Mikhail Bershtein

Topics of Representation Theory (2025–2026)

The course has two parts. In the first, we cover classical results of representation theory. In the second, we turn to more modern developments. Some prior familiarity with algebra and basic representation theory is recommended, though we begin with a short review. We also discuss connections with other areas, including algebraic combinatorics, algebraic geometry, special functions, and knot theory. Some of these topics will be treated only in survey style and may require more advanced prerequisites.

Part I. Fundamentals [1, 2]
Group and algebra representations, subrepresentations, Maschke’s theorem, Schur’s lemma, characters of finite groups, semisimple algebras, density theorem, Jordan–Hölder and Krull–Schmidt theorems.
Part II. Symmetric Group [1, 2, 5, 9]
Irreducible representations of Sn, partitions and Young diagrams, Specht modules and Young symmetrizers, hook length formula, Frobenius character formula, restriction and induction, new (Okounkov–Vershik) approach, branching rules, standard Young tableaux.
Part III. Schur–Weyl Duality and GLn [1, 5]
Double centralizer theorem, Schur–Weyl duality between GLn and Sn, symmetric functions (elementary, complete, power sums, Schur functions), Cauchy identity, Weyl character formula, hook–content formula, Gelfand–Tsetlin patterns, semi-standard Young tableaux, Pieri rules.
Part IV. Lie Algebra sl2 [1, 2, 3]
Universal enveloping algebra, PBW theorem, Verma modules, Category 𝒪, Casimir operator, classification of finite-dimensional irreducibles. Representations in positive characteristic, Frobenius center, indecomposable modules; differential operators on flag varieties, Beilinson–Bernstein localization theorem.
Part V. Real Group SL2(ℝ) [3, 7, 8]
Gauss, Cartan and Iwasawa decompositions, homogeneous spaces, Harish–Chandra modules, principal series, discrete series; Legendre, Jacobi polynomial, hypergeometric functions.
Part VI. Quantum sl2 [4, 6]
Hopf algebras, q-analogs in combinatorics, definition of quantum sl2, representations for generic q, R-matrices; representation theory at roots of unity, , Jones polynomial and knot invariants.

References

  1. P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina. Introduction to Representation Theory. AMS Student Mathematical Library 59, 2011. Free online.
  2. I. Losev. Modern Introduction to Representation Theory. Lecture notes.
  3. P. Etingof. Representations of Lie Groups. MIT course notes, 2023. Free online.
  4. I. Losev. Representation theory: modern introduction. Lecture notes.
  5. W. Fulton. Young Tableaux: With Applications to Representation Theory and Geometry. LMS Student Texts 35, Cambridge Univ. Press, 1997.
  6. C. Kassel, M. Rosso, V. Turaev. Quantum Groups and Knot Invariants. Panoramas et Synthèses 5, SMF, 1997.
  7. N. J. Vilenkin. Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs 22, AMS, 1968.
  8. M. Libine. Introduction to Representations of Real Semisimple Lie Groups. 2016. arXiv:1608.04301.
  9. A. Kleshchev. Linear and Projective Representations of Symmetric Groups. Cambridge Tracts in Mathematics 163, Cambridge Univ. Press, 2005.