Symmetric Functions: A Beginner's Course

Inhaltsverzeichnis

1. Frontmatter

2. Schur Polynomials and Young Tableaux

   1. Frontmatter

   2. Chapter 1. Symmetric Polynomials

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      xA straightforward solution would be as follows: find the roots x1, x2 using the well-known formula, compute their cubes and take their sum. But this is quite a long process, and it is possible to make a mistake during the computations, so it is better to proceed in a different way: observe that, according to Viete’s theorem, we have x1 + x2 = −p, x1x2 = q.

   3. Chapter 2. Schur Polynomials

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      Along with symmetric polynomials we can consider skew-symmetric polynomials. Let us find out some things about them.

   4. Chapter 3. Combinatorial Formula for Schur Polynomials

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      In the previous chapter we made the following observation: in all our examples the coefficients of Schur polynomials were nonnegative integers. As we will see in this chapter, this is always true. Moreover, these coefficients express the numbers of certain combinatorial objects: the Young tableaux.

   5. Chapter 4. The Ring of Symmetric Functions

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      We have already computed this determinant (cf. Problem 2.2), but for the sake of completeness let us do it again.

   6. Chapter 5. The Number of Young Tableaux

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      As we have seen before, a Schur polynomial is equal to the sum of monomials for Young tableaux of a given shape. In this chapter we address the following question: what is the number of such tableaux?

   7. Chapter 6. Problem Set 1

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      Consider the space Ω(n) ⊂ Λ(n) of homogeneous supersymmetric functions of degree n:

3. Arrays and the Littlewood–Richardson Rule

   1. Frontmatter

   2. Chapter 7. Arrays and Condensation Operations

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      This part of our book is devoted to a new tool for working with Young tableaux: arrays. They provide easy proofs of various relations for Schur polynomials. The construction of arrays was proposed by Vladimir Danilov and Gleb Koshevoy in 2005, cf. [DK05]. In our exposition we also largely follow the algebra textbook by Alexei Gorodentsev [Gor17].

   3. Chapter 8. Arrays and Schur Polynomials

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      In this chapter we discuss relations between arrays and Schur polynomials.

      Let a be an array of size m × n. We can transform each of its rows into a weakly increasing sequence by taking its row scan.

   4. Chapter 9. The Littlewood–Richardson Rule

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      In this chapterwe will obtain the rule for expanding products of Schur polynomials as linear combinations of other Schur polynomials.

   5. Chapter 10. Problem Set 2

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      Definition 10.1 The Schensted insertion is an operation producing from a semistandard Young tableau T filled with numbers t ij and a positive integer x a new tableau T′ = (T ← x) consisting of |T| + 1 boxes, according to the following rules:

4. Schubert Polynomials and Pipe Dreams

   1. Frontmatter

   2. Chapter 11. The Symmetric Group

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      This part will be devoted to the study of partially symmetric functions: functions that are symmetric in some variables, while not necessarily being symmetric in the others. More precisely, we will be working with Schubert polynomials; as we will see, they generalize the notion of Schur polynomials.

   3. Chapter 12. Schubert Polynomials

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      This chapter is devoted to partially symmetric polynomials, i.e. polynomials that are symmetric with respect to certain changes of variables: Schubert polynomials. We will mostly follow A. Knutson’s notes [Knu12].

   4. Chapter 13. Combinatorial Presentation of Schubert Polynomials

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      In the previous chapter we defined Schubert polynomials by means of divided difference operators. Despite the fact that the differences are divided, Schubert polynomials are actual polynomials, not rational functions. More surprisingly, even though these are divided difference operators, the coefficients of Schubert polynomials turn out to be positive, as we saw in Corollary 12.17.

   5. Chapter 14. Properties of Pipe Dreams

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      Given a permutation 𝑤 ∈ S_𝑛, how do we draw a pipe dream (at least one of them) with such a shape? A natural idea would be to arrange crosses and elbow joints in the simplest possible way. For example, we can place all crosses into left-adjusted boxes: i.e., in each row we have a sequence of crosses followed by a sequence of elbow joints.

   6. Chapter 15. Problem Set 3

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      This series of exercises presents another proof of the Bereron–Billey–Fomin– Kirillov theorem on equivalence of algebraic and combinatorial definitions of Schubert polynomials (cf. [BJS93]).

   7. Chapter 16. Hints, Answers, and Solutions

      Evgeny Smirnov, Anna Tutubalina

      Abstract

      1.1 Write the system of equations (1.1) in the matrix formand solve it using Cramer’s rule.

5. Backmatter