🧩Representation Theory Unit 1 – Introduction to Group Theory

What's Group Theory All About?

• Group theory studies algebraic structures called groups which consist of a set of elements and a binary operation that combines any two elements to form a third element in the set
• Focuses on the relationships and symmetries within these abstract structures rather than the specific nature of the elements themselves
• Provides a framework for understanding and classifying various mathematical objects and their properties
• Plays a fundamental role in many branches of mathematics including algebra, geometry, topology, and analysis
• Has applications beyond pure mathematics in areas such as physics (quantum mechanics, crystallography), chemistry (molecular symmetry), and computer science (cryptography, coding theory)
• Allows for the development of general theories and techniques that can be applied across different mathematical contexts
• Enables the identification of common patterns and structures across seemingly disparate mathematical objects

Key Concepts and Definitions

• A group $(G, *)$ is a set $G$ together with a binary operation $*$ that satisfies the following axioms:
  • Closure: For all $a, b \in G$, $a * b \in G$
  • Associativity: For all $a, b, c \in G$, $(a * b) * c = a * (b * c)$
  • Identity element: There exists an element $e \in G$ such that for all $a \in G$, $a * e = e * a = a$
  • Inverse element: For each $a \in G$, there exists an element $a^{-1} \in G$ such that $a * a^{-1} = a^{-1} * a = e$
• The order of a group $|G|$ is the number of elements in the set $G$
• A group is called abelian (or commutative) if for all $a, b \in G$, $a * b = b * a$
• The center of a group $Z(G)$ is the set of elements that commute with every element in the group: $Z(G) = \{a \in G : ax = xa \text{ for all } x \in G\}$
• A subgroup $H$ of a group $G$ is a subset of $G$ that forms a group under the same binary operation as $G$
• The left coset of a subgroup $H$ in $G$ with respect to an element $a \in G$ is the set $aH = \{ah : h \in H\}$
• The right coset of a subgroup $H$ in $G$ with respect to an element $a \in G$ is the set $Ha = \{ha : h \in H\}$

Types of Groups and Examples

• Cyclic groups: Groups generated by a single element (generator) where every element can be expressed as a power of the generator
  • Example: The group of integers under addition $(\mathbb{Z}, +)$ is cyclic with generator $1$ (or $-1$)
  • Example: The group of complex $n$-th roots of unity under multiplication is cyclic with generator $e^{2\pi i/n}$
• Symmetric groups: Groups of permutations (bijective functions) on a set of $n$ elements, denoted by $S_n$
  • Example: $S_3$ consists of all permutations of the set $\{1, 2, 3\}$ under the composition of functions
• Dihedral groups: Groups of symmetries of regular polygons, including rotations and reflections, denoted by $D_n$ for an $n$-sided polygon
  • Example: $D_4$ is the group of symmetries of a square, consisting of 4 rotations and 4 reflections
• Matrix groups: Groups of matrices under matrix multiplication
  • Example: The general linear group $GL(n, \mathbb{R})$ consists of all invertible $n \times n$ real matrices
  • Example: The special orthogonal group $SO(3)$ consists of all $3 \times 3$ real orthogonal matrices with determinant 1, representing rotations in 3D space
• Finite groups: Groups with a finite number of elements
  • Example: The Klein four-group $V_4$ is a finite group of order 4 with elements $\{e, a, b, c\}$ satisfying $a^2 = b^2 = c^2 = e$ and $ab = c, bc = a, ca = b$
• Infinite groups: Groups with an infinite number of elements
  • Example: The group of real numbers under addition $(\mathbb{R}, +)$ is an infinite abelian group

Group Operations and Properties

• The binary operation of a group can be represented by a group table (Cayley table) which shows the result of combining any two elements
• A group is called cyclic if it can be generated by a single element, i.e., there exists an element $a \in G$ such that every element of $G$ can be expressed as a power of $a$
• The order of an element $a$ in a group $G$, denoted by $|a|$, is the smallest positive integer $n$ such that $a^n = e$ (if it exists)
  • If no such $n$ exists, the element is said to have infinite order
• Lagrange's theorem states that the order of a subgroup divides the order of the group, i.e., if $H$ is a subgroup of $G$, then $|H|$ divides $|G|$
• The center of a group $Z(G)$ is a subgroup of $G$ and is always abelian
• A group $G$ is called simple if it has no proper normal subgroups (subgroups that are invariant under conjugation by elements of $G$)
• The direct product of two groups $(G \times H, *)$ is a group formed by the Cartesian product of the sets $G$ and $H$ with the operation defined component-wise: $(a, b) * (c, d) = (a*c, b*d)$

Subgroups and Cosets

• A subgroup $H$ of a group $G$ must satisfy the following conditions:
  • $H$ is a non-empty subset of $G$
  • For all $a, b \in H$, $a * b \in H$ (closure under the group operation)
  • For all $a \in H$, $a^{-1} \in H$ (closure under inverses)
• The left cosets of a subgroup $H$ in $G$ partition the group $G$ into disjoint sets of equal size
  • The number of left cosets is called the index of $H$ in $G$, denoted by $[G:H]$
• The right cosets of a subgroup $H$ in $G$ also partition the group $G$ into disjoint sets of equal size
• A subgroup $H$ is called normal if its left and right cosets coincide, i.e., $aH = Ha$ for all $a \in G$
  • Normal subgroups are important because they allow for the construction of quotient groups $G/H$
• The kernel of a group homomorphism $\varphi: G \to G'$ is a normal subgroup of $G$ defined as $\ker(\varphi) = \{a \in G : \varphi(a) = e'\}$, where $e'$ is the identity element of $G'$

Homomorphisms and Isomorphisms

• A group homomorphism is a function $\varphi: G \to G'$ between two groups that preserves the group structure: $\varphi(a * b) = \varphi(a) *' \varphi(b)$ for all $a, b \in G$, where $*$ and $*'$ are the group operations in $G$ and $G'$, respectively
• The kernel of a homomorphism $\ker(\varphi)$ is always a normal subgroup of the domain group $G$
• The image of a homomorphism $\text{Im}(\varphi) = \{\varphi(a) : a \in G\}$ is a subgroup of the codomain group $G'$
• The first isomorphism theorem states that for a group homomorphism $\varphi: G \to G'$, the quotient group $G/\ker(\varphi)$ is isomorphic to the image $\text{Im}(\varphi)$
• A group isomorphism is a bijective homomorphism, i.e., a one-to-one correspondence between two groups that preserves the group structure
  • Isomorphic groups have the same group-theoretic properties and are essentially the same up to relabeling of elements
• The automorphism group $\text{Aut}(G)$ of a group $G$ is the group of all isomorphisms from $G$ to itself under the composition of functions

Applications in Representation Theory

• Representation theory studies the ways in which abstract groups can be represented as linear transformations (matrices) acting on vector spaces
• A representation of a group $G$ on a vector space $V$ is a group homomorphism $\rho: G \to GL(V)$, where $GL(V)$ is the general linear group of invertible linear transformations on $V$
  • The dimension of the representation is the dimension of the vector space $V$
• The character of a representation $\rho$ is the function $\chi_\rho: G \to \mathbb{C}$ defined by $\chi_\rho(g) = \text{tr}(\rho(g))$, where $\text{tr}$ denotes the trace of a matrix
  • Characters are class functions, i.e., they are constant on conjugacy classes of $G$
• Irreducible representations are the building blocks of representation theory; they cannot be decomposed into simpler representations
  • Every representation can be written as a direct sum of irreducible representations
• Schur's lemma states that any linear map between irreducible representations that commutes with the group action is either zero or an isomorphism
• The regular representation of a finite group $G$ is the representation on the vector space $\mathbb{C}[G]$ (the space of complex-valued functions on $G$) given by the left regular action of $G$ on itself
• Representation theory has applications in physics (symmetries in quantum mechanics), chemistry (molecular symmetry and spectroscopy), and harmonic analysis (Fourier analysis on groups)

Common Pitfalls and Tips

• Remember that the group operation must be associative; not all binary operations on a set form a group
• Be careful when dealing with infinite groups; some properties that hold for finite groups may not hold for infinite groups (e.g., not every element has a finite order)
• When proving that a subset is a subgroup, make sure to check all three conditions (non-empty, closed under the operation, and closed under inverses)
• Not all subgroups are normal; be cautious when forming quotient groups
• Lagrange's theorem provides a necessary condition for a subset to be a subgroup, but it is not sufficient (e.g., the alternating group $A_4$ has order 12 but no subgroup of order 6)
• When working with cosets, remember that they are sets, not elements; use set notation and operations accordingly
• Homomorphisms and isomorphisms preserve the group structure but not necessarily the properties of individual elements (e.g., the order of an element may change under a homomorphism)
• In representation theory, the dimension of a representation refers to the dimension of the vector space, not the size of the matrices
• Characters are a powerful tool in representation theory; they can help determine the irreducibility and equivalence of representations
• Schur's lemma is a key result in the study of irreducible representations; use it to simplify proofs and computations involving intertwining maps between representations