2024/2025
Symmetric Functions
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
3, 4 module
Open to:
students of all HSE University campuses
Instructors:
Karine Kuyumzhiyan
Language:
English
ECTS credits:
6
Contact hours:
72
Course Syllabus
Abstract
The theory of symmetric functions is one of the central branches of algebraic combinatorics. Being a rich and beautiful theory by itself, it also has numerous connections with the representation theory and algebraic geometry (especially geometry of homogeneous spaces, such as flag varieties, topic and spherical varieties). In this course we will mostly focus on the combinatorial aspects of the theory of symmetric functions and study the properties of Schur polynomials. In representation theory they appear as characters of representations of $$GL_n$$: they are also closely related with the geometry of Grassmannians. The second half of the course will be devoted to Schubert polynomials, a natural generalization of Schur polynomials, defined as "partially symmetric" functions. Like the Schur functions, they also have a rich structure and admit several nice combinatorial descriptions; geometrically they appear as representatives of Schubert classes in the cohomology ring of a full flag variety. Time permitting, we will also discuss K-theoretic (non - homogeneous) analogues of Schur and Schubert polynomials.