The Geometry and Topology of the Dual Braid Complex
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Affine Partitions and Affine Grassmannians
AFFINE PARTITIONS AND AFFINE GRASSMANNIANS SARA C. BILLEY AND STEPHEN A. MITCHELL Abstract. We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott’s formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of Bousquet-Melou-Eriksson, Eriksson-Eriksson and Reiner. In other types the identities appear to be new. For type An, the affine colored partitions form another family of combinatorial objects in bijection with n + 1-core parti- tions and n-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the affine Grassmannians in terms of affine partitions and a generalization of Young’s lattice which refines weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted. 1. Introduction Let W be a finite irreducible Weyl group associated to a simple connected compact Lie Group G, and let Wf be its associated affine Weyl group. In analogy with the Grassmannian manifolds in classical type A, the quotient Wf /W is the indexing set for the Schubert varieties in the affine Grassmannians LG. Let WfS be the minimal length coset representatives for Wf /W . Much of the geometry and topology for the affine Grassmannians can be studied from the combinatorics of WfS and vice versa. For example, Bott [6] showed that the Poincar´eseries for the cohomology ring for the affine Grassmannian is equal to the length generating function for WfS, and this series can be written in terms of the exponents e1, e2, . -
Diagrams of Affine Permutations and Their Labellings Taedong
Diagrams of Affine Permutations and Their Labellings by Taedong Yun Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2013 c Taedong Yun, MMXIII. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Author.............................................................. Department of Mathematics April 29, 2013 Certified by. Richard P. Stanley Professor of Applied Mathematics Thesis Supervisor Accepted by . Paul Seidel Chairman, Department Committee on Graduate Theses 2 Diagrams of Affine Permutations and Their Labellings by Taedong Yun Submitted to the Department of Mathematics on April 29, 2013, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract We study affine permutation diagrams and their labellings with positive integers. Bal- anced labellings of a Rothe diagram of a finite permutation were defined by Fomin- Greene-Reiner-Shimozono, and we extend this notion to affine permutations. The balanced labellings give a natural encoding of the reduced decompositions of affine permutations. We show that the sum of weight monomials of the column-strict bal- anced labellings is the affine Stanley symmetric function which plays an important role in the geometry of the affine Grassmannian. Furthermore, we define set-valued balanced labellings in which the labels are sets of positive integers, and we investi- gate the relations between set-valued balanced labellings and nilHecke words in the nilHecke algebra. A signed generating function of column-strict set-valued balanced labellings is shown to coincide with the affine stable Grothendieck polynomial which is related to the K-theory of the affine Grassmannian. -
NON-POSITIVELY CURVED CUBE COMPLEXES Henry Wilton Last Updated: June 29, 2012
Cube complexes Winter 2011 NON-POSITIVELY CURVED CUBE COMPLEXES Henry Wilton Last updated: June 29, 2012 1 Some basics of topological and geometric group theory 1.1 Presentations and complexes Let Γ be a discrete group, defined by a presentation P = hai j rji, say, or as the fundamental group of a connected CW-complex X. 0 1 Remark 1.1. Let XP be the CW-complex with a single 0-cell E , one 1-cell Ei 2 for each ai (oriented accordingly), and one 2-cell Ej for each rj, with attaching 2 (1) map @Ej ! XP that reads off the word rj in the generators faig. Then, by the Seifert{van Kampen Theorem, the fundamental group of XP is the group presented by P. Conversely, restricting attention to X(2) and contracting a maximal tree in (1) X , we obtain XP for some presentation P of Γ. So we see that these two situations are equivalent. We will call XP a pre- sentation complex for Γ. Here are two typical questions that we might want to be able to answer about Γ. ∗ Question 1.2. Can we compute the homology H∗(Γ) or cohomology H (Γ)? Question 1.3. Can we solve the word problem in Γ? That is, is there an algorithm to determine whether a word in the generators represents the trivial element in Γ? 1.2 Non-positively curved manifolds In general, the answer to both of these questions is `no'. Novikov (1955) and Boone (1959) constructed a finitely presentable group with an unsolvable word problem, and Cameron Gordon proved that the rank of H2(Γ) is not computable. -
Fundamental Groups of Links of Isolated Singularities
FUNDAMENTAL GROUPS OF LINKS OF ISOLATED SINGULARITIES MICHAEL KAPOVICH AND JANOS´ KOLLAR´ Starting with Grothendieck's proof of the local version of the Lefschetz hyper- plane theorems [Gro68], it has been understood that there are strong parallels between the topology of smooth projective varieties and the topology of links of isolated singularities. This relationship was formulated as one of the guiding prin- ciples in the monograph [GM88, p.26]: \Philosophically, any statement about the projective variety or its embedding really comes from a statement about the sin- gularity at the point of the cone. Theorems about projective varieties should be consequences of more general theorems about singularities which are no longer re- quired to be conical". The aim of this note is to prove the following, which we consider to be a strong exception to this principle. Theorem 1. For every finitely presented group G there is an isolated, 3-dimensi- ∼ onal, complex singularity 0 2 XG with link LG such that π1 LG = G. By contrast, the fundamental groups of smooth projective varieties are rather special; see [ABC+96] for a survey. Even the fundamental groups of smooth quasi projective varieties are quite restricted [Mor78, KM98a, CS08, DPS09]. This shows that germs of singularities can also be quite different from quasi projective varieties. We think of a complex singularity (0 2 X) as a contractible Stein space sitting N 2N−1 in some C . Then its link is link(X) := X \ S , an intersection of X with a small (2N − 1)-sphere centered at 0 2 X. Thus link(X) is a deformation retract of X n f0g. -
Affine Permutations and Rational Slope Parking Functions
AFFINE PERMUTATIONS AND RATIONAL SLOPE PARKING FUNCTIONS EUGENE GORSKY, MIKHAIL MAZIN, AND MONICA VAZIRANI ABSTRACT. We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund’s bijection ζ exchanging the pairs of statistics (area, dinv) and (bounce, area) on Dyck paths, and the Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions. We also relate our combinatorial constructions to representation theory. We derive new formulas for the Poincar´epolynomials of certain affine Springer fibers and describe a connection to the theory of finite dimensional representations of DAHA and nonsymmetric Macdonald polynomials. 1. INTRODUCTION Parking functions are ubiquitous in the modern combinatorics. There is a natural action of the symmetric group on parking functions, and the orbits are labeled by the non-decreasing parking functions which correspond natu- rally to the Dyck paths. This provides a link between parking functions and various combinatorial objects counted by Catalan numbers. In a series of papers Garsia, Haglund, Haiman, et al. [18, 19], related the combinatorics of Catalan numbers and parking functions to the space of diagonal harmonics. There are also deep connections to the geometry of the Hilbert scheme. Since the works of Pak and Stanley [29], Athanasiadis and Linusson [5] , it became clear that parking functions are tightly related to the combinatorics of the affine symmetric group. -
Affine Symmetric Group Joel Brewster Lewis¹*
WikiJournal of Science, 2021, 4(1):3 doi: 10.15347/wjs/2021.003 Encyclopedic Review Article Affine symmetric group Joel Brewster Lewis¹* Abstract The affine symmetric group is a mathematical structure that describes the symmetries of the number line and the regular triangular tesselation of the plane, as well as related higher dimensional objects. It is an infinite extension of the symmetric group, which consists of all permutations (rearrangements) of a finite set. In addition to its geo- metric description, the affine symmetric group may be defined as the collection of permutations of the integers (..., −2, −1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as a group with certain gen- erators and relations. These different definitions allow for the extension of many important properties of the finite symmetric group to the infinite setting, and are studied as part of the fields of combinatorics and representation theory. Definitions Non-technical summary The affine symmetric group, 푆̃푛, may be equivalently Flat, straight-edged shapes (like triangles) or 3D ones defined as an abstract group by generators and rela- (like pyramids) have only a finite number of symme- tions, or in terms of concrete geometric and combina- tries. In contrast, the affine symmetric group is a way to torial models. mathematically describe all the symmetries possible when an infinitely large flat surface is covered by trian- gular tiles. As with many subjects in mathematics, it can Algebraic definition also be thought of in a number of ways: for example, it In terms of generators and relations, 푆̃푛 is generated also describes the symmetries of the infinitely long by a set number line, or the possible arrangements of all inte- gers (..., −2, −1, 0, 1, 2, ...) with certain repetitive pat- 푠 , 푠 , … , 푠 0 1 푛−1 terns. -
The Enumeration of Fully Commutative Affine Permutations
The enumeration of fully commutative affine permutations Christopher R. H. Hanusa, Brant C. Jones To cite this version: Christopher R. H. Hanusa, Brant C. Jones. The enumeration of fully commutative affine permutations. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.457-468. hal-01215077 HAL Id: hal-01215077 https://hal.inria.fr/hal-01215077 Submitted on 13 Oct 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2011, Reykjav´ık, Iceland DMTCS proc. AO, 2011, 457–468 The enumeration of fully commutative affine permutations Christopher R. H. Hanusa1yand Brant C. Jones2z 1Department of Mathematics, Queens College (CUNY), NY, USA 2Department of Mathematics and Statistics, James Madison University, Harrisonburg, VA, USA Abstract. We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci–Del Lungo–Pergola–Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations. -
Arxiv:Math/0601483V1 [Math.CO] 19 Jan 2006 Rmwihtelsoxshuznegrte a Ededuced
A LITTLE BIJECTION FOR AFFINE STANLEY SYMMETRIC FUNCTIONS THOMAS LAM AND MARK SHIMOZONO Abstract. Little [13] developed a combinatorial algorithm to study the Schur- positivity of Stanley symmetric functions and the Lascoux-Sch¨utzenberger tree. We generalize this algorithm to affine Stanley symmetric functions, which were introduced recently in [7]. 1. Introduction A new family of symmetric functions, called affine Stanley symmetric functions were recently introduced in [7]. These symmetric functions F˜w, indexed by affine permutations w ∈ S˜n, are an affine analogue of the Stanley symmetric functions Fw which Stanley [16] introduced to enumerate the reduced decompositions of a per- mutation w ∈ Sn. Stanley symmetric functions were later shown to be stable limits of the Schubert polynomials Sw [12, 1]. In the case that w is a Grassmannian per- mutation the Stanley symmetric function is equal to a Schur function. Shimozono conjectured and Lam [8] recently showed that the symmetric functions F˜w also had a geometric interpretation. When w is affine Grassmannian then F˜w represents a ∗ Schubert class in the cohomology H (G/P) of the affine Grassmannian and F˜w is called an affine Schur function. Affine Schur functions were introduced by Lapointe and Morse in [10], where they were called dual k-Schur functions. A key property of the Stanley symmetric function Fw is that its expansion in terms of the Schur functions sλ involves non-negative coefficients. This was proved by Edelman and Greene [2] using an insertion algorithm and separately by Las- coux and Sch¨utzenberger [12] via transition formulae for Schubert polynomials. -
Lectures on Representations of Surface Groups François Labourie
Lectures on Representations of Surface Groups François Labourie To cite this version: François Labourie. Lectures on Representations of Surface Groups. EMS publishing house, pp.145, 2013, Zurich Lectures in Advanced Mathematics, 978-3-03719-127-9. hal-00627916 HAL Id: hal-00627916 https://hal.archives-ouvertes.fr/hal-00627916 Submitted on 29 Sep 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Copyright Lectures on Representations of Surface Groups Notes of a course given in ETH-Z¨urich Fall 2006, and Orsay Spring 2007-2008. Fran¸coisLABOURIE 1 (Based on notes taken by Tobias HARTNICK) April 12, 2011 1Univ. Paris-Sud, Laboratoire de Math´ematiques,Orsay F-91405 Cedex; CNRS, Orsay cedex, F-91405, The research leading to these results has received funding from the European Research Council under the European Community's seventh Frame- work Programme (FP7/2007-2013)/ERC grant agreement n◦ FP7-246918, as well as well as from the ANR program ETTT (ANR-09-BLAN-0116-01) Chapter 1 Introduction The purpose of this lecture series is to describe the representation variety { or character variety{ of the fundamental group π1(S) of a closed connected surface S of genus greater than 2, with values in a Lie group G. -
The Enumeration of Fully Commutative Affine Permutations
THE ENUMERATION OF FULLY COMMUTATIVE AFFINE PERMUTATIONS CHRISTOPHER R. H. HANUSA AND BRANT C. JONES ABSTRACT. We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci–Del Lungo–Pergola–Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations. 1. INTRODUCTION Let W be a Coxeter group. An element w of W is fully commutative if any reduced expression for w can be obtained from any other using only commutation relations among the generators. For example, if W is simply laced then the fully commutative elements of W are those with no sisjsi factor in any reduced expression, where si and sj are any noncommuting generators. The fully commutative elements form an interesting class of Coxeter group elements with many spe- cial properties relating to smoothness of Schubert varieties [Fan98], Kazhdan–Lusztig polynomials and µ-coefficients [BW01, Gre07], Lusztig’s a(w)-function [BF98, Shi05], and decompositions into cells [Shi03, FS97, GL01]. Some of the properties carry over to the freely braided and maximally clustered elements introduced in [GL02, GL04, Los07]. At the level of the Coxeter group, [Ste96] shows that each fully commutative element w has a unique labeled partial order called the heap of w whose linear extensions encode all of the reduced expressions for w. Stembridge [Ste96] (see also [Fan95, Gra95]) classified the Coxeter groups having finitely many fully commutative elements. -
Affine Permutations and an Affine Coxeter Monoid
Chapter 1 Affine Permutations and an Affine Coxeter Monoid Tom Denton Abstract In this expository paper, we describe results on pattern avoidance arising from the affine Catalan monoid. The schema of affine codes as canonical decompo- sitions in conjunction with two-row moves is detailed, and then applied in studying the Catalan quotient of the 0-Hecke monoid. We prove a conjecture of Hanusa and Jones concerning periodicity in the number of fully-commutative affine permuta- tions. We then re-frame prior results on fully commutative elements using the affine codes. 1.1 Introduction The Hecke Algebra of a Weyl group is a deformation by a parameter q = q1 which q2 for generic values yields an algebra with representation theory equivalent to that of the group algebra of the original Weyl group. Our interest is ultimately in the affine symmetric group, S˜n, which has index set I = f0;1;2;:::;n − 1g = Zn, but we will describe the basic constructions of algebras and monoids in full generality, and then specialize where necessary. At q1 = 0, however, we obtain the 0-Hecke algebra, which can be interpreted as a monoid algebra of the 0-Hecke monoid. If the original Weyl group is gener- ated by some simple reflections si for i int he index set I, then the 0-Hecke monoid is generated by pi for i 2 I. The commutation and braid relations between the pii 2 2 match the relations on the si, but we have pi = pi instead of si = 1. Thus, the 0-Hecke monoid is generated by projections, instead of reflections. -
Canonical Decompositions of Affine Permutations, Affine Codes, And
Canonical Decompositions of Affine Permutations, Affine Codes, and Split k-Schur Functions Tom Denton∗ Department of Mathematics York University Toronto, Ontario, Canada [email protected] Submitted: Apr 12, 2012; Accepted: Oct 8, 2012; Published: Nov 8, 2012 Mathematics Subject Classifications: 05E05, 05E15 Abstract We develop a new perspective on the unique maximal decomposition of an arbitrary affine permutation into a product of cyclically decreasing elements, implicit in work of Thomas Lam [Lam06]. This decomposition is closely related to the affine code, which generalizes the k- bounded partition associated to Grassmannian elements. We also prove that the affine code readily encodes a number of basic combinatorial properties of an affine permutation. As an application, we prove a new special case of the Littlewood-Richardson Rule for k-Schur functions, using the canonical decomposition to control for which permutations appear in the expansion of the k-Schur function in noncommuting variables over the affine nil-Coxeter algebra. ∗Supported by York University and the Fields Institute. the electronic journal of combinatorics 19(4) (2012), #P19 1 Contents 1 Introduction 2 1.1 Further Directions. ............................... 5 1.2 Overview ..................................... 5 1.3 Acknowledgements ............................... 6 2 Background and Definitions 7 2.1 The Affine Nilcoxeter Algebra and Affine Permutations. ........... 7 2.2 Cyclic Elements in Ak. ............................. 11 2.3 k-Schur Functions. ............................... 12 3 Canonical Cyclic Decompositions and k-Codes 14 3.1 Maximal Cyclic Decompositions and k-Codes ................ 15 3.2 Maximizing Moves on k-Codes. ........................ 18 3.3 Insertion Algorithm. .............................. 22 3.4 Bijection Between Affine Permutations and k-Codes. ............ 23 3.5 Relating the Various Cyclic Decompositions of an Affine Permutation.