Spin Kostka Polynomials
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Paths and Kostka–Macdonald Polynomials
Paths and Kostka–Macdonald Polynomials aAnatol N. Kirillov and bReiho Sakamoto a Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan [email protected] b Department of Physics, Graduate School of Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan [email protected] Abstract We give several equivalent combinatorial descriptions of the space of states for the box-ball systems, and connect certain partition functions for these models arXiv:0811.1085v2 [math.QA] 5 Dec 2009 with the q-weight multiplicities of the tensor product of the fundamental rep- resentations of the Lie algebra gl(n). As an application, we give an elementary proof of the special case t = 1 of the Haglund–Haiman–Loehr formula. Also, we propose a new class of combinatorial statistics that naturally generalize the so-called energy statistics. Mathematics Subject Classification (2000) 05E10, 20C35. Key words and phrases: Crystals, Paths, Energy and Tau functions, Box–Ball systems, Kostka–Macdonald polynomials. 1 1 Introduction The purpose of the present paper is two-fold. First of all, we would like to give an introduction to the beautiful Combinatorics related with Box-Ball Systems, and sec- ondly, to relate the latter with the “Classical Combinatorics” revolving around trans- portation matrices, tabloids, the Lascoux–Sch¨utzenberger statistics charge, Macdon- ald polynomials, [31],[35], Haglund–Haiman–Loehr’s formula [8], and so on. As a result of our investigations, we will prove that two statistics naturally appearing in the context of Box-Ball systems, namely energy function and tau-function, have nice combinatorial properties. -
Rigged Configurations and Catalan, Stretched Parabolic Kostka
Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers and Polynomials : Polynomiality, Unimodality and Log-concavity anatol n. kirillov Research Institute of Mathematical Sciences ( RIMS ) Kyoto 606-8502, Japan URL: http://www.kurims.kyoto-u.ac.jp/ ~kirillov and The Kavli Institute for the Physics and Mathematics of the Universe ( IPMU ), 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan Dedicated to Professor Masatoshi NOUMI on the occasion of his 60th Birthday Abstract We will look at the Catalan numbers from the Rigged Configurations point of view originated [9] from an combinatorial analysis of the Bethe Ansatz Equations associated with the higher spin anisotropic Heisenberg models . Our strategy is to take a com- binatorial interpretation of the Catalan number Cn as the number of standard Young 2 tableaux of rectangular shape (n ), or equivalently, as the Kostka number K(n2);12n , as the starting point of our research. We observe that the rectangular (or multidimen- sional) Catalan numbers C(m; n), introduced and studied by P. MacMahon [21], [30], see also [31], can be identified with the corresponding Kostka numbers K(nm);1mn , and therefore can be treated by the Rigged Configurations technique. Based on this tech- nique we study the stretched Kostka numbers and polynomials, and give a proof of a strong rationality of the stretched Kostka polynomials. This result implies a polynomi- ality property of the stretched Kostka and stretched Littlewood{Richardson coefficients [7], [26], [16]. Another application of the Rigged Configuration technique presented, is a new fam- ily of counterexamples to Okounkov's log-concavity conjecture [25]. Finally, we apply Rigged Configurations technique to give a combinatorial proof of the unimodality of the principal specialization of the internal product of Schur func- tions. -
Hall-Littlewood Functions and Kostka-Foulkes Polynomials in Representation Theory
HALL-LITTLEWOOD FUNCTIONS AND KOSTKA-FOULKES POLYNOMIALS IN REPRESENTATION THEORY J. Desarm´ enien,´ B. Leclerc and J.-Y. Thibon Abstract This paper presents a survey of recent applications of Hall-Littlewood functions and Kostka-Foulkes polynomials to the representation theory of the general linear group GL(n; C) and of the symmetric group Sn. The reviewed topics include the q-analogue of Kostant's partition function, vertex operators, generalized exponents of GL(n; C) and Sn-harmonic polynomials. We also give a detailed description of the various combinatorial interpretations of Kostka-Foulkes polynomials. We conclude with the study of Hall-Littlewood functions at roots of unity, which provide a combinatorial description of certain plethysms. R´esum´e Cet article donne un aper¸cude plusieurs applications r´ecentes des fonctions de Hall-Littlewood et des polyn^omesde Kostka-Foulkes aux repr´esentations du groupe lin´eaire GL(n; C) et du groupe sym´etrique Sn. Parmi les sujets abord´esfigurent no- tamment le q-analogue de la fonction de partition de Kostant, les op´erateursvertex, les exposants g´en´eralis´esde GL(n; C) et les polyn^omesharmoniques de Sn. Nous donnons aussi une description d´etaill´eedes diverses interpr´etationscombinatoires des polyn^omesde Kostka-Foulkes. Nous concluons par l'´etudedes fonctions de Hall-Littlewood aux racines de l'unit´e,qui fournit une interpr´etationcombinatoire de certains pl´ethysmes. 1 Introduction The original motivation for defining Hall polynomials, Hall algebra and Hall-Littlewood symmetric functions comes from group theory. It is well known that any finite Abelian p-group G is a direct sum of cyclic subgroups, of orders pλ1 , pλ2 ; : : : ; pλr , say, where we may suppose that λ1 λ2 ::: λr > 0. -
Arxiv:Math/9912094V4
UBIQUITY OF KOSTKA POLYNOMIALS anatol n. kirillov Graduate School of Mathematics, Nagoya University Chikusa–ku, Nagoya 486–8602, Japan and Steklov Mathematical Institute, Fontanka 27, St.Petersburg, 191011, Russia Abstract We report about results revolving around Kostka–Foulkes and parabolic Kostka polynomials and their connections with Representation Theory and Combinatorics. It appears that the set of all parabolic Kostka polynomials forms a semigroup, which we call Liskova semigroup. We show that polynomials frequently appearing in Rep- resentation Theory and Combinatorics belong to the Liskova semigroup. Among such polynomials we study rectangular q–Catalan numbers; generalized exponents polyno- mials; principal specializations of the internal product of Schur functions; generalized q–Gaussian polynomials; parabolic Kostant partition function and its q–analog; cer- tain generating functions on the set of transportation matrices. In each case we apply rigged configurations technique to obtain some interesting and new information about Kostka–Foulkes and parabolic Kostka polynomials, Kostant partition function, MacMa- hon, Gelfand–Tsetlin and Chan–Robbins polytopes. We describe certain connections between generalized saturation and Fulton’s conjectures and parabolic Kostka polyno- mials; domino tableaux and rigged configurations. We study also some properties of l–restricted generalized exponents and the stable behaviour of certain Kostka–Foulkes polynomials. Contents 1 Introduction 2 Kostka–Foulkes polynomials and weight multiplicities -
Arxiv:Math/0404353V2 [Math.CO] 28 Jul 2004 NIVTTO OTEGNRLZDSTRTO CONJECTURE SATURATION GENERALIZED the to INVITATION an 00Mteaissbetcasfiain:0E5 51,05A19
AN INVITATION TO THE GENERALIZED SATURATION CONJECTURE anatol n. kirillov Research Institute of Mathematical Sciences ( RIMS ) Kyoto 606-8502, Japan “ vam ne skaжu za vs Odessu, vs Odessa oqenь velika.” From a famous russian song. “I’m not intending to tell all about Tokyo, The whole Tokyo is too big.” Communicated by T. Kawai. Received March 2, 2004. Revised May 17, 2004. Abstract We report about some results, interesting examples, problems and conjectures re- volving around the parabolic Kostant partition functions, the parabolic Kostka polyno- mials and “saturation” properties of several generalizations of the Littlewood–Richardson numbers. The Contents contains the titles of main topics we are going to discuss in the present paper. arXiv:math/0404353v2 [math.CO] 28 Jul 2004 2000 Mathematics Subject Classifications: 05E05, 05E10, 05A19. 1 Contents 1 Introduction 2 Basic definitions and notation 2.1 Compositions and partitions 2.2 Kostka–Foulkes polynomials 2.3 Skew Kostka–Foulkes polynomials 2.4 Littlewood–Richardson numbers and Saturation Theorem µ 2.5 Internal product of Schur functions, plethysm, and polynomials Lα,β(q) 2.6 Extended and Restricted Littlewood–Richardson numbers 3 Parabolic Kostant partition function and its q-analog 3.1 Definitions: algebraic and combinatorial 3.2 Elementary properties, and explicit formulas for l(η) ≤ 4 3.3 Non-vanishing, degree and saturation theorems 3.4 Rationality and polynomiality theorems 3.5 Parabolic Kostant partition function KΦ(η)(γ) as function of γ 3.6 Reconstruction theorem 4 Parabolic -
Arxiv:2102.09982V1
MACDONALD POLYNOMIALS AND CYCLIC SIEVING JAESEONG OH Abstract. The Garsia–Haiman module is a bigraded Sn-module whose Frobenius image is a Mac- donald polynomial. The method of orbit harmonics promotes an Sn-set X to a graded polynomial ring. The orbit harmonics can be applied to prove cyclic sieving phenomena which is a notion that encapsulates the fixed-point structure of finite cyclic group action on a finite set. By applying this idea to the Garsia–Haiman module, we provide cyclic sieving results regarding the enumeration of matrices that are invariant under certain cyclic row and column rotation and translation of entries. 1. Introduction Since Macdonald [Mac88] defined Macdonald polynomials Hµ(x; q,t) and conjectured the Schur positivity of them, the Macdonald polynomial has been one of the central objects in algebraic e combinatorics. Even though a combinatorial formula for Macdonald polynomials is given [HHL05] and the Schur positivity of Macdonald polynomials is proved [Hai01], not much is known about an explicit combinatorial formula for the (q,t)-Kostka polynomials, which are the Schur coefficients of the Macdonald polynomial. In this paper, we discuss some enumerative results involving the (q,t)-Kostka polynomials in the words of cyclic sieving phenomena. This will provide a series of identities between the number of matrices with certain cyclic symmetries and evaluation of (q,t)-Kostka polynomials at a root of unity, uncovering a part of the mystery of the (q,t)-Kostka polynomials. To begin, we define the cyclic seiving phenomena. Let X be a set with an action of a cyclic group C. -
POLYNOMIAL BEHAVIOUR of KOSTKA NUMBERS Contents 1
POLYNOMIAL BEHAVIOUR OF KOSTKA NUMBERS DINUSHI MUNASINGHE + + − Abstract. Given two standard partitions λ+ = (λ1 ≥ · · · ≥ λs ) and λ− = (λ1 ≥ − · · · ≥ λr ) we write λ = (λ+; λ−) and set sλ(x1; : : : ; xt) := sλt (x1; : : : ; xt) where + + − − λt = (λ1 ; : : : ; λs ; 0;:::; 0; −λr ;:::; −λ1 ) has t parts. We provide two proofs that the Kostka numbers kλµ(t) in the expansion X sλ(x1; : : : ; xt) = kλµ(t)mµ(x1; : : : ; xt) µ demonstrate polynomial behaviour, and establish the degrees and leading coefficients of these polynomials. We conclude with a discussion of two families of examples that introduce areas of possible further exploration. Contents 1. Introduction 2 2. Representations of GLn 3 2.1. Representations 3 2.2. Characters of GLn 4 3. Symmetric Polynomials 5 3.1. Kostka Numbers 6 3.2. Littlewood-Richardson Coefficients 6 3.3. Representations and Characters 7 4. Restrictions 7 4.1. Interlacing Partitions 7 4.2. Branching Rules 8 5. Beyond Polynomial Representations 9 6. Weight Decompositions 11 7. Some Notation 11 8. Stability 12 8.1. Kostka Numbers 12 8.2. Littlewood-Richardson Coefficients 13 9. Asymptotics 15 10. The Polynomial Behaviour of Kostka Numbers 16 11. Further Asymptotics 20 1 2 DINUSHI MUNASINGHE 12. Remarks 23 Appendix: Solving Difference Equations 24 References 24 1. Introduction The Kostka numbers, which we will denote kλµ, were introduced by Carl Kostka in his 1882 work relating to symmetric functions. The Littlewood-Richardson coefficients ν cλµ were introduced by Dudley Ernest Littlewood and Archibald Read Richardson in their 1934 paper Group Characters and Algebra. These two combinatorial objects are ubiquitous, appearing in situations in combinatorics and representation theory, which we will address in this work, as well as algebraic geometry.