DOCSLIB.ORG

Simultaneous Core Partitions for Affine Weyl Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Simultaneous Core Partitions for Affine Weyl Groups Simultaneous core partitions for affine Weyl groups Marko Thiel Universit¨atZ¨urich Joint work with Nathan Williams (LACIM, Montr´eal) Marko Thiel Simultaneous core partitions for affine Weyl groups { Sea-action on a-cores { The geometry of the affine symmetric group Sea { The lattice of cores { Simultaneous (a; b)-cores { Other affine Weyl groups { a-core partitions Outline Marko Thiel Simultaneous core partitions for affine Weyl groups { The geometry of the affine symmetric group Sea { The lattice of cores { Simultaneous (a; b)-cores { Other affine Weyl groups { Sea-action on a-cores Outline { a-core partitions Marko Thiel Simultaneous core partitions for affine Weyl groups { The lattice of cores { Simultaneous (a; b)-cores { Other affine Weyl groups { The geometry of the affine symmetric group Sea Outline { a-core partitions { Sea-action on a-cores Marko Thiel Simultaneous core partitions for affine Weyl groups { Simultaneous (a; b)-cores { Other affine Weyl groups { The lattice of cores Outline { a-core partitions { Sea-action on a-cores { The geometry of the affine symmetric group Sea Marko Thiel Simultaneous core partitions for affine Weyl groups { Other affine Weyl groups { Simultaneous (a; b)-cores Outline { a-core partitions { Sea-action on a-cores { The geometry of the affine symmetric group Sea { The lattice of cores Marko Thiel Simultaneous core partitions for affine Weyl groups { Other affine Weyl groups Outline { a-core partitions { Sea-action on a-cores { The geometry of the affine symmetric group Sea { The lattice of cores { Simultaneous (a; b)-cores Marko Thiel Simultaneous core partitions for affine Weyl groups Outline { a-core partitions { Sea-action on a-cores { The geometry of the affine symmetric group Sea { The lattice of cores { Simultaneous (a; b)-cores { Other affine Weyl groups Marko Thiel Simultaneous core partitions for affine Weyl groups An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Integer partitions Marko Thiel Simultaneous core partitions for affine Weyl groups Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Integer partitions An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Marko Thiel Simultaneous core partitions for affine Weyl groups Ferrers diagram of λ has λi boxes in row i: 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Integer partitions An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Marko Thiel Simultaneous core partitions for affine Weyl groups 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Integer partitions An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: Marko Thiel Simultaneous core partitions for affine Weyl groups 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Integer partitions An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: Marko Thiel Simultaneous core partitions for affine Weyl groups 6 4 2 1 1 Integer partitions An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Marko Thiel Simultaneous core partitions for affine Weyl groups Integer partitions An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: 6 7 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Marko Thiel Simultaneous core partitions for affine Weyl groups An a-core is an integer partition with no hook length equal to a. Example: λ = (6; 4; 2; 1; 1) is a 3-core. 107 5 4 2 1 8 4 2 1 4 1 2 1 Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic. a-core partitions Marko Thiel Simultaneous core partitions for affine Weyl groups Example: λ = (6; 4; 2; 1; 1) is a 3-core. 107 5 4 2 1 8 4 2 1 4 1 2 1 Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic. a-core partitions An a-core is an integer partition with no hook length equal to a. Marko Thiel Simultaneous core partitions for affine Weyl groups 107 5 4 2 1 8 4 2 1 4 1 2 1 Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic. a-core partitions An a-core is an integer partition with no hook length equal to a. Example: λ = (6; 4; 2; 1; 1) is a 3-core. Marko Thiel Simultaneous core partitions for affine Weyl groups 107 5 4 2 1 8 4 2 1 4 1 2 1 Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic. a-core partitions An a-core is an integer partition with no hook length equal to a. Example: λ = (6; 4; 2; 1; 1) is a 3-core. Marko Thiel Simultaneous core partitions for affine Weyl groups 107 5 4 2 1 8 4 2 1 4 1 2 1 a-core partitions An a-core is an integer partition with no hook length equal to a. Example: λ = (6; 4; 2; 1; 1) is a 3-core. Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic. Marko Thiel Simultaneous core partitions for affine Weyl groups The a-residue of a box in row i and column j is r(i; j) = j i mod a: − 0 1 2 0 1 2 2 0 1 2 1 2 3-residues of a 3-core 0 2 removable box: Can remove this box and still have a Ferrers shape. Adding/Removing boxes Marko Thiel Simultaneous core partitions for affine Weyl groups 0 1 2 0 1 2 2 0 1 2 1 2 3-residues of a 3-core 0 2 removable box: Can remove this box and still have a Ferrers shape. Adding/Removing boxes The a-residue of a box in row i and column j is r(i; j) = j i mod a: − Marko Thiel Simultaneous core partitions for affine Weyl groups 0 1 2 0 1 2 2 0 1 2 1 2 3-residues of a 3-core 0 2 removable box: Can remove this box and still have a Ferrers shape. Adding/Removing boxes The a-residue of a box in row i and column j is r(i; j) = j i mod a: − Marko Thiel Simultaneous core partitions for affine Weyl groups 0 1 2 0 1 2 2 0 1 2 1 2 3-residues of a 3-core 0 2 Adding/Removing boxes The a-residue of a box in row i and column j is r(i; j) = j i mod a: − removable box: Can remove this box and still have a Ferrers shape. Marko Thiel Simultaneous core partitions for affine Weyl groups addable box: Can add this box and still have a Ferrers shape. Adding/Removing boxes The a-residue of a box in row i and column j is r(i; j) = j i mod a: − 0 1 2 0 1 2 2 0 1 2 1 2 0 2 removable box: Can remove this box and still have a Ferrers shape. Marko Thiel Simultaneous core partitions for affine Weyl groups Adding/Removing boxes The a-residue of a box in row i and column j is r(i; j) = j i mod a: − 0 1 2 0 1 2 2 0 1 2 1 2 0 2 removable box: Can remove this box and still have a Ferrers shape. addable box: Can add this box and still have a Ferrers shape. Marko Thiel Simultaneous core partitions for affine Weyl groups Note: There are never both addable and removable boxes of the same residue. Adding/Removing boxes The a-residue of a box in row i and column j is r(i; j) = j i mod a: − 0 1 2 0 1 2 0 2 0 1 2 0 1 2 0 0 1 2 1 removable box: Can remove this box and still have a Ferrers shape.
Load more

Recommended publications
  • 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, .
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Of Symmetric Functions Indexed by Permutations
    STANLEY SYMMETRIC FUNCTIONS AND PETERSON ALGEBRAS THOMAS LAM Abstract. These are (mostly) expository notes for lectures on affine Stanley symmetric functions given at the Fields Institute in 2010. We focus on the algebraic and combinatorial parts of the theory. The notes contain a number of exercises and open problems. Stanley symmetric functions are a family fFw j w 2 Sng of symmetric functions indexed by permutations. They were invented by Stanley [Sta] to enumerate the reduced words of elements of the symmetric group. The most important properties of the Stanley symmetric functions are their symmetry, established by Stanley, and their Schur positivity, first proven by Edelman and Greene [EG], and by Lascoux and Sch¨utzenberger [LSc82]. Recently, a generalization of Stanley symmetric functions to affine permutations was developed in [Lam06]. These affine Stanley symmetric functions turned out to have a natural geometric interpretation [Lam08]: they are pullbacks of the cohomol- ogy Schubert classes of the affine flag variety LSU(n)=T to the affine Grassmannian (or based loop space) ΩSU(n) under the natural map ΩSU(n) ! LSU(n)=T . The combinatorics of reduced words and the geometry of the affine homogeneous spaces are connected via the nilHecke ring of Kostant and Kumar [KK], together with a remarkable commutative subalgebra due to Peterson [Pet]. The symmetry of affine Stanley symmetric functions follows from the commutativity of Peterson's subalgebra, and the positivity in terms of affine Schur functions is established via the relationship between affine Schubert calculus and quantum Schubert calculus [LS10, LL]. The affine-quantum connection was also discovered by Peterson.
    [Show full text]
  • Matrix-Ball Construction of Affine Robinson-Schensted Correspondence
    Matrix-Ball Construction of affine Robinson-Schensted correspondence Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina To cite this version: Michael Chmutov, Pavlo Pylyavskyy, Elena Yudovina. Matrix-Ball Construction of affine Robinson- Schensted correspondence. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. hal-02173388 HAL Id: hal-02173388 https://hal.archives-ouvertes.fr/hal-02173388 Submitted on 4 Jul 2019 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 2016 Vancouver, Canada DMTCS proc. BC, 2016, 335–346 Matrix-Ball Construction of affine Robinson-Schensted correspondence Michael Chmutovy, Pavlo Pylyavskyyz, and Elena Yudovina Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Abstract. In his study of Kazhdan-Lusztig cells in affine type A, Shi has introduced an affine analog of Robinson- Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combi- natorial realization of Shi’s algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Honeywill, we extend the algorithm to a bijection between extended affine symmetric group and triples (P; Q; ρ) where P and Q are tabloids and ρ is a dominant weight.
    [Show full text]
  • Actions of the 0-Hecke Monoids of Affine Symmetric Groups
    Séminaire Lotharingien de Combinatoire 80B (2018) Proceedings of the 30th Conference on Formal Power Article #65, 12 pp. Series and Algebraic Combinatorics (Hanover) Actions of the 0-Hecke monoids of affine symmetric groups Eric Marberg∗1 1Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Abstract. There are left and right actions of the 0-Hecke monoid of the affine sym- metric group S˜n on involutions whose cycles are labeled periodically by nonnegative integers. Using these actions we construct two bijections, which are length-preserving in an appropriate sense, from the set of involutions in S˜n to the set of N-weighted matchings in the n-element cycle graph. As an application, we show that the bivariate generating function counting the involutions in S˜n by length and absolute length is a rescaled Lucas polynomial. The 0-Hecke monoid of S˜n also acts on involutions (with- out any cycle labelling) by Demazure conjugation. The atoms of an involution z 2 S˜n are the minimal length permutations w which transform the identity to z under this action. We prove that the set of atoms for an involution in S˜n is naturally a bounded, graded poset, and give a formula for the set’s minimum and maximum elements. Keywords: Affine symmetric groups, 0-Hecke monoids, involutions, Bruhat order 1 Introduction For each integer n ≥ 1, let S˜n be the affine symmetric group of rank n, consisting of the bijections w : Z ! Z with w(i + n) = w(i) + n for all i 2 Z and w(1) + w(2) + ··· + n+1 ˜ ˜ w(n) = ( 2 ).
    [Show full text]