Gröbner Geometry of Schubert Polynomials
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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. -
Combinatorial Results on (1, 2, 1, 2)-Avoiding $ GL (P,\Mathbb {C
COMBINATORIAL RESULTS ON (1, 2, 1, 2)-AVOIDING GL(p, C) × GL(q, C)-ORBIT CLOSURES ON GL(p + q, C)/B ALEXANDER WOO AND BENJAMIN J. WYSER Abstract. Using recent results of the second author which explicitly identify the “(1, 2, 1, 2)- avoiding” GL(p, C)×GL(q, C)-orbit closures on the flag manifold GL(p+q, C)/B as certain Richardson varieties, we give combinatorial criteria for determining smoothness, lci-ness, and Gorensteinness of such orbit closures. (In the case of smoothness, this gives a new proof of a theorem of W.M. McGovern.) Going a step further, we also describe a straightforward way to compute the singular locus, the non-lci locus, and the non-Gorenstein locus of any such orbit closure. We then describe a manifestly positive combinatorial formula for the Kazhdan-Lusztig- Vogan polynomial Pτ,γ (q) in the case where γ corresponds to the trivial local system on a (1, 2, 1, 2)-avoiding orbit closure Q and τ corresponds to the trivial local system on any orbit ′ Q contained in Q. This combines the aforementioned result of the second author, results of A. Knutson, the first author, and A. Yong, and a formula of Lascoux and Sch¨utzenberger which computes the ordinary (type A) Kazhdan-Lusztig polynomial Px,w(q) whenever w ∈ Sn is cograssmannian. 1. Introduction Let G be a complex semisimple reductive group and B a Borel subgroup of G. The opposite Borel subgroup B− (as well as B itself) acts on the flag variety G/B with finitely many orbits. -
Non-Openness and Non-Equidimensionality in Algebraic Quotients
Pacific Journal of Mathematics NON-OPENNESS AND NON-EQUIDIMENSIONALITY IN ALGEBRAIC QUOTIENTS JOHN DUSTIN DONALD Vol. 41, No. 2 December 1972 PACIFIC JOURNAL OF MATHEMATICS Vol. 41, No. 2, 1972 NON-OPENNESS AND NON-EQUIDIMENSIONALΠΎ IN ALGEBRAIC QUOTIENTS JOHN D. DONALD Suppose given an equivalence relation R on an algebraic variety V and the associated fibering of V by a family of subvarieties. This paper treats the question of the existence of a quotient structure for this situation when the fibering is non-equidimensional. For this purpose a general definition of quotient variety for algebraic equivalence relations is used which contains no topological requirements. The results are of two types. In §1 it is shown that certain maps into nonsingular varieties are quotient maps for the induced equivalence relation whenever the union of the excessive orbits has codimension ^ 2. This theorem yields many examples of non-equidimensional quotients. Section 2 contains a converse showing that no excessive orbit containing a normal hypersurface can be fitted into a quotient. This theorem depends on a stronger and less conceptual field- theoretic result which fails without the normality hypothesis. Section 3 contains a counterexample. These results are a first attempt at answering the question: when can a good algebraic structure be assigned to a non-equidimensional fibering of VΊ Analogously, one might ask whether there is a good sense in which an equidimensional family of subvarieties of V can degenerate into subvarieties of different dimension. It is a standard result that no fibers of a morphism have dimension less than that of the generic fiber, so that we must concern ourselves only with degeneration into higher-dimensional subvarieties. -
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. -
Introduction to Schubert Varieties Lms Summer School, Glasgow 2018
INTRODUCTION TO SCHUBERT VARIETIES LMS SUMMER SCHOOL, GLASGOW 2018 MARTINA LANINI 1. The birth: Schubert's enumerative geometry Schubert varieties owe their name to Hermann Schubert, who in 1879 pub- lished his book on enumerative geometry [Schu]. At that time, several people, such as Grassmann, Giambelli, Pieri, Severi, and of course Schubert, were inter- ested in this sort of questions. For instance, in [Schu, Example 1 of x4] Schubert asks 3 1 How many lines in PC do intersect 4 given lines? Schubert's answer is two, provided that the lines are in general position: How did he get it? And what does general position mean? Assume that we can divide the four lines `1; `2; `3; `4 into two pairs, say f`1; `2g and f`3; `4g such that the lines in each pair intersect in exactly one point and do not touch the other two. Then each pair generates a plane, and we assume that these two planes, say π1 and π2, are not parallel (i.e. the lines are in general position). Then, the planes intersect in a line. If this happens, there are exactly two lines intersecting `1; `2; `3; `4: the one lying in the intersection of the two planes and the one passing through the two intersection points `1 \`2 and `3 \ `4. Schubert was using the principle of special position, or conservation of num- ber. This principle says that the number of solutions to an intersection problem is the same for any configuration which admits a finite number of solutions (if this is the case, the objects forming the configuration are said to be in general position). -
The Algebraic Geometry of Kazhdan-Lusztig-Stanley Polynomials
The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials Nicholas Proudfoot Department of Mathematics, University of Oregon, Eugene, OR 97403 [email protected] Abstract. Kazhdan-Lusztig-Stanley polynomials are combinatorial generalizations of Kazhdan- Lusztig polynomials of Coxeter groups that include g-polynomials of polytopes and Kazhdan- Lusztig polynomials of matroids. In the cases of Weyl groups, rational polytopes, and realizable matroids, one can count points over finite fields on flag varieties, toric varieties, or reciprocal planes to obtain cohomological interpretations of these polynomials. We survey these results and unite them under a single geometric framework. 1 Introduction Given a Coxeter group W along with a pair of elements x; y 2 W , Kazhdan and Lusztig [KL79] defined a polynomial Px;y(t) 2 Z[t], which is non-zero if and only if x ≤ y in the Bruhat order. This polynomial has a number of different interpretations in terms of different areas of mathematics: • Combinatorics: There is a purely combinatorial recursive definition of Px;y(t) in terms of more elementary polynomials, called R-polynomials. See [Lus83a, Proposition 2], as well as [BB05, x5.5] for a more recent account. • Algebra: The Hecke algebra of W is a q-deformation of the group algebra C[W ], and the polynomials Px;y(t) are the entries of the matrix relating the Kazhdan-Lusztig basis to the standard basis of the Hecke algebra [KL79, Theorem 1.1]. • Geometry: If W is the Weyl group of a semisimple Lie algebra, then Px;y(t) may be interpreted as the Poincar´epolynomial of a stalk of the intersection cohomology sheaf on a Schubert variety in the associated flag variety [KL80, Theorem 4.3]. -
On the Product in Negative Tate Cohomology for Finite Groups 2
ON THE PRODUCT IN NEGATIVE TATE COHOMOLOGY FOR FINITE GROUPS HAGGAI TENE Abstract. Our aim in this paper is to give a geometric description of the cup product in negative degrees of Tate cohomology of a finite group with integral coefficients. By duality it corresponds to a product in the integral homology of BG: Hn(BG, Z) ⊗ Hm(BG, Z) → Hn+m+1(BG, Z) for n,m > 0. We describe this product as join of cycles, which explains the shift in dimensions. Our motivation came from the product defined by Kreck using stratifold homology. We then prove that for finite groups the cup product in negative Tate cohomology and the Kreck product coincide. The Kreck product also applies to the case where G is a compact Lie group (with an additional dimension shift). Introduction For a finite group G one defines Tate cohomology with coefficients in a Z[G] module M, denoted by H∗(G, M). This is a multiplicative theory: b Hn(G, M) ⊗ Hm(G, M ′) → Hn+m(G, M ⊗ M ′) b b b and the product is called cup product. For n > 0 there is a natural isomor- phism Hn(G, M) → Hn(G, M), and for n < −1 there is a natural isomorphism b Hn(G, M) → H (G, M). We restrict ourselves to coefficients in the triv- b −n−1 ial module Z. In this case, H∗(G, Z) is a graded ring. Also, in this case the b group cohomology and homology are actually the cohomology and homology of a topological space, namely BG, the classifying space of principal G bundles - n n arXiv:0911.3014v2 [math.AT] 27 Jul 2011 H (G, Z) =∼ H (BG, Z) and Hn(G, Z) =∼ Hn(BG, Z). -
Fine and Coarse Moduli Spaces in the Representation Theory of Finite Dimensional Algebras
FINE AND COARSE MODULI SPACES IN THE REPRESENTATION THEORY OF FINITE DIMENSIONAL ALGEBRAS B. Huisgen-Zimmermann Dedicated to Ragnar-Olaf Buchweitz on the occasion of his seventieth birthday Abstract. We discuss the concepts of fine and coarse moduli spaces in the context of fi- nite dimensional algebras over algebraically closed fields. In particular, our formulation of a moduli problem and its potential strong or weak solution is adapted to classification prob- lems arising in the representation theory of such algebras. We then outline and illustrate a dichotomy of strategies for concrete applications of these ideas. One method is based on the classical affine variety of representations of fixed dimension, the other on a projective variety parametrizing the same isomorphism classes of modules. We state sample results and give numerous examples to exhibit pros and cons of the two lines of approach. The juxtaposition highlights differences in techniques and attainable goals. 1. Introduction and notation The desire to describe/classify the objects of various algebro-geometric categories via collections of invariants is a red thread that can be traced throughout mathematics. Promi- nent examples are the classification of similarity classes ofmatricesintermsofnormal forms, the classification of finitely generated abelian groups in terms of annihilators of their indecomposable direct summands, and the classification of varieties of fixed genus and dimension up to isomorphism or birational equivalence, etc., etc. – the reader will readily extend the list. In each setting, one selects an equivalence relation on the collection of objects to be sorted; the “invariants” one uses to describetheobjectsarequantitiesnot depending on the choice of representatives from the considered equivalence classes; and the chosen data combine to finite parcels that identify these classes, preferably without redundancy. -
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. -
Invariants of a Maximal Unipotent Subgroup and Equidimensionality
November 7, 2011 INVARIANTS OF A MAXIMAL UNIPOTENT SUBGROUP AND EQUIDIMENSIONALITY DMITRI I. PANYUSHEV INTRODUCTION The ground field | is algebraically closed and of characteristic zero. Let G be a semisimple algebraic group. Fix a maximal unipotent subgroup U ⊂ G and a maximal torus T ⊂ B = NG(U). Let G act on an affine variety X. As was shown by Hadziev,ˇ the algebra of U- invariants, |[X]U , is finitely generated. He constructed a finitely generated |-algebra A, equipped with a locally finite representation of G (“the universal algebra”), such that U G (0·1) |[X] ' (|[X] ⊗ A) ; [6, Theorem 3.1]. Actually, A is isomorphic to |[G]U and using our notation, the variety Spec (A) is denoted by C(X+), where X+ is the monoid of dominant weights with respect to (T;U), see Section1. This variety has a dense G-orbit and the stabiliser of any x 2 C(X+) contains a maximal unipotent subgroup of G. Affine varieties having these two properties have been studied in [14]. They are called S-varieties. There is a bijection between the G- isomorphism classes of S-varieties and the finitely generated monoids in X+. If S ⊂ X+ is a finitely generated monoid, then C(S) stands for the corresponding S-variety. This note contains two contributions to the theory of U-invariants related to the equidi- mensionality property of quotient morphisms. C-1) Set Z = X ×C(X+). Isomorphism (0·1) means that the corresponding affine varieties, X==U and Z==G, are isomorphic. This suggests that there can be a relationship between fibres of two quotient morphisms πX;U : X ! X==U and πZ;G : Z ! Z==G. -
Introduction to Affine Grassmannians
Introduction to Affine Grassmannians Sara Billey University of Washington http://www.math.washington.edu/∼billey Connections for Women: Algebraic Geometry and Related Fields January 23, 2009 0-0 Philosophy “Combinatorics is the equivalent of nanotechnology in mathematics.” 0-1 Outline 1. Background and history of Grassmannians 2. Schur functions 3. Background and history of Affine Grassmannians 4. Strong Schur functions and k-Schur functions 5. The Big Picture New results based on joint work with • Steve Mitchell (University of Washington) arXiv:0712.2871, 0803.3647 • Sami Assaf (MIT), preprint coming soon! 0-2 Enumerative Geometry Approximately 150 years ago. Grassmann, Schubert, Pieri, Giambelli, Severi, and others began the study of enumerative geometry. Early questions: • What is the dimension of the intersection between two general lines in R2? • How many lines intersect two given lines and a given point in R3? • How many lines intersect four given lines in R3 ? Modern questions: • How many points are in the intersection of 2,3,4,. Schubert varieties in general position? 0-3 Schubert Varieties A Schubert variety is a member of a family of projective varieties which is defined as the closure of some orbit under a group action in a homogeneous space G/H. Typical properties: • They are all Cohen-Macaulay, some are “mildly” singular. • They have a nice torus action with isolated fixed points. • This family of varieties and their fixed points are indexed by combinatorial objects; e.g. partitions, permutations, or Weyl group elements. 0-4 Schubert Varieties “Honey, Where are my Schubert varieties?” Typical contexts: • The Grassmannian Manifold, G(n, d) = GLn/P . -
Properties Defined on the Basis of Coincidence in GFO-Space
Properties Defined on the Basis of Coincidence in GFO-Space Ringo BAUMANN a, Frank LOEBE a;1 and Heinrich HERRE b a Computer Science Institute, University of Leipzig, Germany b IMISE, University of Leipzig, Germany Abstract. The General Formal Ontology (GFO) is a top-level ontology that in- cludes theories of time and space, two domains of entities of fundamental charac- ter. In this connection the axiomatic theories of GFO are based on the relation of coincidence, which can apply to the boundaries of time and space entities. Two co- incident, but distinct boundaries account for distinct temporal or spatial locations of no distance. This is favorable for modeling or grasping certain situations. At a second glance and especially concerning space, in combination with axioms on boundaries and mereology coincidence also gives rise to a larger variety of entities. This deserves ontological scrutiny. In the present paper we introduce a new notion (based on coincidence) referred to as ‘continuousness’ in order to capture and become capable of addressing an aspect of space entities that, as we argue, is commonly implicitly assumed for basic kinds of entities (such as ‘line’ or ‘surface’). Carving out that definition first leads us to adopting a few new axioms for GFO-Space. Together with the two further properties of connectedness and ordinariness we then systematically investigate kinds of space entities. Keywords. top-level ontology, ontology of space, coincidence, GFO 1. Introduction The General Formal Ontology (GFO) [1] is one of those top-level ontologies that aims at accounting for entities of time and space by means of axiomatic theories about these do- mains.