Algebraic Combinatorics Course | Möbius Inversion, Symmetric Functions, LGV Lemma
Course Details
| Exam Registration | 22 |
|---|---|
| Course Status | Ongoing |
| Course Type | Elective |
| Language | English |
| Duration | 12 weeks |
| Categories | Algebra, Mathematics |
| Credit Points | 3 |
| Level | Undergraduate/Postgraduate |
| Start Date | 19 Jan 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 20 Feb 2026 |
| Exam Date | 17 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
Unlocking the Power of Structure: An Introduction to Algebraic Combinatorics
Welcome to the fascinating world of Algebraic Combinatorics, a field where the abstract beauty of algebra meets the concrete world of counting and arrangement. This 12-week course, designed for advanced undergraduates and postgraduates, serves as a gateway to some of the most powerful paradigms in modern mathematics and theoretical computer science. Guided by experts from The Institute of Mathematical Sciences (IMSc), Chennai, this journey will equip you with tools to see deep algebraic structures within combinatorial objects.
Meet Your Instructors
This course is led by two distinguished faculty members from IMSc Chennai, both with deep expertise in representation theory and its combinatorial facets.
- Prof. Amritanshu Prasad: A faculty member at IMSc, his research interests lie in representation theory. He is also the author of An introduction to Schur polynomials, a key resource for this course.
- Prof. S. Viswanath: Also a faculty member at IMSc, his research focuses on representation theory, providing a strong algebraic foundation for the combinatorial concepts covered.
Who Should Take This Course?
This course is meticulously designed for a diverse group of learners passionate about advanced mathematics and its applications.
- M.Sc./M.Tech. students in Mathematics or Computer Science.
- PhD students undertaking coursework in these fields.
- Advanced undergraduates (B.Sc., B.E., B.Tech.) with a strong mathematical background.
Prerequisites: Building on a Solid Foundation
To fully benefit from this course, you should be comfortable with:
- Undergraduate-level set theory, linear algebra, and abstract algebra.
- Basic discrete mathematics: mathematical induction, graph and tree theory, elementary counting techniques, recurrence relations.
Course Layout: A 12-Week Roadmap
The course is structured to build your knowledge from fundamental concepts to advanced theorems, week by week.
| Week | Topic |
|---|---|
| 1 | Partially ordered sets and Möbius inversion |
| 2 | Finite distributive lattices |
| 3 | Combinatorial classes and their ordinary generating functions |
| 4 | Operations on combinatorial classes |
| 5 | Combinatorial specifications |
| 6 | Groups of permutations, cycle structure, orbit counting method |
| 7 | Polya’s enumeration theorem, the cycle index generating function |
| 8 | Combinatorial species and associated generating functions |
| 9 | Operations on species |
| 10 | The Lindström-Gessel-Viennot (LGV) lemma |
| 11 | Symmetric functions – elementary, complete, power sum |
| 12 | Symmetric functions – Schur functions |
Core Themes and Learning Outcomes
By the end of this course, you will have gained a working understanding of several cornerstone techniques in algebraic combinatorics:
- Möbius Inversion: Learn to invert summations over partially ordered sets, a generalization of the principle of inclusion-exclusion.
- Combinatorial Classes & Species: Understand how to define families of combinatorial objects (like graphs, trees, permutations) and use generating functions to enumerate them. The theory of species provides a categorical framework for this.
- Polya’s Theory: Master the orbit-counting method and Polya’s Enumeration Theorem to count objects up to symmetry, crucial in chemistry, network theory, and design.
- The LGV Lemma: Explore this powerful determinant-based method for counting non-intersecting lattice paths, with applications in plane partitions and random matrices.
- Symmetric Functions: Delve into the elegant algebra of symmetric polynomials, culminating in the rich theory of Schur functions, which bridge representation theory, geometry, and combinatorics.
Essential Reading and Resources
To complement the lectures, the following texts are highly recommended:
- Enumerative Combinatorics (Volumes 1 & 2) by Richard Stanley - The definitive treatise on the subject.
- Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick - Focuses on generating functions and asymptotic analysis.
- Combinatorial Species and Tree-like Structures by Bergeron, Labelle, and Leroux - The primary reference for the theory of species.
- An Introduction to Schur Polynomials by Amritanshu Prasad - A concise and accessible resource available on arXiv.
- Algebraic Combinatorics: Walks, Trees, Tableaux, and More by Richard Stanley - A more focused text covering several key topics.
Why Study Algebraic Combinatorics?
Algebraic Combinatorics is not just an abstract pursuit. Its techniques are vital in:
- Computer Science: Algorithm analysis, database theory, and the design of combinatorial algorithms.
- Statistical Physics: Modeling phase transitions and lattice models.
- Chemistry: Enumerating isomers and molecular structures.
- Representation Theory: Understanding the symmetric group and related algebras, as highlighted by the instructors' expertise.
- Data Science: Understanding symmetries in data and network structures.
This course offers a unique opportunity to learn these interconnected topics from researchers at the forefront of the field. Whether you aim to pursue pure mathematics, theoretical computer science, or related disciplines, the tools of algebraic combinatorics will provide you with a profound and practical perspective on the nature of discrete structures.
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