We will start out exploring the theory of Young tableaux. Truly elementary objects with a rich algebro-combinatorial life, we will see them appearing in a number of classical theories, e.g., the study of symmetric functions, the representation theory of symmetric groups, and Schubert calculus on Grassmannians and flag manifolds.
Besides the exposition of this rather classical circle of ideas, another goal of this course is to present the recent proof of the Saturation Conjecture (see W. Fulton's article Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. 37 (2000), 209-249, for a survey, and R. Stanley's article Recent progress in algebraic combinatorics, Bull. Amer. Math. Soc. 40 (2003), 55-68, for a brief account). Originally a statement about positivity of certain structure coefficients, the Saturation Conjecture relates to a number of other topics, in fact, it implies the long standing Horn's conjecture on the characterization of eigenvalues of sums of Hermitian matrices.
Time permitting, we will discuss even more recent developments where the acquired algebro-combinatorial machinery is put to work, e.g., for describing equivariant Schubert calculus.
References: Our main reference for the first part of this course will be
In the second part we will rely on research papers.
- William Fulton: Young Tableaux;
London Math. Soc. Student Texts 35, 1997.Prerequisites: No more than basic courses in algebra. We will start on an elementary level. Following demand and interest, I will be happy to provide the necessary background from representation theory, algebraic topology, and algebraic geometry.
Language: English or German.