Macdonald Operators Research Articles

Computing the R-matrix of the quantum toroidal algebra

We consider the problem of the R-matrix of the quantum toroidal algebra Uq,t(gl..1) in the Fock representation. Using the connection between the R-matrix R(u) (u being the spectral parameter) and the theory of Macdonald operators, we obtain explicit formulas for R(u) in the operator and matrix forms. These formulas are expressed in terms of the eigenvalues of a certain Macdonald operator, which completely describe the functional dependence of R(u) on the spectral parameter u. We then consider the geometric R-matrix (obtained from the theory of K-theoretic stable bases on moduli spaces of framed sheaves), which is expected to coincide with R(u) and thus gives another approach to the study of the poles of the R-matrix as a function of u.

Elliptic generalisation of integrable q-deformed anisotropic Haldane–Shastry long-range spin chain

We describe integrable elliptic q-deformed anisotropic long-range spin chain. The derivation is based on our recent construction for commuting anisotropic elliptic spin Ruijsenaars–Macdonald operators. We prove that the Polychronakos freezing trick can be applied to these operators, thus providing the commuting set of Hamiltonians for long-range spin chain constructed by means of the elliptic Baxter-Belavin -matrix. Namely, we show that the freezing trick is reduced to a set of elliptic function identities, which are then proved. These identities can be treated as conditions for equilibrium position in the underlying classical spinless Ruijsenaars–Schneider model. Trigonometric degenerations are studied as well. For example, in M = 2 case our construction provides q-deformation for anisotropic XXZ Haldane–Shastry model. The standard Haldane–Shastry model and its Uglov’s q-deformation based on XXZ R-matrix are included into consideration by separate verification.

Wave Function for GL(n,ℝ) Hyperbolic Sutherland Model II. Dual Hamiltonians

Abstract Recently, we found Mellin–Barnes integrals, representing the wave function for $GL(n,{\mathbb {R}})$ hyperbolic Sutherland model. In present paper, we establish bispectral properties of this wave function with respect to dual Ruijesenaars–Macdonald operators.

Capped vertex with descendants for zero dimensional A∞ quiver varieties

In this paper, we study the capped vertex functions associated to certain zero-dimensional type-A Nakajima quiver varieties. The insertion of descendants into the vertex functions can be expressed by the Macdonald operators, which leads to explicit combinatorial formulas for the capped vertex functions.We determine the monodromy of the vertex functions and show that it coincides with the elliptic R-matrix of symplectic dual variety. We apply our results to give the vertex functions and the characters of the tautological bundles on the quiver varieties formed from arbitrary stability conditions.

Macdonald operator function and the incomplete Cauchy problem for the Euler-Poisson-Darboux equation

Macdonald operator function and the incomplete Cauchy problem for the Euler-Poisson-Darboux equation

Generalized Macdonald Functions on Fock Tensor Spaces and Duality Formula for Changing Preferred Direction

An explicit formula is obtained for the generalized Macdonald functions on the $N$-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property of the arbitrary matrix elements of the multi-valent intertwining operator (or refined topological vertex operator) associated with the Ding--Iohara--Miki algebra (DIM algebra) with respect to the generalized Macdonald functions, which was conjectured by Awata, Feigin, Hoshino, Kanai, Yanagida and one of the authors. Our proof is based on the combinatorial and analytic properties of the asymptotic eigenfunctions of the ordinary Macdonald operator of $A$-type, and the Euler transformation formula for Kajihara and Noumi's multiple basic hypergeometric series. That factorization formula provides us with a reasonable algebraic description of the 5D (K-theoretic) Alday-Gaiotto-Tachikawa (AGT) correspondence, and the interpretation of the invariance under the preferred direction from the point of view of the $SL(2,\mathbb{Z})$ duality of the DIM algebra.

($${{\mathbf {t}}},{{\mathbf {q}}}$$)-Deformed Q-Systems, DAHA and Quantum Toroidal Algebras via Generalized Macdonald Operators

We introduce the natural (t, q)-deformation of the Q-system algebra in type A. The q-Whittaker limit $$t\rightarrow \infty $$ gives the quantum Q-system algebra of Di Francesco and Kedem (Lett Math Phys 107(2):301–341, [DFK17]), a deformation of the Groethendieck ring of finite dimensional Yangian modules, compatible with graded tensor products (Hatayama et al. in: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Volume 248 of Contemporary Mathematics, Amer. Math. Soc., Providence, [HKO+99]; Feigin and Loktev in: Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Volume 194 of Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, [FL99]; Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, [DFK14]). We show that the (q, t)-deformed algebra is isomorphic to the spherical double affine Hecke algebra of type $${\mathfrak {gl}}_N$$ . Moreover, we describe the kernel of the surjective homomorphism from the quantum toroidal algebra (Miki in J Math Phys 48(12):123520, [Mik07]) and the elliptic Hall algebra (Schiffmann and Vasserot in Compos Math 147(1):188–234, [SV11]) to this new algebra. It is generated by (q, t)-determinants, new objects which are a deformation of the quantum determinant associated with the quantum Q-system. The functional representation of the algebra is generated by generalized Macdonald operators, obtained from the usual Macdonald operators by the $$SL_2({\mathbb {Z}})$$ -action on the spherical Double Affine Hecke Algebra. The generating function for generalized Macdonald operators acts by plethysms on the space of symmetric functions. We give the relation to the plethystic operators from Macdonald theory of Bergeron et al. (J Comb 7(4):671–714, [BGLX16]) in the limit $$N\rightarrow \infty $$ . Thus, the (q, t)-deformation of the Q-system cluster algebra leads directly to Macdonald theory.

Interpolation Macdonald operators at infinity

We study the interpolation Macdonald functions, remarkable inhomogeneous generalizations of Macdonald functions, and a sequence A1,A2,… of commuting operators whose common eigenfunctions are the interpolation Macdonald functions. Such a sequence of operators arises in the projective limit of finite families of commuting q-difference operators studied by Okounkov, Knop and Sahi. The main theorem is an explicit formula for the operators Ak. Our formula involves the family of Hall–Littlewood functions and a new family of inhomogeneous Hall–Littlewood functions, for which we give an explicit construction and identify as a degeneration of the interpolation Macdonald functions when q→0. This article is inspired by the recent papers of Nazarov–Sklyanin on Macdonald and Sekiguchi–Debiard operators, and our main theorem is an extension of their results.

A representation-theoretic proof of the branching rule for Macdonald polynomials

We give a new representation-theoretic proof of the branching rule for Macdonald polynomials using the Etingof-Kirillov Jr. expression for Macdonald polynomials as traces of intertwiners of U_q(gl_n). In the Gelfand-Tsetlin basis, we show that diagonal matrix elements of such intertwiners are given by application of Macdonald's operators to a simple kernel. An essential ingredient in the proof is a map between spherical parts of double affine Hecke algebras of different ranks based upon the Dunkl-Kasatani conjecture.

Elliptic Ding–Iohara Algebra and the Free Field Realization of the Elliptic Macdonald Operator

The Ding–Iohara algebra is a quantum algebra arising from the free field realization of the Macdonald operator. Starting from the elliptic kernel function introduced by Komori, Noumi and Shiraishi, we define an elliptic analog of the Ding–Iohara algebra. The free field realization of the elliptic Macdonald operator is also constructed.

Commutative Families of the Elliptic Macdonald Operator

In the paper [J. Math. Phys. 50 (2009), 095215, 42 pages, arXiv:0904.2291], Feigin, Hashizume, Hoshino, Shiraishi, and Yanagida constructed two families of commuting operators which contain the Macdonald operator (commutative families of the Macdonald operator). They used the Ding-Iohara-Miki algebra and the trigonometric Feigin-Odesskii algebra. In the previous paper [arXiv:1301.4912], the present author constructed the elliptic Ding-Iohara-Miki algebra and the free field realization of the elliptic Macdonald operator. In this paper, we show that by using the elliptic Ding-Iohara-Miki algebra and the elliptic Feigin-Odesskii algebra, we can construct commutative families of the elliptic Macdonald operator. In Appendix, we will show a relation between the elliptic Macdonald operator and its kernel function by the free field realization.

An M-theoretic derivation of a 5d and 6d AGT correspondence, and relativistic and elliptized integrable systems

We generalize our analysis in [ arXiv:1301.1977 ], and show that a 5d and 6d AGT correspondence for SU(N) — which essentially relates the relevant 5d and 6d Nekrasov instanton partition functions to the integrable representations of a q-deformed and elliptic affine $ {{\mathcal{W}}_N} $ -algebra — can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. Via an appropriate defect, we also derive a “fully-ramified” version of the 5d and 6d AGT correspondence where integrable representations of a quantum and elliptic affine SU(N)-algebra at the critical level appear on the 2d side, and argue that the relevant “fully-ramified” 5d and 6d Nekrasov instanton partition functions are simultaneous eigenfunctions of commuting operators which define relativistic and elliptized integrable systems. As an offshoot, we also obtain various mathematically novel and interesting relations involving the double loop algebra of SU(N), elliptic Macdonald operators, equivariant elliptic genus of instanton moduli space, and more.

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.

Polynomials and Parking Functions

In a 2010 paper Haglund, Morse, and Zabrocki studied the family of polynomials $\nabla C_{p1}\dots C_{pk}1$ , where $p=(p_1,\ldots,p_k)$ is a composition, $\nabla$ is the Bergeron-Garsia Macdonald operator and the $C_\alpha$ are certain slightly modified Hall-Littlewood vertex operators. They conjecture that these polynomials enumerate a composition indexed family of parking functions by area, dinv and an appropriate quasi-symmetric function. This refinement of the nearly decade old ``Shuffle Conjecture,'' when combined with properties of the Hall-Littlewood operators can be shown to imply the existence of certain bijections between these families of parking functions. In previous work to appear in her PhD thesis, the author has shown that the existence of these bijections follows from some relatively simple properties of a certain family of polynomials in one variable x with coefficients in $\mathbb{N}[q]$. In this paper we introduce those polynomials, explain their connection to the conjecture of Haglund, Morse, and Zabrocki, and explore some of their surprising properties, both proven and conjectured. Dans un article de 2010, Haglund, Morse et Zabrocki étudient la famille de polynômes $\nabla C_{p1}\dots C_{pk}1$ où $p=(p_1,\ldots,p_k)$ est une composition, $\nabla$ est l’opérateur de Bergeron-Garsia et les $C_\alpha$ sont des opérateurs ``vertex'' de Hall-Littlewood légèrement altérés. Il posent la conjecture que ces polynômes donnent l’énumération d'une famille de fonctions ``parking'', indexées par des compositions, par aire, le ``dinv'' et une fonction quasi-symétrique associée. Cette conjecture raffine la conjecture ``Shuffle'', qui est âgée de presque dix ans. On peut montrer, a partir de cette conjecture, que les propriétés des opérateurs de Hall-Littlewood, impliquent l'existence de certaines bijections entre ces familles de fonctions ``parking''. Dans un précédent travail , qui fait partie de sa thèse de doctorat, l'auteur montre que l’existence de ces bijections découle de certaines propriétés relativement simples d'une famille de polynômes à une variable x, avec coefficients dans $\mathbb{N}[q]$. Dans cet article, on introduit ces polynômes, on explique leur connexion avec la conjecture de Haglund, Morse et Zabrocki, et on explore certaines de leurs propriétés surprenantes, qu'elles soient prouvées ou seulement conjecturées.

Macdonald operators and homological invariants of the colored Hopf link

Using a power sum (boson) realization for the Macdonald operators, we investigate the Gukov, Iqbal, Kozçaz and Vafa (GIKV) proposal for the homological invariants of the colored Hopf link, which include Khovanov–Rozansky homology as a special case. We prove the polynomiality of the invariants obtained by GIKV’s proposal for arbitrary representations. We derive a closed formula of the invariants of the colored Hopf link for antisymmetric representations. We argue that a little amendment of GIKV’s proposal is required to make all the coefficients of the polynomial non-negative integers.

On some special solutions to periodic Benjamin-Ono equation with discrete Laplacian

We investigate a periodic version of the Benjamin-Ono (BO) equation associated with a discrete Laplacian. We find some special solutions to this equation, and calculate the values of the first two integrals of motion I1 and I2 corresponding to these solutions. It is found that there exists a strong resemblance between them and the spectra for the Macdonald q-difference operators. To better understand the connection between these classical and quantum integrable systems, we consider the special degenerate case corresponding to q=0 in more detail. Namely, we give general solutions to this degenerate periodic BO, obtain explicit formulas representing all the integrals of motions In (n=1,2,…), and successfully identify it with the eigenvalues of Macdonald operators in the limit q→0, i.e. the limit where Macdonald polynomials tend to the Hall–Littlewood polynomials.

Higher Order Macdonald Operators with Stability Properties and Their Applications

The higher order Macdonald operators which have compatibility with the restriction homomorphism are defined. As an application of these operators the new derivation of the transition coefficients between the Hall-Littlewood functions P(r)(x;t ± k) indexed by one part partitions and Macdonald functions is given. A family of q,t-identities is also obtained from the specialization of the resultant expansion formulae.

Macdonald functions associated to complex reflection groups

Let W be the complex reflection group S n ⋉( Z /e Z ) n . In the author's previous paper [J. Algebra 245 (2001) 650–694], Hall–Littlewood functions associated to W were introduced. In the special case where W is a Weyl group of type B n , they are closely related to Green polynomials of finite classical groups. In this paper, we introduce a two variables version of the above Hall–Littlewood functions, as a generalization of Macdonald functions associated to symmetric groups. A generalization of Macdonald operators is also constructed, and we characterize such functions by making use of Macdonald operators, assuming a certain conjecture.

Elliptic Ruijsenaars operators and functional equations

We study mutually commutative difference operators introduced by Ruijsenaars, which are regarded as elliptic analogs of Macdonald operators and consist of the Jacobi theta functions and shift operators. These operators are constructed in terms of R-operators due to Shibukawa and Ueno or root algebras introduced by Cherednik, which give rise to a set of functional equations. We investigate solutions of these functional equations for generalizations of elliptic Ruijsenaars operators.

Macdonald Polynomials and Algebraic Integrability

We construct explicitly (nonpolynomial) eigenfunctions of the difference operators by Macdonald in the case t=qk, k∈Z. This leads to a new, more elementary proof of several Macdonald conjectures, proved first by Cherednik. We also establish the algebraic integrability of Macdonald operators at t=qk (k∈Z), generalizing the result of Etingof and Styrkas. Our approach works uniformly for all root systems including the BCn case and related Koornwinder polynomials. Moreover, we apply it for a certain deformation of the An root system where the previously known methods do not work.