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Symmetric Functions

Category 'Best Course for New Knowledge and Skills'
Type: Optional course (faculty)
When: 1, 2 module
Instructors: Evgeny Smirnov
Language: English
ECTS credits: 6

Course Syllabus


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.
Learning Objectives

Learning Objectives

  • To get acquaintance to fundamental notions of this theory, as well as their connections with algebra, algebraic geometry, and representation theory
  • To learn about the main tools and methods of modern algebraic combinatorics and will be able to apply these methods
  • To be able to understand research papers in this subject
Expected Learning Outcomes

Expected Learning Outcomes

  • The students are expected to know the definitions of elementary and complete symmetric functions and power sums, to be able to operate with their generating functions and to prove the fundamental theorem on symmetric polynomials.
  • The students are expected to know combinatorial and algebraic definitions of Schur functions, to be able to prove their equivalence and to operate with these notions.
  • The students are expected to know Pieri formulas and to be able to use them for computations in the ring of symmetric functions.
  • The students are expected to be able to apply methods of the theory of symmetric functions to various problems of enumerative combinatorics, such as computing the numbers of plane partitions and their generating functions. They are expected to understand the relation between these methods and elementary combinatorial methods, such as the Gessel-Viennot trick.
  • The students should know the main notions of the array theory (horizontal and vertical operations, condensation, the fiber product theorem) and to perform computations with arrays, including explicit computations related to the RSK-correspondence.
  • The students are expected to know the Littlewood-Richardson rule in terms of array condensation and skew Young tableaux, as well as to be able to compute the Littlewood-Richardson coefficients in the simple cases manually.
  • The students are expected to learn the fundamental notions of the theory of symmetric groups and to be able to find, for a given permutation, its sign, length, reduced decompositions, as well as compare permutations with respect to the Bruhat order.
  • The students are expected to know definitions of Schubert polynomials and to perform simple computations with them
  • The students are expected to be able to perform computations with Schubert polynomials using Monk's formula and the Lascoux transition formula
  • The students should be able to compute Schubert polynomials using their combinatorial definition, write down explicitly the corresponding pipe dreams, and apply this technique to solving problems about Schubert polynomials.
  • The students are expected to be able to know the main facts and to perform computations with double Schubert and Grothendieck polynomials
Course Contents

Course Contents

  • Symmetric polynomials
    Bases in the ring of symmetric polynomials: complete and elementary symmetric polynomials, monomial symmetric polynomials, power sums. Transition formulas.
  • Schur functions
    Schur functions. Algebraic definition. Jacobi - Trudi formula. Combinatorial definition, equivalence with the algebraic definition. Young tableaux.
  • Pieri rule. Kostka numbers
    Pieri rule. Kostka numbers. Enumeration of semistandard and standard Young tableaux, hook length formula
  • Applications to combinatorics
    Applications to combinatorics: enumerating plane partitions. MacMahon formula
  • Multiplications of Schur functions. Littlewood - Richardson rule.
    Multiplications of Schur functions. Littlewood - Richardson rule
  • Symmetric group, its Coxeter presentation
    Symmetric group, its Coxeter presentation. The Bruhat order. The Lehmer code and the essential set of a permutation
  • "Partially symmetric" polynomials. Divided difference operators. Schubert polynomials.
  • Properties of Schubert polynomials
    Properties of Schubert polynomials. Monk's formula, Lascoux transition formula.
  • Combinatorial presentation of Schubert polynomials
    Combinatorial presentation of Schubert polynomials. Pipe dreams. Positivity. Fomin - Kirillov theorem.
  • Generalizations
    Generalizations: double Schubert polynomials. Stanley symmetric functions, Grothendieck polynomials.
  • Enumeration of arrays
    Enumeration of arrays (after Danilov and Koshevoy). Dense arrays. RSK - correspondence
Assessment Elements

Assessment Elements

  • non-blocking Midterm written exam
  • non-blocking Final written exam
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    0.6 * Final written exam + 0.4 * Midterm written exam


Recommended Core Bibliography

  • Danilov, V., & Koshevoy, G. (2005). Arrays and the octahedron recurrence. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.393FE4BA
  • Fomin, S., & Kirillov, A. N. (1996). The Yang-Baxter equation, symmetric functions, and Schubert polynomials. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.47EB5BFF
  • Fulton, W. (1997). Young Tableaux : With Applications to Representation Theory and Geometry. Cambridge [England]: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=570403
  • Nantel Bergeron, & Sara Billey. (1993). RC-Graphs and Schubert Polynomials. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.D701F5B1
  • Nantel Bergeron, & Sara Billey. (n.d.). 4. Double Schubert Polynomials. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.B5710E55
  • V. I. Danilov, & G. A. Koshevoy. (2005). Arrays and the octahedron recurrence. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.51E59E17

Recommended Additional Bibliography

  • Stanley, R. P. (1999). Enumerative Combinatorics: Volume 2. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=502394