Cubature rules from Hall–Littlewood polynomials

2020 ◽  
Author(s):  
J F van Diejen ◽  
E Emsiz
Keyword(s):  
Haar Measure ◽  
Symplectic Group ◽  
Symmetric Functions ◽  
Hyperplane Arrangement ◽  
Cubature Formulas ◽  
Littlewood Polynomials ◽  
Orthogonality Relations ◽  
Unitary Ensemble ◽  
Determinantal Expression ◽  
Complex Hyperplane

Abstract Discrete orthogonality relations for Hall–Littlewood polynomials are employed so as to derive cubature rules for the integration of homogeneous symmetric functions with respect to the density of the circular unitary ensemble (which originates from the Haar measure on the special unitary group $SU(n;\mathbb{C})$). By passing to Macdonald’s hyperoctahedral Hall–Littlewood polynomials, we moreover find analogous cubature rules for the integration with respect to the density of the circular quaternion ensemble (which originates in turn from the Haar measure on the compact symplectic group $Sp (n;\mathbb{H})$). The cubature formulas under consideration are exact for a class of rational symmetric functions with simple poles supported on a prescribed complex hyperplane arrangement. In the planar situations (corresponding to $SU(3;\mathbb{C})$ and $Sp (2;\mathbb{H})$), a determinantal expression for the Christoffel weights enables us to write down compact cubature rules for the integration over the equilateral triangle and the isosceles right triangle, respectively.

scholarly journals Combinatorial Expansions in $K$-Theoretic Bases

10.37236/2320 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Jason Bandlow ◽  
Jennifer Morse
Keyword(s):  
Generating Functions ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Geometric Information ◽  
Combinatorial Description ◽  
Littlewood Polynomials ◽  
Stanley Symmetric Functions ◽  
K Theory ◽  
Combinatorial Representation

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape.  Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space.  In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.

Poincaré series of Milnor algebras and free arrangements

2010 ◽  
Vol 47 (4) ◽  
pp. 513-521
Author(s):  
Shahid Ahmad
Keyword(s):  
Hyperplane Arrangement ◽  
Betti Numbers ◽  
Poincaré Series ◽  
Poincare Series ◽  
Complex Hyperplane

We show that for a free complex hyperplane arrangement \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{A}$$ \end{document}: f = 0, the Poincaré series of the graded Milnor algebra M(f) and the Betti numbers of the arrangement complement M(\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{A}$$ \end{document}) determine each other. Examples show that this is false if we drop the freeness assumption.

scholarly journals Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

2012 ◽  
Vol 206 ◽  
pp. 75-97 ◽  
Author(s):  
Alexandru Dimca
Keyword(s):  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Monodromy Operator ◽  
Low Dimensions ◽  
Milnor Fiber ◽  
Necessary And Sufficient ◽  
Complex Hyperplane

AbstractThe order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

scholarly journals Circle-valued Morse theory for complex hyperplane arrangements

2015 ◽  
Vol 27 (4) ◽  
Author(s):  
Toshitake Kohno ◽  
Andrei Pajitnov
Keyword(s):  
Morse Theory ◽  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Complex Hyperplane

AbstractLet 𝒜 be an essential complex hyperplane arrangement in

scholarly journals Hyperplane Arrangements and Diagonal Harmonics

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Drew Armstrong
Keyword(s):  
Symmetric Functions ◽  
Hilbert Series ◽  
Hyperplane Arrangement ◽  
Type A ◽  
Hyperplane Arrangements ◽  
Combinatorial Interpretation ◽  
Nabla Operator ◽  
Affine Weyl Group ◽  
Diagonal Harmonics ◽  
International Audience

International audience In 2003, Haglund's bounce statistic gave the first combinatorial interpretation of the q,t-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type A. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement — which we call the Ish arrangement. We prove that our statistics are equivalent to the area' and bounce statistics of Haglund and Loehr. In this setting, we observe that bounce is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to "extended'' Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions. En 2003, la statistique bounce de Haglund a donné la première interprétation combinatoire de la somme des nombres q,t-Catalan et de la série de Hilbert des harmoniques diagonaux. Dans cet article nous proposons une nouvelle interprétation combinatoire à partir du groupe de Weyl affine de type A. En particulier, nous définissons deux statistiques sur les permutations affines; l'une à partir de l'arrangement d'hyperplans Shi, et l'autre à partir d'un nouvel arrangement — que nous appelons l'arrangement Ish. Nous prouvons que nos statistiques sont équivalentes aux statistiques area' et bounce de Haglund et Loehr. Dans ce contexte, nous observons que bounce s'exprime naturellement comme une statistique sur le réseau des racines. Nous prolongeons nos statistiques dans deux directions: arrangements Shi "étendus'', et chambres bornées associées. Cela conduit à une interprétation (conjecturale) combinatoire pour toutes les puissances entières de l'opérateur nabla de Bergeron-Garsia appliqué aux fonctions symétriques élémentaires.

scholarly journals On the Schur Expansion of Hall-Littlewood and Related Polynomials via Yamanouchi Words

10.37236/6732 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
Gamma Delta ◽  
Colored Graph ◽  
Littlewood Polynomials ◽  
Dual Equivalence ◽  
Leading Term ◽  
Definition Of

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $\delta \subset {\mathbb Z} \times {\mathbb Z}$, written as $\widetilde H_{\delta}(X;q,t)$ and $\widetilde H_{\delta}(X;0,t)$, respectively. We then give an explicit Schur expansion of $\widetilde H_{\delta}(X;0,t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_{\gamma,\delta}(X)$ as a refinement of $\widetilde H_{\delta}(X;0,t)$ and similarly describe its Schur expansion. We then analyze $R_{\gamma,\delta}(X)$ to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of $\widetilde H_{\delta}(X;q,t)$. To gain these results, we use a construction from the 2007 work of Sami Assaf to associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_\delta$. In the case where a subgraph of $\mathcal{H}_\delta$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.

scholarly journals The expansion of Hall-Littlewood functions in the dual Grothendieck polynomial basis

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Jason Bandlow ◽  
Jennifer Morse
Keyword(s):  
Symmetric Functions ◽  
Polynomial Basis ◽  
Dual Basis ◽  
Nous Obtenons ◽  
Plane Partitions ◽  
Generalize Charge ◽  
Littlewood Polynomials ◽  
International Audience ◽  
Reverse Plane ◽  
Grothendieck Polynomials

International audience A combinatorial expansion of the Hall-Littlewood functions into the Schur basis of symmetric functions was first given by Lascoux and Schützenberger, with their discovery of the charge statistic. A combinatorial expansion of stable Grassmannian Grothendieck polynomials into monomials was first given by Buch, using set-valued tableaux. The dual basis of the stable Grothendieck polynomials was given a combinatorial expansion into monomials by Lam and Pylyavskyy using reverse plane partitions. We generalize charge to set-valued tableaux and use all of these combinatorial ideas to give a nice expansion of Hall-Littlewood polynomials into the dual Grothendieck basis. \par En associant une charge à un tableau, une formule combinatoire donnant le développement des polynômes de Hall-Littlewood en termes des fonctions de Schur a été obtenue par Lascoux et Schützenberger. Une formule combinatoire donnant le développement des polynômes de Grothendieck Grassmanniens stables en termes des fonctions monomiales a quant à elle été obtenue par Buch à l'aide de tableaux à valeurs sur des ensembles. Finalement, une formule faisant intervenir des partitions planaires inverses a été obtenue par Lam et Pylyavskyy pour donner le développement de la base duale aux polynômes de Grothendieck stables en termes de monômes. Nous généralisons le concept de charge aux tableaux à valeurs sur des ensembles et, en nous servant de toutes ces notions combinatoires, nous obtenons une formule élégante donnant le développement des polynômes de Hall-Littlewood en termes de la base de Grothendieck duale.

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