Constant Term Identities Research Articles

The t-analog of the basic string function for twisted affine Kac–Moody algebras

We study Lusztigʼs t-analog of weight multiplicities associated to level one representations of twisted affine Kac–Moody algebras. An explicit closed form expression is obtained for the corresponding t-string function using constant term identities of Macdonald and Cherednik. The closed form involves the generalized exponents of the graded pieces of the twisted affine algebra, considered as modules for the underlying finite dimensional simple Lie algebra. This extends previous work on level 1 t-string functions for the untwisted simply-laced affine Kac–Moody algebras.

The t-analog of the level one string function for twisted affine Kac–Moody algebras

We study Lusztigʼs t-analog of weight multiplicities, or affine Kostka–Foulkes polynomials, associated to level one representations of twisted affine Kac–Moody algebras. We obtain an explicit closed form expression for the unique t-string function, using constant term identities of Macdonald and Cherednik. This extends previous work on t-string functions for the untwisted simply-laced affine Kac–Moody algebras.

On [formula omitted]-algebra extensions of [formula omitted] minimal models: [formula omitted

This is a continuation of Adamović and Milas (2010) [5], where, among other things, we classified irreducible representations of the triplet vertex algebra W 2 , 3 . In this part we extend the classification to W 2 , p , for all odd p > 3 . We also determine the structure of the center of the Zhu algebra A ( W 2 , p ) which implies the existence of a family of logarithmic modules having L ( 0 ) -nilpotent ranks 2 and 3. A logarithmic version of Macdonald–Morris constant term identity plays a key role in the paper.

On Zeilberger's Constant Term for Andrews' TSSCPP Theorem

This paper studies Zeilberger's two prized constant term identities. For one of the identities, Zeilberger asked for a simple proof that may give rise to a simple proof of Andrews theorem for the number of totally symmetric self complementary plane partitions. We obtain an identity reducing a constant term in $2k$ variables to a constant term in $k$ variables. As applications, Zeilberger's constant terms are converted to single determinants. The result extends for two classes of matrices, the sum of all of whose full rank minors is converted to a single determinant. One of the prized constant term problems is solved, and we give a seemingly new approach to Macdonald's constant term for root system of type BC.

The structure of Zhuʼs algebras for certain [formula omitted]-algebras

We introduce a new approach that allows us to determine the structure of Zhuʼs algebra for certain vertex operator (super)algebras which admit horizontal Z -grading. By using this method and an earlier description of Zhuʼs algebra for the singlet W -algebra, we completely describe the structure of Zhuʼs algebra for the triplet vertex algebra W ( p ) . As a consequence, we prove that Zhuʼs algebra A ( W ( p ) ) and the related Poisson algebra P ( W ( p ) ) have the same dimension. We also completely describe Zhuʼs algebras for the N = 1 triplet vertex operator superalgebra SW ( m ) . Moreover, we obtain similar results for the c = 0 triplet vertex algebra W 2 , 3 , important in logarithmic conformal field theory. Because our approach is “internal” we had to employ several constant term identities for purposes of getting right upper bounds on dim ( A ( V ) ) . This work is, in a way, a continuation of the results published in Adamović and Milas (2008) [4].

On W-Algebras Associated to (2, p) Minimal Models and Their Representations

For every odd , we investigate the representation theory of the vertex algebra associated to (2, p) minimal models for the Virasoro algebras. We demonstrate that vertex algebras are C 2 cofinite and irrational. Complete classification of irreducible representations for is obtained, while the classification for is subject to certain constant term identities. These identities can be viewed as “logarithmic deformations” of Dyson and Selberg constant term identities, and are of independent interest.

AFFINE HALL-LITTLEWOOD FUNCTIONS FOR AFormula AND SOME CONSTANT TERM IDENTITIES OF CHEREDNIK-MACDONALD-MEHTA TYPE

We study t-analogs of string functions for integrable highest weight representations of the affine Kac-Moody algebra A(1)((1)). We obtain closed form formulas for certain t-string functions of levels 2 and 4. As corollaries, we obtain explicit identities for the corresponding affine Hall-Littlewood functions, as well as higher level generalizations of Cherednik's Macdonald and Macdonald-Mehta constant term identities.

A family of q-Dyson style constant term identities

By generalizing Gessel–Xin's Laurent series method for proving the Zeilberger–Bressoud q-Dyson Theorem, we establish a family of q-Dyson style constant term identities. These identities give explicit formulas for certain coefficients of the q-Dyson product, including three conjectures of Sills' as special cases and generalizing Stembridge's first layer formulas for characters of SL ( n , C ) .

A unified elementary approach to the Dyson, Morris, Aomoto, and Forrester constant term identities

We introduce an elementary method to give unified proofs of the Dyson, Morris, and Aomoto identities for constant terms of Laurent polynomials. These identities can be expressed as equalities of polynomials and thus can be proved by verifying them for sufficiently many values, usually at negative integers where they vanish. Our method also proves some special cases of the Forrester conjecture.

Two Coefficients of the Dyson Product

In this paper, the closed-form expressions for the coefficients of ${x_r^2\over x_s^2}$ and ${x_r^2\over x_sx_t}$ in the Dyson product are found by applying an extension of Good's idea. As consequences, we find several interesting Dyson style constant term identities.

Generalized Cherednik-Macdonald identities

We derive generalizations of the Cherednik-Macdonald constant term identities associated to root systems which depend, besides on the usual multiplicity function, symmetrically on two additional parameters $\omega_{\pm}$. They are natural analogues of the Cherednik-Macdonald constant term $q$-identities in which the deformation parameter $q=\exp(2\pi i\omega_+/\omega_-)$ is allowed to have modulus one. They unite the Cherednik-Macdonald constant term $q$-identities with closely related Jackson $\widetilde{q}$-integral identities due to Macdonald, where the deformation parameter $\widetilde{q}=\exp(-2\pi i\omega_-/\omega_+)$ is related to $q$ by modular inversion.

Constant terms, jagged partitions, and partitions with difference two at distance two

In this paper we establish a group of constant term identities that contain generating functions for ?jagged? partitions and interpret these identities as theorems about partitions with difference two at distance two.

Random matrix theory, the exceptional Lie groups andL-functions

There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of L-functions. These connections are here extended to non-classical groups. We focus on an explicit example: the exceptional Lie group G_2. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional representations of G_2, defined with respect to the uniform invariant (Haar) measure, are calculated using two of the Macdonald constant term identities. A one parameter family of L-functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the 7-dimensional representation of G_2. The random matrix calculations extend to all exceptional Lie groups

Forrester's Conjectured Constant Term Identity II

We continue our study on Forrester's conjectured constant term identity which is equivalent to a new kind of generalization of the Selberg integral. The special cases N 1 = 2,3 of the conjecture have been verified in our previous paper [6]. We show the conjecture holds in the other extreme case N 1 = N-1. The proof is based on the integration formula of Jack polynomials and the Chu-Vandermonde formula for the generalized binomial coefficients.

Elliptic Beta Integrals and Mudular Hypergeometric Sums: An Overview

Recent results on elliptic generalizations of various beta integrals are reviewed. Firstly, a single variable Askey-Wilson type integral describing an elliptic extension of the Nassrallah-Rahman integral is presented. Then a multiple Selberg-type integral defining an elliptic extension of the Macdonald-Morris constant term identities for nonreduced root systems is described. The Frenkel-Turaev sum and its multivariable generalization, conjectured recently by Warnaar, follow from these integrals through residue calculus. A new elliptic Selberg-type integral, from which the previous one can be derived via a technique due to Gustafson, is defined. Residue calculus applied to this integral yields an elliptic generalization of the Denis-Gustafson sum-a modular extension of the Milne-type multiple basic hypergeometric sums.

PROOF OF BAKER-FORRESTER'S CONSTANT TERM CONJECTURE FOR THE CASES <i>N</i><sub>1</sub> = 2, 3

The Baker-Forrester's constant term conjecture is an extension of the q-Morris constant term identity. We give a proof of the conjecture in the cases N1 = 2,3 by using the q-integration formula of Macdonald polynomial.

On the evaluation formula for Jack polynomials with prescribed symmetry

The Jack polynomials with prescribed symmetry are obtained from the nonsymmetric polynomials via the operations of symmetrization, antisymmetrization and normalization. After dividing out the corresponding antisymmetric polynomial of smallest degree, a symmetric polynomial results. Of interest in applications is the value of the latter polynomial when all the variables are set equal. Dunkl has obtained this evaluation, making use of a certain skew symmetric operator. We introduce a simpler operator for this purpose, thereby obtaining a new derivation of the evaluation formula. An expansion formula of a certain product in terms of Jack polynomials with prescribed symmetry implied by the evaluation formula is used to derive a generalization of a constant term identity due to Macdonald, Kadell and Kaneko. Although we don't give the details in this work, the operator introduced here can be defined for any reduced crystallographic root system, and used to provide an evaluation formula for the corresponding Heckman-Opdam polynomials with prescribed symmetry.

Elliptic Selberg integrals

We introduce new Selberg-type multidimensional integrals built of Ruijsenaars' elliptic gamma functions. We show that the vanishing of our integrals for a specific parameter hypersurface implies closed evaluation formulas valid for the full parameter space. The resulting integration formulas contain the Macdonald-Morris constant term identities for nonreduced root systems as special limiting cases.

Aomoto’s machine and the Dyson constant term identity

Aomoto’s machine and the Dyson constant term identity

Generalizations of theq-Morris Constant Term Identity

Theq-Morris constant term identity gives the constant term in the Laurent polynomial expansion ofΦ≔∏Nl=1(wl;q)a(q/wl;q)b∏1⩽j<k⩽N(wj/wk;q)λ(qwk/wj;q)λ. A conjecture is given for the constant term ofΦwhen multiplied by ∏N0+1⩽j<k⩽N(1−qλ(wj/wk))(1−qλ+1(wk/wj)). This conjecture is proved in the casesa=λ(generalN0,N1,b,λ),a=b=0 (generalN0,N1,λ), andN1=2 (generala,b,λ,N0), whereN0+N1=N. Also, a general identity relatingq-Morris-type constant terms andq-Selberg-type integrals is derived and is used to rewrite the conjecture as aq-Selberg-type integral evaluation.