PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Combinatorics → Algebraic Combinatorics ↓


Representation Theory


Representation theory is a field of mathematics that studies the symmetries of algebraic structures to find out how these structures can act on vector spaces. This approach connects algebra to geometry, providing tools for solving problems by analyzing the ways algebraic objects can be represented as linear transformations of vector spaces. It has many areas of application, including theoretical physics, chemistry, and computer science. In this explanation, we will focus on the combinatorial aspects of representation theory, particularly in the context of algebraic combinatorics.

Basics of representation theory

In representation theory, an algebraic object such as a group, algebra or Lie algebra is represented by matrices and matrix operations. Let's start with an intuitive example that will guide us through the basic concepts.

Simple example: symmetric group

Consider the symmetric group S 3, which consists of all permutations of three elements. There are a total of six possible permutations. A specific element of this group can be written as a permutation, such as:

(1 2 3) -> (2 1 3)

In representation theory, our goal is to find a set of matrices such that the group operation mimics matrix multiplication. For S 3, we can find a 3x3 matrix representation for each permutation of the group that acts on a vector space.

For example, the permutation (1 2 3) -> (2 1 3) can be represented by the matrix:

[0 1 0]
[1 0 0]
[0 0 1]

This rearranges the components of a vector to reflect a permutation of the matrix indices.

Key concepts in representation theory

Modules and vector spaces

We use vector spaces to study how algebraic structures act on vector spaces. A representation is essentially a map from an algebraic structure to a set of matrices that act on a vector space. More formally, if G is a group, then a representation of G is an isomorphism:

φ : G → GL(V)

where GL(V) denotes the general linear group of a vector space V. This means that every element g in G is associated with a linear transformation in the vector space V.

Irreducible representations

Much of representation theory is concerned with breaking down representations into simpler components, called irreducible representations, which cannot be further decomposed into smaller representations.

Visualizing this is like breaking down a complex structure into its simplest building blocks. For example, consider a three-dimensional object such as a cube. You can imagine pushing or pulling on the cube using a grid, where each line in the grid represents a dimension. Similarly, in representation theory, each "line" would correspond to an invariant representation. Solving a complex group representation often involves finding these simpler, indivisible components.

Characters of a representation

The character of a representation is a powerful tool that assigns a number to each group element, and yields essential information about the representation. The character of an element g in a representation φ is given by the trace of the matrix representing g:

χ(g) = trace(φ(g))

Characters are very useful because they remain unchanged when the representation is changed by a similarity transformation, providing invariance that simplifies the study of representations. They form a nice bridge between representation theory and combinatorics.

Combinatorial aspects of representation theory

Young tableaus and standard tableaus

A powerful visualization tool in the combinatorial side of representation theory is the Young tableau. Young tableaus are combinatorial objects used to study symmetric groups and serve as indices for irreducible representations of these groups.

Consider again the symmetric group S 3. Its representations are indexed by partitions of 3. These partitions can be thought of as ways of distributing 3 equal objects into groups.

1 2 3

The table above shows the partition of S 3 as (2,1), which captures one of the ways to break the index symmetry of S 3. Each cell in the table can be filled with numbers satisfying strict rules that suggest different ways to permute or arrange the group elements.

Standard Young tableaux are those where the numbers increase up each row and down each column. They provide a systematic way to look at symmetries and count them.

Schur function

Another important concept is the Schur function, which emerges naturally in the study of symmetric functions and is closely related to representation theory. They are particularly important in quantum mechanics and algebraic geometry.

Schur functions can be expressed in terms of determinants. For example, if you have a partition λ = (λ 1 , λ 2 ,..., λ n ), then the Schur function s λ can be written as:

s λ( x 1, x 2, ..., x n ) = det(x i λ j + nj)

The power and beauty of Schur functions lie in their symmetry properties and their decomposition into corresponding invariant representations of the symmetric group.

Applications of representation theory

Physics: Quantum mechanics

Representation theory has profound implications in physics, particularly in quantum mechanics where the symmetries of quantum systems often determine their physical properties. Particles in quantum mechanics are often classified according to their representations of groups such as the rotation group SO(3) or the unitary group SU(2).

Coding theory and cryptography

Representation theory plays an important role in coding theory, which is crucial for error detection and correction in data transmission, and is also important in cryptography, where symmetries and group representations are fundamental to creating and understanding secure codes.

Chemical composition analysis

In chemistry, molecules are often studied based on their symmetry properties. Representation theory provides a mathematical basis for understanding molecular vibrations and spectra, which can be analyzed with group theoretic concepts that emerge from representation theoretic frameworks.

Conclusion

Representation theory connects together elements from different mathematical domains, providing deeper insights into the symmetric structures of algebraic objects. It serves as a bridge between abstract algebra and practical applications in science and engineering. By using tools from combinatorics and aligning mathematical structures through vector space representations, it opens up avenues for solving complex problems using practical mathematical concepts.


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Representation Theory

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