A LITTLEWOOD-RICHARDSON RULE for FACTORIAL SCHUR FUNCTIONS 1. Introduction As Λ Runs Over All Partitions with Length L(Λ)
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Comultiplication Rules for the Double Schur Functions and Cauchy Identities
Comultiplication rules for the double Schur functions and Cauchy identities A. I. Molev School of Mathematics and Statistics University of Sydney, NSW 2006, Australia [email protected] Submitted: Aug 27, 2008; Accepted: Jan 17, 2009; Published: Jan 23, 2009 Mathematics Subject Classifications: 05E05 Abstract The double Schur functions form a distinguished basis of the ring Λ(xjja) which is a multiparameter generalization of the ring of symmetric functions Λ(x). The canonical comultiplication on Λ(x) is extended to Λ(xjja) in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood{Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood{Richardson coefficients provide a multiplication rule for the dual Schur functions. Contents 1 Introduction 2 2 Double and supersymmetric Schur functions 6 2.1 Definitions and preliminaries . 6 2.2 Analogues of classical bases . 9 2.3 Duality isomorphism . 10 2.4 Skew double Schur functions . 11 3 Cauchy identities and dual Schur functions 14 3.1 Definition of dual Schur functions and Cauchy identities . 14 3.2 Combinatorial presentation . 17 3.3 Jacobi{Trudi-type formulas . 20 3.4 Expansions in terms of Schur functions . 22 the electronic journal of combinatorics 16 (2009), #R13 1 4 Dual Littlewood{Richardson polynomials 29 5 Transition matrices 33 5.1 Pairing between the double and dual symmetric functions . -
A Combinatorial Proof That Schubert Vs. Schur Coefficients Are Nonnegative
A COMBINATORIAL PROOF THAT SCHUBERT VS. SCHUR COEFFICIENTS ARE NONNEGATIVE SAMI ASSAF, NANTEL BERGERON, AND FRANK SOTTILE Abstract. We give a combinatorial proof that the product of a Schubert polynomial by a Schur polynomial is a nonnegative sum of Schubert polynomials. Our proof uses Assaf’s theory of dual equivalence to show that a quasisymmetric function of Bergeron and Sottile is Schur-positive. By a geometric comparison theorem of Buch and Mihalcea, this implies the nonnegativity of Gromov-Witten invariants of the Grassmannian. Dedicated to the memory of Alain Lascoux Introduction A Littlewood-Richardson coefficient is the multiplicity of an irreducible representa- tion of the general linear group in a tensor product of two irreducible representations, and is thus a nonnegative integer. Littlewood and Richardson conjectured a formula for these coefficients in 1934 [25], which was proven in the 1970’s by Thomas [33] and Sch¨utzenberger [31]. Since Littlewood-Richardson coefficients may be defined combina- torially as the coefficients of Schur functions in the expansion of a product of two Schur functions, these proofs of the Littlewood-Richardson rule furnish combinatorial proofs of the nonnegativity of Schur structure constants. The Littlewood-Richardson coefficients are also the structure constants for expressing products in the cohomology of a Grassmannian in terms of its basis of Schubert classes. Independent of the Littlewood-Richardson rule, these Schubert structure constants are known to be nonnegative integers through geometric arguments. The integral cohomol- ogy ring of any flag manifold has a Schubert basis and again geometry implies that the corresponding Schubert structure constants are nonnegative integers. -
The Contributions of Stanley to the Fabric of Symmetric and Quasisymmetric Functions
THE CONTRIBUTIONS OF STANLEY TO THE FABRIC OF SYMMETRIC AND QUASISYMMETRIC FUNCTIONS SARA C. BILLEY AND PETER R. W. MCNAMARA A E S A D I Dedicated to L N T C H R on the occasion of his 70th birthday. Y R Abstract. We weave together a tale of two rings, SYM and QSYM, following one gold thread spun by Richard Stanley. The lesson we learn from this tale is that \Combinatorial objects like to be counted by quasisymmetric functions." 1. Introduction The twentieth century was a remarkable era for the theory of symmetric func- tions. Schur expanded the range of applications far beyond roots of polynomials to the representation theory of GLn and Sn and beyond. Specht, Hall and Mac- donald unified the algebraic theory making it far more accessible. Lesieur recog- nized the connection between Schur functions and the topology of Grassmannian manifolds spurring interest and further developments by Borel, Bott, Bernstein{ Gelfand{Gelfand, Demazure and many others. Now, symmetric functions routinely appear in many aspects of mathematics and theoretical physics, and have significant importance in quantum computation. In that era of mathematical giants, Richard Stanley's contributions to symmet- ric functions are shining examples of how enumerative combinatorics has inspired and influenced some of the best work of the century. In this article, we focus on a few of the gems that continue to grow in importance over time. Specifically, we survey some results and applications for Stanley symmetric functions, chromatic symmetric functions, P -partitions, generalized Robinson{Schensted{Knuth corre- spondence, and flag symmetry of posets. -
Tropicalization, Symmetric Polynomials, and Complexity
TROPICALIZATION, SYMMETRIC POLYNOMIALS, AND COMPLEXITY ALEXANDER WOO AND ALEXANDER YONG ABSTRACT. D. Grigoriev-G. Koshevoy recently proved that tropical Schur polynomials have (at worst) polynomial tropical semiring complexity. They also conjectured tropical skew Schur polynomials have at least exponential complexity; we establish a polynomial complexity upper bound. Our proof uses results about (stable) Schubert polynomials, due to R. P. Stanley and S. Billey-W. Jockusch-R. P. Stanley, together with a sufficient condition for polynomial complexity that is connected to the saturated Newton polytope property. 1. INTRODUCTION The tropicalization of a polynomial X i1 i2 in f = ci1;:::;in x1 x2 ··· xn 2 C[x1; x2; : : : ; xn] n (i1;i2;:::;in)2Z≥0 (with respect to the trivial valuation val(a) = 0 for all a 2 C∗) is defined to be (1) Trop(f) := max fi1x1 + ··· + inxng: i1;:::;in This is a polynomial over the tropical semiring (R; ⊕; ), where a ⊕ b = max(a; b) and a b = a + b respectively denote tropical addition and multiplication, respectively. We refer to the books [ItMiSh09, MaSt15] for more about tropical mathematics. Let Symn denote the ring of symmetric polynomials in x1; : : : ; xn. A linear basis of Symn is given by the Schur polynomials. These polynomials are indexed by partitions λ (identi- fied with their Ferrers/Young diagrams). They are generating series over semistandard Young tableaux T of shape λ with entries from [n] := f1; 2; : : : ; ng: X T T Y #i’s in T sλ(x1; : : : ; xn) := x where x := xi . T i The importance of this basis stems from its applications to, for example, enumerative and algebraic combinatorics, the representation theory of symmetric groups and general linear groups, and Schubert calculus on Grassmannians; see, for example, [Fu97, St99]. -
Combinatorics of Schubert Polynomials
COMBINATORICS OF SCHUBERT POLYNOMIALS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Avery J St. Dizier August 2020 c 2020 Avery J St. Dizier ALL RIGHTS RESERVED COMBINATORICS OF SCHUBERT POLYNOMIALS Avery J St. Dizier, Ph.D. Cornell University 2020 In this thesis, we study several aspects of the combinatorics of various im- portant families of polynomials, particularly focusing on Schubert polynomials. Schubert polynomials arise as distinguished representatives of cohomology classes in the cohomology ring of the flag variety. As polynomials, they enjoy a rich and well-studied combinatorics. Through joint works with Fink and M´esz´aros,we connect the supports of Schubert polynomials to a class of polytopes called gener- alized permutahedra. Through a realization of Schubert polynomials as characters of flagged Weyl modules, we show that the exponents of a Schubert polynomial are exactly the integer points in a generalized permutahedron. We also prove a combinatorial description of this permutahedron. We then study characters of flagged Weyl modules more generally and give an interesting inequality on their coefficients. We next shift our focus onto the coefficients of Schubert polynomials. We describe a construction due to Magyar called orthodontia. We use orthodontia to- gether with the previous inequality for characters to give a complete description of the Schubert polynomials that have only zero and one as coefficients. Through joint work with Huh, Matherne, and M´esz´aros,we next show a discrete log-concavity property of the coefficients of Schubert polynomials. -
The Prism Tableau Model for Schubert Polynomials
THE PRISM TABLEAU MODEL FOR SCHUBERT POLYNOMIALS ANNA WEIGANDT AND ALEXANDER YONG ABSTRACT. The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1; x2;:::]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Grobner¨ geometry of matrix Schubert varieties. CONTENTS 1. Introduction 2 1.1. Overview 2 1.2. The main result 2 1.3. Organization 5 2. Main idea of the model and its proof 5 3. Proof of the Theorem 1.1 7 3.1. Matrix Schubert varieties and Schubert polynomials 7 3.2. Multi-plus diagrams 9 3.3. Local moves on plus diagrams 10 3.4. Proof of Theorem 3.6 15 3.5. Conclusion of the proof of Theorem 1.1 16 4. Further discussion 18 4.1. Comparisons to the literature 18 4.2. Details of the reduction to semistandard tableaux 19 4.3. Stable Schubert polynomials 20 4.4. An overlay interpretation of chute and ladder moves 20 4.5. Future work 21 Acknowledgments 22 References 22 Date: September 8, 2015. 1 1. INTRODUCTION 1.1. Overview. A. Lascoux–M.-P. Schutzenberger¨ [LaSh82a] recursively defined an inte- gral basis of Pol = Z[x1; x2;:::] given by the Schubert polynomials fSw : w 2 S1g. If w0 is n−1 n−2 the longest length permutation in the symmetric group Sn then Sw0 := x1 x2 ··· xn−1. -
Arxiv:1705.09649V2 [Math.CO]
A DEMAZURE CRYSTAL CONSTRUCTION FOR SCHUBERT POLYNOMIALS SAMI ASSAF AND ANNE SCHILLING Abstract. Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur functions and Demazure characters for the general linear group. We establish this connection by imposing a Demazure crystal structure on key tableaux, recently introduced by the first author in connection with Demazure characters and Schubert polynomials, and linking this to the type A crystal structure on reduced word factorizations, recently introduced by Morse and the second author in connection with Stanley symmetric functions. 1. Introduction Schubert polynomials Sw were first introduced by Bernstein et al. [BGG73] as certain polynomial repre- sentatives of cohomology classes of Schubert cycles Xw in flag varieties. They were extensively studied by Lascoux and Sch¨utzenberger [LS82] using an explicit definition in terms of difference operators ∂w. Subse- quently, a combinatorial expression for Schubert polynomials as the generating polynomial for compatible se- quences for reduced expressions of a permutation w was discovered by Billey, Jockusch, and Stanley [BJS93]. In the special case of the Grassmannian subvariety, Schubert polynomials are Schur polynomials, which also arise as the irreducible characters for the general linear group. The Stanley symmetric functions Fw were introduced by Stanley [Sta84] in the pursuit of enumerations of the reduced expressions of permutations, in particular of the long permutation w0. They are defined combinatorially as the generating functions of reduced factorizations of permutations. Stanley symmetric functions are the stable limit of Schubert polynomials [Mac91a, Mac91b], precisely (1.1) Fw(x1, x2,...) = lim S1m×w(x1, x2,...,xn+m). -
The Theory of Schur Polynomials Revisited
THE THEORY OF SCHUR POLYNOMIALS REVISITED HARRY TAMVAKIS Abstract. We use Young’s raising operators to give short and uniform proofs of several well known results about Schur polynomials and symmetric func- tions, starting from the Jacobi-Trudi identity. 1. Introduction One of the earliest papers to study the symmetric functions later known as the Schur polynomials sλ is that of Jacobi [J], where the following two formulas are found. The first is Cauchy’s definition of sλ as a quotient of determinants: λi+n−j n−j (1) sλ(x1,...,xn) = det(xi )i,j . det(xi )i,j where λ =(λ1,...,λn) is an integer partition with at most n non-zero parts. The second is the Jacobi-Trudi identity (2) sλ = det(hλi+j−i)1≤i,j≤n which expresses sλ as a polynomial in the complete symmetric functions hr, r ≥ 0. Nearly a century later, Littlewood [L] obtained the positive combinatorial expansion (3) s (x)= xc(T ) λ X T where the sum is over all semistandard Young tableaux T of shape λ, and c(T ) denotes the content vector of T . The traditional approach to the theory of Schur polynomials begins with the classical definition (1); see for example [FH, M, Ma]. Since equation (1) is a special case of the Weyl character formula, this method is particularly suitable for applica- tions to representation theory. The more combinatorial treatments [Sa, Sta] use (3) as the definition of sλ(x), and proceed from there. It is not hard to relate formulas (1) and (3) to each other directly; see e.g. -
The Permutahedral Variety, Mixed Eulerian Numbers, and Principal Specializations of Schubert Polynomials
THE PERMUTAHEDRAL VARIETY, MIXED EULERIAN NUMBERS, AND PRINCIPAL SPECIALIZATIONS OF SCHUBERT POLYNOMIALS PHILIPPE NADEAU AND VASU TEWARI Abstract. We compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants aw are expressed as a sum of normalized mixed Eulerian numbers indexed naturally by reduced words of w. The description implies that the aw are positive for all permutations w 2 Sn of length n − 1, thereby answering a question of Harada, Horiguchi, Masuda and Park. We use the same expression to establish the invariance of aw under taking inverses and conjugation by the longest word, and subsequently establish an intriguing cyclic sum rule for the numbers. We then move toward a deeper combinatorial understanding for the aw by exploiting in addition the relation to Postnikov's divided symmetrization. Finally, we are able to give a combinatorial interpretation for aw when w is vexillary, in terms of certain tableau descents. It is based in part on a relation between the numbers aw and principal specializations of Schubert polynomials. Along the way, we prove results and raise questions of independent interest about the combina- torics of permutations, Schubert polynomials and related objects. We also sketch how to extend our approach to other Lie types, highlighting in particular an identity of Klyachko. Contents 1. Introduction and statement of results 2 2. Preliminaries 6 3. Formulas for aw 11 4. Mixed Eulerian numbers 13 5. Properties of the numbers aw 16 6. Combinatorial interpretation of aw in special cases 20 7. The case of vexillary permutations 25 8. -
Boolean Product Polynomials, Schur Positivity, and Chern Plethysm
BOOLEAN PRODUCT POLYNOMIALS, SCHUR POSITIVITY, AND CHERN PLETHYSM SARA C. BILLEY, BRENDON RHOADES, AND VASU TEWARI Abstract. Let k ≤ n be positive integers, and let Xn = (x1; : : : ; xn) be a list of n variables. P The Boolean product polynomial Bn;k(Xn) is the product of the linear forms i2S xi where S ranges over all k-element subsets of f1; 2; : : : ; ng. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials Bn;k(Xn) for certain k to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate Bn;n−1(Xn) to a bigraded action of the symmetric group Sn on a divergence free quotient of superspace. 1. Introduction The symmetric group Sn of permutations of [n] := f1; 2; : : : ; ng acts on the polynomial ring C[Xn] := C[x1; : : : ; xn] by variable permutation. Elements of the invariant subring Sn (1.1) C[Xn] := fF (Xn) 2 C[Xn]: w:F (Xn) = F (Xn) for all w 2 Sn g are called symmetric polynomials. Symmetric polynomials are typically defined using sums of products of the variables x1; : : : ; xn. Examples include the power sum, the elementary symmetric polynomial, and the homogeneous symmetric polynomial which are (respectively) (1.2) k k X X pk(Xn) = x1 + ··· + xn; ek(Xn) = xi1 ··· xik ; hk(Xn) = xi1 ··· xik : 1≤i1<···<ik≤n 1≤i1≤···≤ik≤n Given a partition λ = (λ1 ≥ · · · ≥ λk > 0) with k ≤ n parts, we have the monomial symmetric polynomial X (1.3) m (X ) = xλ1 ··· xλk ; λ n i1 ik i1; : : : ; ik distinct as well as the Schur polynomial sλ(Xn) whose definition is recalled in Section 2. -
FRG Report: the Crystal Structure of the Plethysm of Schur Functions
FRG report: The crystal structure of the plethysm of Schur functions Mike Zabrocki, Franco Saliola Apr 1-8, 2018 Our focused research group consisted of Laura Colmenarejo, Rosa Orellana, Franco Saliola, Anne Schilling and Mike Zabrocki. We worked at the BIRS station from April 1 until April 8 (except Rosa Orellana who left on April 6). 1 The problem The irreducible polynomial representations of GLn are indexed by partitions with at most n parts. Given such a representation indexed by the partition λ, its character is the Schur polynomial X weight(T ) sλ = x : (1) T 2SSYT(λ) Composition of these representations becomes composition of their characters, denoted sλ[sµ] and this op- eration is known as the operation of plethysm. Composition of characters is a symmetric polynomial which Littlewood [Lit44] called the (outer) plethysm. The objective of our focused research group project is the resolution of the following well known open problem. ν Problem 1 Find a combinatorial interpretation of the coefficients aλ,µ in the expansion X ν sλ[sµ] = aλ,µsν : (2) ν In the last more than a century of research in representation theory, the basic problem of understanding ν the coefficients aλ,µ has stood as a measure of progress in the field. A related question that we felt was an important first step in the resolution of Problem 1 is the following second approach to understanding the underlying representation theory. Problem 2 Find a combinatorial interpretation for the multiplicity of an irreducible Sn module indexed by a partition λ in an irreducible polynomial GLn module indexed by a partition µ. -
SYMMETRIC POLYNOMIALS and Uq(̂Sl2)
REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 4, Pages 46{63 (February 7, 2000) S 1088-4165(00)00065-0 b SYMMETRIC POLYNOMIALS AND Uq(sl2) NAIHUAN JING Abstract. We study the explicit formula of Lusztig's integral forms of the b level one quantum affine algebra Uq(sl2) in the endomorphism ring of sym- metric functions in infinitely many variables tensored with the group algebra of Z. Schur functions are realized as certain orthonormal basis vectors in the vertex representation associated to the standard Heisenberg algebra. In this picture the Littlewood-Richardson rule is expressed by integral formulas, and −1 b is used to define the action of Lusztig's Z[q; q ]-form of Uq(sl2)onSchur polynomials. As a result the Z[q; q−1]-lattice of Schur functions tensored with the group algebra contains Lusztig's integral lattice. 1. Introduction The relation between vertex representations and symmetric functions is one of the interesting aspects of affine Kac-Moody algebras and the quantum affine alge- bras. In the late 1970's to the early 1980's the Kyoto school [DJKM] found that the polynomial solutions of KP hierarchies are obtained by Schur polynomials. This breakthrough was achieved in formulating the KP and KdV hierarchies in terms of affine Lie algebras. On the other hand, I. Frenkel [F1] identified the two construc- tions of the affine Lie algebras via vertex operators, which put the boson-fermion correspondence in a rigorous formulation. I. Frenkel [F2] further showed that the boson-fermion correspondence can give the Frobenius formula of the irreducible characters for the symmetric group Sn (see also [J1]).