On certain identities for ratios of theta-functions and some new modular equations of mixed degree

In this paper, we derive certain identities for ratios of theta-functions. As applications of the identities, we establish certain new modular equations of mixed degree in the theory of signature 3, which are analogous to Ramanujan-Weber type modular equations and Ramanujan-Schläfli type mixed modular equations.     

The homology of graded infinite-dimensional Lie algebras in connection with Macdonald identities

D. Fucks' monograph devoted to the cohomology of infinite-dimensional Lie algebras contains an error in calculating the homology of a graded affine Kac-Moody algebra of type A _n ⁽¹⁾ , so that the proof of the corresponding Macdonald identity, which is based on that calculation, is incorrect. In the present paper, a revised proof is suggested.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1996, pp. 5–17.The author is grateful to A. M. Vershik, who suggested the topic of this paper, for his constant attention and useful comments. The author also thanks M. A. Vsemirnov for his information on non-coincidence of the coefficients in two sides of the equality in [4, §3.2.3] for n=7, which he observed while proving Macdonald identities in a new (analytical) way, and on the sign of a summand on the right (see his paper in this issue).     

Cubic symplectic theta-functions

The existence of two cubic theta-functions defined on S(4,)/S(4) is announced.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 162, pp. 186–188, 1987.     

An elementary proof of the Macdonald identities for A(1)l

In this paper it is shown that the Macdonald identities for A⁽¹⁾_l are a natural consequence of the recent multivariable generalization of classical basic hypergeometric series known as basic hypergeometric series in U(n). More precisely, a U(n) multiple series generalization of the q-binomial theorem is derived and used to generalize Cauchy's elegant proof of Jacobi's triple product identity and to give a direct, elementary proof of the Macdonald identities for A⁽¹⁾_l.     

A note on a generalization of Macdonald’s identities forA l andB l

Let η(q) denote the Dedekind’s η-function. Macdonald obtained identities for η(q) ^{dim g} where g is complex simple finite dimensional Lie algebra. The aim of this paper is to obtain generalization of the above identities in the case of g=A _l andB _l. We also get new formulas for the generating functions of the Ramanujan’s τ-function and ?_α-functions.     

Combinatorial formula for Macdonald polynomials and generic Macdonald polynomials

We give a direct proof of the combinatorial formula for interpolation Macdonald polynomials by introducing certain polynomials, which we call generic Macdonald polynomials, and which depend on d additional parameters and specialize to all Macdonald polynomials of degree d. The form of these generic polynomials is that of a Bethe eigenfunction and they imitate, on a more elementary level, the R-matrix construction of quantum immanants.     

Macdonald processes

Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters $q,t \in [0,1)$ . We prove several results about these processes, which include the following. (1) We explicitly evaluate expectations of a rich family of observables for these processes. (2) In the case $t=0$ , we find a Fredholm determinant formula for a $q$ -Laplace transform of the distribution of the last part of the Macdonald-random partition. (3) We introduce Markov dynamics that preserve the class of Macdonald processes and lead to new “integrable” 2d and 1d interacting particle systems. (4) In a large time limit transition, and as $q$ goes to 1, the particles of these systems crystallize on a lattice, and fluctuations around the lattice converge to O’Connell’s Whittaker process that describe semi-discrete Brownian directed polymers. (5) This yields a Fredholm determinant for the Laplace transform of the polymer partition function, and taking its asymptotics we prove KPZ universality for the polymer (free energy fluctuation exponent $1/3$ and Tracy-Widom GUE limit law). (6) Under intermediate disorder scaling, we recover the Laplace transform of the solution of the KPZ equation with narrow wedge initial data. (7) We provide contour integral formulas for a wide array of polymer moments. (8) This results in a new ansatz for solving quantum many body systems such as the delta Bose gas.     

Jacobi theta-functions and systems of integral shifts of Gaussian functions

This paper is devoted to studying the problems of orthogonalization and interpolation for the system of integral shifts of Gaussian functions. Special attention is paid to the analysis of the role played by the Jacobi theta-functions in solving the problem considered, and also specific features of the numerical implementation of the algorithms being applied.     

Symmetric and nonsymmetric Macdonald polynomials

The symmetric Macdonald polynomials may be constructed from the nonsymmetric Macdonald polynomials. This allows us to develop the theory of the symmetric Macdonald polynomials by first developing the theory of their nonsymmetric counterparts. In taking this approach we are able to obtain new results as well as simpler and more accessible derivations of a number of the known fundamental properties of both kinds of polynomials.Supported by an APA scholarship.     

Asymptotic properties of the weierstrass sigma-functions and the Jacobi theta-functions

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 707–710, May, 1989.     

Colorful combinatorics and Macdonald polynomials

The non-negative integer cocharge statistic on words was introduced in the 1970s by Lascoux and Schützenberger to combinatorially characterize the Hall–Littlewood polynomials. Cocharge has since been used to explain phenomena ranging from the graded decomposition of Garsia–Procesimodules to the cohomology structure of the Grassmann variety. Although its application to contemporary variations of these problems had been deemed intractable, we prove that the two-parameter, symmetric Macdonald polynomials are generating functions of a distinguished family of colored words. Cocharge adorns one parameter and the second measure its deviation from cocharge on words without color. We use the same framework to expand the plactic monoid, apply Kashiwara’s crystal theory to various Garsia–Haiman modules, and to address problems in $K$-theoretic Schubert calculus.     

q-Hypergeometric Series and Macdonald Functions

We derive a duality formula for two-row Macdonald functions by studying their relation with basic hypergeometric functions. We introduce two parameter vertex operators to construct a family of symmetric functions generalizing Hall-Littlewood functions. Their relation with Macdonald functions is governed by a very well-poised q-hypergeometric functions of type ₄₃, for which we obtain linear transformation formulas in terms of the Jacobi theta function and the q-Gamma function. The transformation formulas are then used to give the duality formula and a new formula for two-row Macdonald functions in terms of the vertex operators. The Jack polynomials are also treated accordingly.     

Macdonald operators at infinity

We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables x ₁,x ₂,… and of two parameters q,t are their eigenfunctions. These operators are defined as limits at N→∞ of renormalized Macdonald operators acting on symmetric polynomials in the variables x ₁,…,x _N . They are differential operators in terms of the power sum variables \(p_{n}=x_{1}^{n}+x_{2}^{n}+\cdots\) and we compute their symbols by using the Macdonald reproducing kernel. We express these symbols in terms of the Hall–Littlewood symmetric functions of the variables x ₁,x ₂,…. Our result also yields elementary step operators for the Macdonald symmetric functions.     

Cyclic homology and the Macdonald conjectures

Summary LetA+(k) denote the ring [t]/t ^k+1 and letG be a reductive complex Lie algebra with exponentsm ₁, ...,m n. This paper concerns the Lie algebra cohomology ofGA ₊(k) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we callweight, is inherited from the obvious grading ofGA ₊(k)). We conjecture that this Lie algebra cohomology is an exterior algebra withk+1 generators of homological degree 2m _s +1 fors=1,2, ...,n. Of thesek+1 generators of degree 2m _s +1, one has weight 0 and the others have weights (k+1)m _s +t fort=1,2, ...,k.It is shown that this conjecture about the Lie algebra cohomology of A ₊(k) implies the Macdonald root system conjectures. Next we consider the case thatG is a classical Lie algebra with root systemA _n ,B _n ,C _n , orD _n. It is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofA+(k). Lastly we discuss the relevance of this limiting case to the case of finiten in this situation.Partially supported by NSF grant number MCS-8401718 and a Bantrell Fellowship