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Combinatorial Aspects of Generalizations of Schur Functions

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Combinatorial Aspects of Generalizations of Schur Functions Combinatorial aspects of generalizations of Schur functions A Thesis Submitted to the Faculty of Drexel University by Derek Heilman in partial fulfillment of the requirements for the degree of Doctor of Philosophy March 2013 CONTENTS ii Contents Abstract iv 1 Introduction 1 2 General background 3 2.1 Symmetric functions . .4 2.2 Schur functions . .6 2.3 The Hall inner product . .9 2.4 The Pieri rule for Schur functions . 10 3 The Pieri rule for the dual Grothendieck polynomials 16 3.1 Grothendieck polynomials . 16 3.2 Dual Grothendieck polynomials . 17 3.3 Elegant fillings . 18 3.4 Pieri rule for the dual Grothendieck polynomials . 21 4 Insertion proof of the dual Grothendieck Pieri rule 24 4.1 Examples . 25 4.2 Insertion algorithm . 27 4.3 Sign changing involution on reverse plane partitions and XO-diagrams 32 4.4 Combinatorial proof of the dual Grothendieck Pieri rule . 41 5 Factorial Schur functions and their expansion 42 5.1 Definition of a factorial Schur polynomial . 43 5.2 The expansion of factorial Schur functions in terms of Schur functions 47 6 A reverse change of basis 52 6.1 Change of basis coefficients . 52 CONTENTS iii 6.2 A combinatorial involution . 55 6.3 Reverse change of basis . 57 References 59 ABSTRACT iv Abstract Combinatorial aspects of generalizations of Schur functions Derek Heilman Jennifer Morse, Ph.D The understanding of the space of symmetric functions is gained through the study of its bases. Certain bases can be defined by purely combinatorial methods, some- times enabling important properties of the functions to fall from carefully constructed combinatorial algorithms. A classic example is given by the Schur basis, made up of functions that can be defined using semi-standard Young tableaux. The Pieri rule for multiplying an important special case of Schur functions is proven using an insertion algorithm on tableaux that was defined by Robinson, Schensted, and Knuth. Further- more, the transition matrices between Schur functions and other symmetric function bases are often linked to representation theoretic multiplicities. The description of these matrices can sometimes be given combinatorially as the enumeration of a set of objects such as tableaux. A similar combinatorial approach is applied here to a basis for the symmetric function space that is dual to the Grothendieck polynomial basis. These polynomials are defined combinatorially using reverse plane partitions. Bijecting reverse plane partitions with a subset of semi-standard Young tableaux over a doubly-sized alphabet enables the extension of RSK-insertion to reverse plane partitions. This insertion, paired with a sign changing involution, is used to give the desired combinatorial proof of the Pieri rule for this basis. Another basis of symmetric functions is given by the set of factorial Schur functions. While their expansion into Schur functions can ABSTRACT v be described combinatorially, the reverse change of basis had no such formulation. A new set of combinatorial objects is introduced to describe the expansion coefficients, and another sign changing involution is used to prove that these do in fact encode the transition matrices. 1 INTRODUCTION 1 1 Introduction Symmetric functions play a large role in many mathematical fields including group theory, Lie algebras, and algebraic geometry. There are many different bases for the ring of symmetric functions, one of the most fundamental is the Schur functions. Schur functions are indexed by partitions and are directly connected to other mathematical fields including geometry and representation theory. The most usual definition of a Schur function is the combinatorial one (Section 2). This definition uses combinatorial objects allowing many instrumental mathematical proofs. These combinatorial proofs can be seen visually acting on these objects. One example is the RSK insertion proof, which proves the Pieri rule for Schur functions. Another set of functions that form a basis for the ring of symmetric functions are the Grothendieck polynomials, they have many similar properties to those of Schur functions [6; 8; 15; 18]. Lascoux and Schutzenberger introduced Grothendieck polynomials [14]. These are inhomogeneous polynomials representing classes of structure sheaves of Schubert varieties in the Grothendieck ring of the flag varieties. Fomin and Kirillov continued the study of these polynomials giving a combinatorial construction of Grothendieck polynomials in terms of rc-graphs [5]. Buch continued work on these polynomials and developed the Littlewood-Richardson rule for them [3]. Similar to the Littlewood- Richardson rule for Schur functions, this rule defines the coefficients for the expansion of the product of two Grothendieck polynomials in terms of Grothendieck polynomi- als. The set of polynomials that are dual to the Grothendieck polynomials is known as the dual Grothendieck polynomials. The Littlewood-Richardson coefficients for these dual polynomials are also derived in Buch's paper by means of the coproduct [3]. The dual Grothendieck polynomials were first studied directly and called dual stable Grothendieck polynomials [11]. They can be defined combinatorially using 1 INTRODUCTION 2 objects known as reverse plane partitions. Recall that the Pieri rule for Schur func- tions has a very elegant proof using RSK insertion. However, there was no such elegant proof for the dual Grothendieck polynomials due to the structure of the re- verse plane partitions. There does exists a bijection between reverse plane partitions and pairs of semi-standard Young tableaux and elegant fillings. Lam and Pylyavskyy defined one bijection and Bandlow and Morse defined another slightly more intuitive approach [11; 1]. These topics will be covered in Section 3. Using these pairs of semi-standard Young tableaux and elegant fillings, a similar insertion method will be used to construct an insertion based proof for the Pieri rule for the dual Grothendieck polynomials (Section 4). The proof will also require a sign changing involution to be defined to account for the remaining terms (section 4). Schur functions, when limited to n variables, can be generalized to a different type of functions. One set of these functions are the factorial Schur functions. They are a generalization of Schur functions introducing another set of variables a. When Biedenharn and Louck first discovered them, they fixed the values of the variables a to ai = 1 − i [2]. This was done to decompose tensor products of representations when using particular bases. Factorial Schur functions are special cases of double Schu- bert polynomials for Grassmannian permutations [14; 13]. Knutson and Tao also showed that factorial Schur functions are the equivariant cohomology of Grassmanni- ans [9]. Chen and Louck gave new foundations based on divided difference operators [4]. Goulden and Hamel further developed the analogy between Schur functions and factorial Schur functions [7]. Molev and Sagan found a Littlewood-Richardson rule for factorial Schur functions as well as other useful results [17]. Kreiman later dis- covered more interesting facts for factorial Schur functions [10]. One of these facts is the the change of basis formula for expanding factorial Schur functions in terms of Schur functions, as well as Schur functions in terms of factorial Schur functions. Molev also gave an easy combinatorial method for computing the coefficients of the 2 GENERAL BACKGROUND 3 expansion of factorial Schur functions in terms of Schur functions, which is covered in Section 5 [16]. Although this method can be used to compute the reverse expansion, it requires more mechanics. Section 6 will give a simple combinatorial method of describing these reverse change of basis coefficients and an elegant proof. 2 General background This section will describe symmetric functions, bases for the ring of symmetric func- tions, and tools for Schur functions from algebraic combinatorics. The theory of sym- metric functions applies to enumerative combinatorics. These applications branch out to many other fields of mathematics including group theory, Lie algebras, algebraic geometry, and representation theory. Partitions and Ferrers diagrams will play an important role in indexing and describing bases for the ring of symmetric functions. Fillings of Ferrers diagrams and particularly semi-standard Young tableaux are the primary combinatorial objects used to both define Schur functions and appear as a tool in various combinatorial proofs. Monomial symmetric functions and the complete homogeneous symmetric polynomials are two bases for the ring of symmetric func- tions. The Hall inner product is defined using the the monomial symmetric functions and the complete homogeneous symmetric polynomials. This Hall inner product is used to define duality between two bases for the ring of symmetric functions. The Schur functions are the only set of functions that is dual to itself. Expanding products of Schur functions in terms of Schur functions was a classical problem in the field of algebraic combinatorics. A simplified version of this problem is the Pieri rule for the Schur functions. The classical proof of the Pieri rule for Schur functions was done using the RSK algorithm [6; 8; 15; 18]. 2 GENERAL BACKGROUND 4 2.1 Symmetric functions For an m-tuple of non-negative integers, γ = (γ1; γ2; ··· ; γm), and n independent variables, (x1; x2; : : : ; xn), define γ (γ1,γ2,...,γm) γ1 γ2 γm x = x = x1 x2 : : : xm . For an infinite sequence of non-negative integers ,γ = (γ1; γ2;:::), and infinite inde- pendent variables, (x1; x2;:::), define γ (γ1,γ2;:::) γ1 γ2 x = x = x1 x2 ··· . (2;1;3;0;4) 2 1 3 0 4 2 3 4 Example. x = x1x2x3x4x5 = x1x2x3x5. P α For a set of n independent variables x = (x1; x2; : : : ; xn), given f(x) = cαx , where α ranges over n-tuples of non-negative integers and cα 2 R, f(x) is a symmetric function of n variables if f(x) = f(!(x)) 8! 2 Sn, where Sn is the symmetric group of degree n. Therefore f(x) is invariant by any permutation on the variables x.Λn is the ring formed from all the symmetric functions of n variables.
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