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Math 263A Notes: Algebraic Combinatorics and Symmetric Functions

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MATH 263A NOTES: ALGEBRAIC COMBINATORICS AND SYMMETRIC FUNCTIONS AARON LANDESMAN CONTENTS 1. Introduction 4 2. 10/26/16 5 2.1. Logistics 5 2.2. Overview 5 2.3. Down to Math 5 2.4. Partitions 6 2.5. Partial Orders 7 2.6. Monomial Symmetric Functions 7 2.7. Elementary symmetric functions 8 2.8. Course Outline 8 3. 9/28/16 9 3.1. Elementary symmetric functions eλ 9 3.2. Homogeneous symmetric functions, hλ 10 3.3. Power sums pλ 12 4. 9/30/16 14 5. 10/3/16 20 5.1. Expected Number of Fixed Points 20 5.2. Random Matrix Groups 22 5.3. Schur Functions 23 6. 10/5/16 24 6.1. Review 24 6.2. Schur Basis 24 6.3. Hall Inner product 27 7. 10/7/16 29 7.1. Basic properties of the Cauchy product 29 7.2. Discussion of the Cauchy product and related formulas 30 8. 10/10/16 32 8.1. Finishing up last class 32 8.2. Skew-Schur Functions 33 8.3. Jacobi-Trudi 36 9. 10/12/16 37 1 2 AARON LANDESMAN 9.1. Eigenvalues of unitary matrices 37 9.2. Application 39 9.3. Strong Szego limit theorem 40 10. 10/14/16 41 10.1. Background on Tableau 43 10.2. KOSKA Numbers 44 11. 10/17/16 45 11.1. Relations of skew-Schur functions to other fields 45 11.2. Characters of the symmetric group 46 12. 10/19/16 49 13. 10/21/16 55 13.1. Review 55 13.2. Completing the example from last class 55 13.3. Completing the example; back to the Schur functions 57 14. 10/24/16 58 15. 10/26/16 61 16. 10/28/16 66 16.1. Plane partitions, RSK, and MacMahon’s generating function 66 17. 10/31/16 71 17.1. Announcements and Review 71 18. 11/2/16 74 18.1. Overview 74 18.2. P-partitions 74 18.3. The order polynomial 75 19. 11/4/16 78 19.1. Review 78 19.2. A possibly non-politically correct example 78 19.3. More on descent 79 19.4. Shuffling Cards 79 20. 11/7/16 81 21. 11/9/16 83 21.1. Algebra of the Ai 83 21.2. Quasi-Symmetric Functions 84 22. 11/11/16 86 22.1. Application to symmetric function theory 87 22.2. Connection of quasi-symmetric functions to card shuffling 88 22.3. Applications 89 23. 11/14/16 90 23.1. Combinatorial Hopf Algebras 90 23.2. Examples of Hopf Algebras 91 MATH 263A NOTES: ALGEBRAIC COMBINATORICS AND SYMMETRIC FUNCTIONS3 23.3. What did Hopf do? 92 24. 11/16/16 93 24.1. Definition of combinatorial Hopf algebras 93 24.2. Examples 94 25. 11/18/16 95 25.1. What do Hopf algebras have to do with card shuffling? 95 25.2. Lyndon words 97 25.3. The standard bracketing of Lyndon words 97 26. 11/28/16 98 27. 11/30/16 100 28. 12/2/16 102 28.1. Macdonald Polynomials 102 28.2. Proof of Theorem 28.3 103 29. 12/5/16 106 29.1. Review 106 29.2. Defining D 107 29.3. Examples of Macdonald polynomials 107 29.4. Understanding the operator D in an alternate manner 108 30. 12/7/16 110 30.1. School 1 110 30.2. School 2 111 30.3. Persi’s next project 112 4 AARON LANDESMAN 1. INTRODUCTION Persi Diaconis taught a course (Math 263A) on Algebraic Combi- natorics and Symmetric Function Theory at Stanford in Fall 2016. These are my “live-TEXed“ notes from the course. Conventions are as follows: Each lecture gets its own “chapter,” and appears in the table of contents with the date. Of course, these notes are not a faithful representation of the course, either in the mathematics itself or in the quotes, jokes, and philo- sophical musings; in particular, the errors are my fault. By the same token, any virtues in the notes are to be credited to the lecturer and not the scribe. Thanks to Lisa Sauermann for taking notes on October 17, when I missed class. 1 Please email suggestions to aaronlandesman@ gmail.com. 1This introduction has been adapted from Akhil Matthew’s introduction to his notes, with his permission. MATH 263A NOTES: ALGEBRAIC COMBINATORICS AND SYMMETRIC FUNCTIONS5 2. 10/26/16 2.1. Logistics. (1) Math 263A (2) Algebraic Combinatorics (3) Persi Diaconis (4) office hours Tuesday 2-4, 383 D (5) (No email) 2.2. Overview. This is a course in algebraic combinatorics and sym- metric function theory. We’ll talk about what we want to cover in the course. Combinatorics is pretty hard to define. It deals with things like sets Xn which are finite, permutations, partitions, graphs, trees. We might try to estimate jXnj, functions T : Xn R or j fx 2 Xn : T(x) = yg j. ! Here’s a slogan: Symmetric function theory “makes math” out of lots of classical combinatorics. We’ll try to cover (1) Chapter I of MacDonald’s book symmetric functions and Hall polynomials (2) More things that weren’t mentioned. Remark 2.1. We’ll have many digressions into “why are we studying this” and “what is it good for.” 2.3. Down to Math. Definition 2.2. For n 2 Z, a weak composition of n is a partition of n into (a1, a2, :::) with k=1 ak = n with ak ≥ 0. Definition 2.3. Let R beP1 a commutative ring. For example, R = Z, Q, Z[x], ::: Suppose we have infinitely many variables x1, x2, x3, ::: then a homogeneous symmetric function of degree n is a formal power series α f(x1, x2, :::) = f(x) = cαx α X where (1) α ranges over all weak compositions of n. (2) cα 2 R α α1 α2 (3) x = x1 x2 ··· (4) f(x1, x2, :::) = f(xσ(1), xσ(2), ::: for σ 2 S . 1 6 AARON LANDESMAN (5) Every term has the same degree. Example 2.4. (1) f(x) = n=1 xi is a symmetric function. 2 2 2 2 (2) f(x) = x1x2 + x1x3 + ···1+ x2x2 + x2x3 + ··· is another sym- metric function. P n Definition 2.5. Let ΛR be all symmetric functions of degree n over R. Remark 2.6. We often omit the R subscript when it is understood or clear from context. Remark 2.7. We have Λn · Λm ⊂ Λn+m. n Definition 2.8. Define ΛR := ⊕n=0ΛR. 2.4. Partitions. 1 Definition 2.9. λ is a partition of n, written λ ` n if λ = (λ1, λ2, :::) with λ1 ≤ λ2 ≤ · · · and λi = n. i X Write jλj = n, `(λ) := the number of nonzero parts of λ . Example 2.10. The partitions of 5 are f5, 41, 311, 32, 2111, 11111, 221g . We will often write λ = 1n1(λ)2n2(λ) with ni(λ) equal to the number of parts equal to i. For example, 221 = 122. These satisfy ini(λ) = n. i X Example 2.11. We can also draw young diagrams with dots or young tableaux with boxes. Here is the partition 13223 = 322111 ` 10. MATH 263A NOTES: ALGEBRAIC COMBINATORICS AND SYMMETRIC FUNCTIONS7 Definition 2.12. If λ is a partition of n, λ0 (the transpose) is what you obtain when flipping the diagram. 2.5. Partial Orders. Example 2.13. We can define the partial order by λ ≤ µ if λi ≤ µi for all i. Example 2.14. One can write down a partial orders. For example, we can define a partial order on these diagrams by saying one can get from one partition to another by moving dots to adjacent rows so that at every stage one arrives at a partition. Algebraically, the partial j j order is majorization order, where λ ≤ µ if i=1 λi ≥ i=1 µi for all j. P P Fact 2.15. We have λ ≤ µ µ0 ≤ λ0. This isn’t too hard, but it’s a little bit finicky, and we’ll come back to proving it later. () Example 2.16. Take λ < µ if jλj < µ or λ1 = µ1, ::: , λn = µn, λn+1 < µn+1. This is a lexicographic ordering. 2.6. Monomial Symmetric Functions. Suppose λ is some partition α λ = (λ1, λ2, :::) and mλ = α x , the sum over all distinct permuta- tions of λ. P 2 2 Example 2.17. We have m21 = i<j xixj + xi xj. We have m2 = x2. We have m = x x . j j 11 i<j i j P n LemmaP 2.18. For mλ ofP size jλj = n, these form a basis for Λ . Corollary 2.19. We have dim Λn = p(n). Proof. Apply the preceding lemma. 8 AARON LANDESMAN 2.7. Elementary symmetric functions. Definition 2.20. The elementary symmetric functions ej = xi1 xi2 ··· xij , i <···<i 1 X j n1(λ) n2(λ) so e1 = i xi, e2 = i<j xixj, :::. For λ ` n, we define eλ = e1 e2 ··· . Fact 2.21.PWe’ll seeP that feλg as λ ` n form a basis of Λ over Z. Lemma 2.22. We have eλ = Mλµmµ µX`n where mλµ is the number of matrices with row sums λ and column sums µ. Proof. Say λ = λ1 ··· λr, µ = µ1 ··· µs. Example 2.23. [Darwin’s data (see Persi’s paper “sequential Monte Carlo methods for statistical analysis of tables”)] Look at the set of all tables with the same row sums and the same column sums (this is the Mµλ) and see where Darwin’s original table fits in.
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