\rho (g))$. Characters:

• Are class functions (constant on conjugacy classes).
• $\chi (e)=\dim V$.
• $\chi (g^{-1})=\overline{\chi (g)}$ for unitary $\rho$.

12.2 Orthogonality of characters

For finite $G$ over $\mathbb{C}$, define an inner product on class functions: $⟨\chi ,\psi ⟩=\frac{1}{|G|}{\sum }_{g\in G}\chi (g)\overline{\psi (g)}.$ Then:

• First orthogonality: $⟨{\chi }_i,{\chi }_j⟩={\delta }_{ij}$ for irreducible characters.
• Second orthogonality: ${\sum }_i{\chi }_i(g)\overline{{\chi }_i(h)}=|C_G(g)|\cdot [g,h\text{ conjugate}]$.

12.3 Character table

Square table with rows indexed by irreducible characters and columns by conjugacy classes. Examples:

$S_3$ has three conjugacy classes (identity, transpositions, 3-cycles) and three irreducibles (trivial, sign, standard 2-dim):

	$e$	$(12)$	$(123)$
trivial $1$	1	1	1
sign	1	$-1$	1
std	2	0	$-1$

Dimensions check: $1^2+1^2+2^2=6=|S_3|$.

12.4 Decomposition

The number of copies of irreducible ${\chi }_i$ in $\chi$ is $⟨\chi ,{\chi }_i⟩$. A representation is determined by its character (over $\mathbb{C}$).

13. Representations of $S_n$ and $A_n$

13.1 Young diagrams and partitions

Irreducible representations of $S_n$ are indexed by partitions $\lambda ⊢n$ — non-increasing sequences ${\lambda }_1\ge {\lambda }_2\ge \cdots$ summing to $n$. Each partition is drawn as a Young diagram.

13.2 Specht modules

Wilhelm Specht 1935 constructed the irreducible representation $S^{\lambda }$ from Young tableaux. Dimension by the hook length formula (Frame-Robinson-Thrall 1954): $\dim S^{\lambda }=\frac{n!}{{\prod }_{c\in \lambda }h(c)}$ where $h(c)$ is the hook length of the cell $c$.

13.3 Murnaghan-Nakayama rule

Computes character values ${\chi }^{\lambda }(\mu )$ for partition $\mu$ as a sum over rim-hook tableaux: ${\chi }^{\lambda }(\mu )={\sum }_T(-1{)}^{ht(T)}.$

13.4 Schur-Weyl duality

Hermann Weyl 1939. Joint action of $\text{GL}(V)$ and $S_n$ on $V^{\otimes n}$ decomposes: $V^{\otimes n}={⨁}_{\lambda ⊢n,\ell (\lambda )\le \dim V}{W}_{\lambda }^{\text{GL}}\otimes S^{\lambda }$ where $W_{\lambda }^{\text{GL}}$ is the irreducible $\text{GL}(V)$ representation indexed by $\lambda$. The two actions are mutual centralisers.

This bridges $S_n$ representation theory and ${\text{GL}}_n$ representation theory and is the model for many “duality” constructions in geometric representation theory.

14. Representations of compact and Lie groups

14.1 Compact groups

A topological group is compact if its underlying space is compact. Examples: $S^1,\text{SU}(n),\text{SO}(n),\text{Sp}(n)$ (compact symplectic, not the same as ${\text{Sp}}_{2n}(\mathbb{R})$).

Haar measure: a compact group has a unique translation-invariant Borel probability measure. Averaging over Haar measure plays the role of $\frac{1}{|G|}\sum$ for finite groups.

14.2 Peter-Weyl theorem

Hermann Weyl and Fritz Peter 1927: for compact $G$, the matrix coefficients of finite-dim irreducible representations form an orthogonal basis of $L^2(G)$: $L^2(G)\cong {\widehat{⨁}}_{\pi \in \hat{G}}{V}_{\pi }^{\ast }\otimes V_{\pi }$ where $\hat{G}$ is the unitary dual.

Consequences:

• Every irreducible representation of compact $G$ is finite-dimensional.
• Every continuous representation decomposes as a Hilbert direct sum of irreducibles.

Special case $G=S^1$: irreducibles are $z\mapsto z^n$ for $n\in \mathbb{Z}$; Peter-Weyl is Fourier series on the circle.

14.3 Weyl’s character formula

Hermann Weyl 1925-26. For a compact connected Lie group $G$ with maximal torus $T$ and root system $\Phi$: ${\chi }_{\lambda }(t)=\frac{{\sum }_{w\in W}(-1{)}^{\ell (w)}{e}^{w(\lambda +\rho )}(t)}{{\prod }_{\alpha \in {\Phi }^{+}}(e^{\alpha /2}(t)-e^{-\alpha /2}(t))}$ where $W$ is the Weyl group, $\rho =\frac{1}{2}{\sum }_{\alpha \in {\Phi }^{+}}\alpha$, and $\lambda$ is the highest weight.

For $\text{SU}(2)$: irreducibles are indexed by non-negative integers $n$ (spin $n/2$). $\dim V_n=n+1$. Character: ${\chi }_n(e^{i\theta })=\frac{\sin ((n+1)\theta )}{\sin \theta }.$

14.4 Borel-Weil theorem

Armand Borel and André Weil 1954. Irreducible representations of a compact Lie group $G$ are realised as global sections of holomorphic line bundles on the flag variety $G_{\mathbb{C}}/B$, where $B$ is a Borel subgroup. Geometric construction.

14.5 Highest weight theory

Every finite-dimensional irreducible representation of a complex semisimple Lie algebra is determined by its highest weight, a dominant element of the weight lattice. Cartan’s classification of complex semisimple Lie algebras by Dynkin diagrams: types $A_n,B_n,C_n,D_n$ (classical) and $E_6,E_7,E_8,F_4,G_2$ (exceptional). See lie-groups-so3-se3 and lie-group-catalog.

15. Applications

15.1 Crystallographic groups

A crystallographic group in ${\mathbb{R}}^n$ is a discrete subgroup of $E(n)={\mathbb{R}}^n⋊O(n)$ with compact quotient. In dimension 2 there are 17 wallpaper groups (proven by Fedorov 1891 and others). In dimension 3 there are 230 space groups (Schoenflies, Fedorov 1891). The point groups (quotients by the translation lattice) are finite subgroups of $O(3)$: 32 crystallographic point groups in 3D.

The crystallographic restriction theorem (Hessel 1830, Bravais 1851): in 2D/3D the only orders of rotation in a crystallographic group are $1,2,3,4,6$ — no 5-fold or 7-fold symmetry in a periodic crystal. Quasi-crystals (Shechtman 1984, Nobel 2011) circumvent by being aperiodic.

15.2 Particle physics

The Standard Model gauge group is $\text{SU}(3)\times \text{SU}(2)\times \text{U}(1)$:

• $\text{SU}(3{)}_C$: colour. The 8 gluons are the adjoint representation. Quarks live in the fundamental $3$; antiquarks in $\bar{3}$.
• $\text{SU}(2{)}_L$: weak isospin. Acts on left-handed doublets $({\nu }_e,e^{-}{)}_L$, $(u,d{)}_L$, etc.
• $\text{U}(1{)}_Y$: hypercharge.

After electroweak symmetry breaking ($\text{SU}(2)\times \text{U}(1)\to \text{U}(1{)}_{em}$), the photon is the unbroken generator.

Quark content of hadrons: baryons = $3\otimes 3\otimes 3=10\oplus 8\oplus 8\oplus 1$, hence the baryon decuplet and two octets in the SU(3)-flavour approximation.

Grand unification candidates: $\text{SU}(5)$ (Georgi-Glashow 1974), $\text{SO}(10)$, $E_6$ — embedding the Standard Model into a simple Lie group.

15.3 Galois theory

The fundamental connection between number theory, algebra, and group theory.

Fundamental theorem of Galois theory (Galois 1832): for a finite Galois extension $L/K$, there is an inclusion-reversing bijection between

• intermediate fields $K\subseteq M\subseteq L$, and
• subgroups $H\le \text{Gal}(L/K)$,

with $M=L^H$ and $H=\text{Gal}(L/M)$. Normal subgroups correspond to normal (= Galois) intermediate extensions.

Examples:

• $\text{Gal}(\mathbb{Q}({\zeta }_n)/\mathbb{Q})\cong (\mathbb{Z}/n\mathbb{Z}{)}^{\times }$ — cyclotomic.
• $\text{Gal}(\mathbb{Q}(\sqrt[3]{2},\omega )/\mathbb{Q})\cong S_3$.
• $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ — the absolute Galois group of $\mathbb{Q}$, a profinite group of cardinality $2^{{\aleph }_0}$; its structure is the central object of the Langlands program.

Inverse Galois problem: is every finite group realisable as $\text{Gal}(L/\mathbb{Q})$? Known for solvable groups (Shafarevich 1954), symmetric and alternating groups, many sporadic groups, but open in general.

See number-theory for class field theory and the abelianisation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.

15.4 Molecular and spectroscopic symmetry

Point groups classify molecular symmetry: methane (CH${}_4$) has $T_d$ symmetry (full tetrahedral, order 24), benzene has $D_{6h}$ symmetry (order 24), water has $C_{2v}$ symmetry (order 4). Character tables of point groups determine selection rules in IR/Raman spectroscopy and degeneracies of molecular orbitals. Standard reference: Cotton Chemical Applications of Group Theory (3rd ed., 1990).

15.5 Robotics and computer graphics

The rotation group $\text{SO}(3)$ and the rigid-motion group $\text{SE}(3)={\mathbb{R}}^3⋊\text{SO}(3)$ describe rigid-body kinematics. Group exponentials and Lie algebras give differentially smooth interpolation (quaternion slerp, SE(3) integration). See lie-groups-so3-se3 for details and applications to filtering and control.

15.6 Cryptography

• Discrete log in $(\mathbb{Z}/p\mathbb{Z}{)}^{\times }$ and elliptic-curve groups — see number-theory.
• Lattice-based crypto uses abelian groups ${\mathbb{Z}}^n/L$ for a lattice $L$.
• Group signatures, blind signatures, ring signatures: built on group-theoretic constructions.
• Identity-based encryption via pairings on elliptic-curve groups.

15.7 Combinatorics

• Sn-acting symmetric functions and Schur polynomials.
• Burnside / Pólya enumeration — see §4.4-4.5.
• Quasi-randomness in graphs (Chung-Graham-Wilson 1989) connects to the spectrum of the adjacency operator and group representations.
• Random walks on groups: mixing times via representation theory (Diaconis-Shahshahani 1981 for $S_n$; rate of card shuffling).

16. Open problems

• Inverse Galois problem — see §15.3.
• Aschbacher-Smith quasi-thin completion of CFSG is regarded as complete; a “second-generation” simplified proof (Gorenstein-Lyons-Solomon) is ongoing.
• Burnside problem: is every finitely generated group with all elements of bounded order finite? Negative in general (Adyan-Novikov 1968 for sufficiently large odd exponent). Restricted Burnside problem (positive answer): Efim Zelmanov 1991 (Fields 1994).
• Andrews-Curtis conjecture: about presentations of the trivial group; long-standing.
• Kazhdan’s property (T) for various groups: large-scale ergodic and representation-theoretic implications.
• Geometric Langlands program over $\mathbb{C}$ — major recent claimed progress (Gaitsgory et al. 2024).
• Quantum groups and categorification: ${\mathcal{U}}_q(\mathfrak{g})$ and their categorical lifts (Khovanov-Lauda, Rouquier).

17. Computational tools

• GAP (Groups, Algorithms, Programming): the standard system for finite groups.
• Magma (commercial): strong for group theory, number theory, and algebraic geometry.
• SageMath: open Python-based system; uses GAP and Pari under the hood.
• LiE: Lie group / Lie algebra computations.
• Sage’s group cohomology databases: $H^{\ast }(G,{\mathbb{F}}_p)$ for many small groups.
• The Atlas of Finite Groups (Conway, Curtis, Norton, Parker, Wilson 1985) and online ATLAS at brauer.maths.qmul.ac.uk: character tables, presentations, maximal subgroups of finite simple groups.

Further reading

• Dummit, D. S. and R. M. Foote. 2004. Abstract Algebra (3rd ed.). Wiley.
• Lang, S. 2002. Algebra (3rd ed.). Springer GTM 211.
• Rotman, J. J. 1995. An Introduction to the Theory of Groups (4th ed.). Springer GTM 148.
• Aschbacher, M. 2000. Finite Group Theory (2nd ed.). Cambridge.
• Robinson, D. J. S. 1996. A Course in the Theory of Groups (2nd ed.). Springer GTM 80.
• Isaacs, I. M. 2006. Character Theory of Finite Groups. Dover reprint.
• Serre, J.-P. 1977. Linear Representations of Finite Groups. Springer GTM 42.
• Fulton, W. and J. Harris. 1991. Representation Theory: A First Course. Springer GTM 129. The standard introduction including Lie algebras.
• Curtis, C. W. and I. Reiner. 1962. Representation Theory of Finite Groups and Associative Algebras. AMS Chelsea.
• Sagan, B. E. 2001. The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions (2nd ed.). Springer GTM 203.
• James, G. D. 1978. The Representation Theory of the Symmetric Groups. Springer LNM 682.
• Macdonald, I. G. 1995. Symmetric Functions and Hall Polynomials (2nd ed.). Oxford.
• Knapp, A. W. 2002. Lie Groups Beyond an Introduction (2nd ed.). Birkhäuser.
• Bröcker, T. and T. tom Dieck. 1985. Representations of Compact Lie Groups. Springer GTM 98.
• Brown, K. S. 1982. Cohomology of Groups. Springer GTM 87.
• Cotton, F. A. 1990. Chemical Applications of Group Theory (3rd ed.). Wiley.
• Conway, J. H. et al. 1985. Atlas of Finite Groups. Oxford.

Adjacent

• linear-algebra-essentials
• algebraic-geometry-foundations
• lie-groups-so3-se3
• lie-group-catalog
• number-theory
• graph-theory
• combinatorial-optimization
• complex-analysis