arXiv subjects

Jasper Stokman

Publications and source records attributed to Jasper Stokman.

16 recordsLinked to original sources

arXiv2024

Graphical calculus for quantum vertex operators, II: q-KZB and coordinate Macdonald-Ruijsenaars equations

We extend the graphical calculus developed in the first part of this paper to the parametrising spaces of quantum vertex operators. This involves a graphical implementation of the dynamical twist functor, which is a strict monoidal functor that describes how a morphism acting on the spin space of a quantum vertex operator $\Phi$ is transported to a morphism on the parametrising space of $\Phi$. The monoidal structure of the underlying nonstrict monoidal functor, considered before by Etingof and Varchenko in 1999, is given in terms of dynamical fusion operators, which are operators that describe the fusion of quantum vertex operators on the level of parametrising spaces. In the second part of the paper we use the extended graphical calculus to give intuitive, graphical derivations of various systems of difference equations for universal multipoint weighted trace functions. This includes the dual $q$-Knizhnik-Zamolodchikov-Bernard (KZB) and the dual Macdonald-Ruijsenaars (MR) equations, earlier obtained by Etingof and Varchenko in 2000, as well as an extension of the dual MR equations called dual coordinate MR equations. We use a known symmetry property of the universal weighted trace function, involving the exchange of its geometric and spectral parameter, to derive non-dual versions of these equations.

math.QA↗

arXiv2024

Quasi-Polynomial Extensions of Nonsymmetric Macdonald-Koornwinder Polynomials

In a recent joint paper with S. Sahi and V. Venkateswaran (2025), families of actions of the double affine Hecke algebra on spaces of quasi-polynomials were introduced. These so-called quasi-polynomial representations led to the introduction of quasi-polynomial extensions of the nonsymmetric Macdonald polynomials, which reduce to metaplectic Iwahori-Whittaker functions in the $\mathfrak{p}$-adic limit. In this paper, these quasi-polynomial representations are extended to Sahi's 5-parameter double affine Hecke algebra, and the quasi-polynomial extensions of the nonsymmetric Koornwinder polynomials are introduced.

math.RT↗

arXiv2023

Quantum superintegrable spin systems on graph connections

In this paper we construct certain quantum spin systems on moduli spaces of $G$-connections on a connected oriented finite graph, with $G$ a simply connected compact Lie group. We construct joint eigenfunctions of the commuting quantum Hamiltonians in terms of local invariant tensors. We determine sufficient conditions ensuring superintegrability of the quantum spin system using irreducibility criteria for Harish-Chandra modules due to Harish-Chandra and Lepowsky & McCollum. The resulting class of quantum superintegrable spin systems includes the quantum periodic and open spin Calogero-Moser spin chains as special cases. In the periodic case the description of the joint eigenfunctions in terms of local invariant tensors are multipoint generalised trace functions, in the open case multipoint spherical functions on compact symmetric spaces.

math.RT↗

arXiv2022

N-point spherical functions and asymptotic boundary KZB equations

Let $G$ be a split real connected Lie group with finite center. In the first part of the paper we define and study formal elementary spherical functions. They are formal power series analogues of elementary spherical functions on $G$ in which the role of the quasi-simple admissible $G$-representations is replaced by Verma modules. For generic highest weight we express the formal elementary spherical functions in terms of Harish-Chandra series and integrate them to spherical functions on the regular part of $G$. We show that they produce eigenstates for spin versions of quantum hyperbolic Calogero-Moser systems. In the second part of the paper we define and study special subclasses of global and formal elementary spherical functions, which we call global and formal $N$-point spherical functions. Formal $N$-point spherical functions arise as limits of correlation functions for boundary Wess-Zumino-Witten conformal field theory on the cylinder when the position variables tend to infinity. We construct global $N$-point spherical functions in terms of compositions of equivariant differential intertwiners associated with principal series representations, and express them in terms of Eisenstein integrals. We show that the eigenstates of the spin quantum Calogero-Moser system associated to $N$-point spherical functions are also common eigenfunctions of a commuting family of first-order differential operators, which we call asymptotic boundary Knizhnik-Zamolodchikov-Bernard operators. These operators are explicitly given in terms of $θ$-folded classical dynamical $r$-matrices and associated dynamical $k$-matrices.

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arXiv2022

Graphical calculus for quantum vertex operators, I: The dynamical fusion operator

This paper is the first in a series on graphical calculus for quantum vertex operators. We establish in great detail the foundations of graphical calculus for ribbon categories and braided monoidal categories with twist. We illustrate the potential of this approach by applying it to various categories of quantum group modules, in particular to derive an extension of the linear operator equation for dynamical fusion operators, due to Arnaudon, Buffenoir, Ragoucy and Roche, to a system of linear operator equations of $q$-KZ type.

math.QA↗

arXiv2022

Quasi-polynomial representations of double affine Hecke algebras

We introduce an explicit family of representations of the double affine Hecke algebra $\mathbb{H}$ acting on spaces of quasi-polynomials, defined in terms of truncated Demazure-Lusztig type operators. We show that these quasi-polynomial representations provide concrete realizations of a natural family of cyclic $Y$-parabolically induced $\mathbb{H}$-representations. We recover Cherednik's well-known polynomial representation as a special case. The quasi-polynomial representation gives rise to a family of commuting operators acting on spaces of quasi-polynomials. These generalize the Cherednik operators, which are fundamental in the study of Macdonald polynomials. We provide a detailed study of their joint eigenfunctions, which may be regarded as quasi-polynomial, multi-parametric generalizations of nonsymmetric Macdonald polynomials. We also introduce generalizations of symmetric Macdonald polynomials, which are invariant under a multi-parametric generalization of the standard Weyl group action. We connect our results to the representation theory of metaplectic covers of reductive groups over non-archimedean local fields. We introduce root system generalizations of the metaplectic polynomials from our previous work by taking a suitable restriction and reparametrization of the quasi-polynomial generalizations of Macdonald polynomials. We show that metaplectic Iwahori-Whittaker functions can be recovered by taking the Whittaker limit of these metaplectic polynomials.

math.RT↗

arXiv2021

Folded and contracted solutions of coupled classical dynamical Yang-Baxter and reflection equations

In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space $\mathfrak{a}$ satisfying three coupled classical dynamical Yang-Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on $\mathfrak{a}$, generalising asymptotic boundary Knizhnik-Zamolodchikov-Bernard (KZB) equations. The recipe involves folding and contracting $\mathfrak{a}$-invariant and $θ$-twisted symmetric classical dynamical $r$-matrices along an involutive automorphism $θ$. In case of the universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$ we determine the Etingof-Schiffmann classical dynamical $r$-matrices which are $\mathfrak{a}$-invariant and $θ$-twisted symmetric. The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories.

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arXiv2020

Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains

Asymptotic boundary KZB equations describe the consistency conditions of degenerations of correlation functions for boundary Wess-Zumino-Witten-Novikov conformal field theory on a cylinder. In the first part of the paper we define asymptotic boundary KZB operators for connected real semisimple Lie groups G with finite center. We prove their main properties algebraically using coordinate versions of Harish-Chandra's radial component map. We show that their commutativity is governed by a system of equations involving coupled versions of classical dynamical Yang-Baxter equations and reflection equations. We use the coordinate radial components maps to introduce a new class of quantum superintegrable systems, called quantum Calogero-Moser spin chains. A quantum Calogero-Moser spin chain is a mixture of a quantum spin Calogero-Moser system associated to the restricted root system of G and an one-dimensional spin chain with two-sided reflecting boundaries. The asymptotic boundary KZB operators provide explicit expressions for its first order quantum Hamiltonians. We also explicitly describe the Schr\"odinger operator.

math.RT↗

arXiv2019

Towers of solutions of qKZ equations and their applications to loop models

Cherednik's type A quantum affine Knizhnik-Zamolodchikov (qKZ) equations form a consistent system of linear $q$-difference equations for $V_n$-valued meromorphic functions on a complex $n$-torus, with $V_n$ a module over the GL${}_n$-type extended affine Hecke algebra $\mathcal{H}_n$. The family $(\mathcal{H}_n)_{n\geq 0}$ of extended affine Hecke algebras forms a tower of algebras, with the associated algebra morphisms $\mathcal{H}_n\rightarrow\mathcal{H}_{n+1}$ the Hecke algebra descends of arc insertion at the affine braid group level. In this paper we consider qKZ towers $(f^{(n)})_{n\geq 0}$ of solutions, which consist of twisted-symmetric polynomial solutions $f^{(n)}$ ($n\geq 0$) of the qKZ equations that are compatible with the tower structure on $(\mathcal{H}_n)_{n\geq 0}$. The compatibility is encoded by so-called braid recursion relations: $f^{(n+1)}(z_1,\ldots,z_{n},0)$ is required to coincide up to a quasi-constant factor with the push-forward of $f^{(n)}(z_1,\ldots,z_{n})$ by an intertwiner $μ_{n}: V_{n}\rightarrow V_{n+1}$ of $\mathcal{H}_{n}$-modules, where $V_{n+1}$ is considered as an $\mathcal{H}_{n}$-module through the tower structure on $(\mathcal{H}_n)_{n\geq 0}$. We associate to the dense loop model on the half-infinite cylinder with nonzero loop weights a qKZ tower $(f^{(n)})_{n\geq 0}$ of solutions. The solutions $f^{(n)}$ are constructed from specialised dual non-symmetric Macdonald polynomials with specialised parameters using the Cherednik-Matsuo correspondence. In the special case that the extended affine Hecke algebra parameter is a third root of unity, $f^{(n)}$ coincides with the (suitably normalized) ground state of the inhomogeneous dense $O(1)$ loop model on the half-infinite cylinder with circumference $n$.

math-ph↗

arXiv2019

Some remarks on non-symmetric interpolation Macdonald polynomials

We provide elementary identities relating the three known types of non-symmetric interpolation Macdonald polynomials. In addition we derive a duality for non-symmetric interpolation Macdonald polynomials. We consider some applications of these results, in particular for binomial formulas involving non-symmetric interpolation Macdonald polynomials.

math.QA↗

arXiv2016

Integral solutions to boundary quantum Knizhnik-Zamolodchikov equations

We construct integral representations of solutions to the boundary quantum Knizhnik-Zamolodchikov equations. These are difference equations taking values in tensor products of Verma modules of quantum affine $\mathfrak{sl}_2$, with the K-operators acting diagonally. The integrands in question are products of scalar-valued elliptic weight functions with vector-valued trigonometric weight functions (boundary Bethe vectors). These integrals give rise to a basis of solutions of the boundary qKZ equations over the field of quasi-constant meromorphic functions in weight subspaces of the tensor product.

math.QA↗

arXiv2015

Koornwinder polynomials and the XXZ spin chain

Nonsymmetric Koornwinder polynomials are multivariable extensions of nonsymmetric Askey-Wilson polynomials. They naturally arise in the representation theory of (double) affine Hecke algebras. In this paper we discuss how nonsymmetric Koornwinder polynomials naturally arise in the theory of the Heisenberg XXZ spin-$\frac{1}{2}$ chain with general reflecting boundary conditions. A central role in this story is played by an explicit two-parameter family of spin representations of the two-boundary Temperley-Lieb algebra. These spin representations have three different appearances. Their original definition relates them directly to the XXZ spin chain, in the form of matchmaker representations they relate to Temperley-Lieb loop models in statistical physics, while their realization as principal series representations leads to the link with nonsymmetric Koornwinder polynomials. The nonsymmetric difference Cherednik-Matsuo correspondence allows to construct for special parameter values Laurent-polynomial solutions of the associated reflection quantum KZ equations in terms of nonsymmetric Koornwinder polynomials. We discuss these aspects in detail by revisiting and extending work of De Gier, Kasatani, Nichols, Cherednik, the first author and many others.

math.QA↗

arXiv2014

Boundary quantum Knizhnik-Zamolodchikov equations and fusion

In this paper we extend our previous results concerning Jackson integral solutions of the boundary quantum Knizhnik-Zamolodchikov equations with diagonal K-operators to higher-spin representations of quantum affine $\mathfrak{sl}_2$. First we give a systematic exposition of known results on $R$-operators acting in the tensor product of evaluation representations in Verma modules over quantum $\mathfrak{sl}_2$. We develop the corresponding fusion of $K$-operators, which we use to construct diagonal $K$-operators in these representations. We construct Jackson integral solutions of the associated boundary quantum Knizhnik-Zamolodchikov equations and explain how in the finite-dimensional case they can be obtained from our previous results by the fusion procedure.

math.QA↗

arXiv2014

Boundary quantum Knizhnik-Zamolodchikov equations and Bethe vectors

Solutions to boundary quantum Knizhnik-Zamolodchikov equations are constructed as bilateral sums involving "off-shell" Bethe vectors in case the reflection matrix is diagonal and only the 2-dimensional representation of $U_q(\hat{\frak{sl}(2)})$ is involved. We also consider their rational and classical degenerations.

math.QA↗

arXiv2006

Quantum dimensions and their non-Archimedean degenerations

We derive explicit dimension formulas for irreducible $M_F$-spherical $K_F$-representations where $K_F$ is the maximal compact subgroup of the general linear group $GL(d,F)$ over a local field $F$ and $M_F$ is a closed subgroup of $K_F$ such that $K_F/M_F$ realizes the Grassmannian of $n$-dimensional $F$-subspaces of $F^d$. We explore the fact that $(K_F,M_F)$ is a Gelfand pair whose associated zonal spherical functions identify with various degenerations of the multivariable little $q$-Jacobi polynomials. As a result, we are led to consider generalized dimensions defined in terms of evaluations and quadratic norms of multivariable little $q$-Jacobi polynomials, which interpolate between the various classical dimensions. The generalized dimensions themselves are shown to have representation theoretic interpretations as the quantum dimensions of irreducible spherical quantum representations associated to quantum complex Grassmannians.

math.QA↗

arXiv1999

Fourier transforms on the quantum SU(1,1) group

The main goal is to interpret the Askey-Wilson function and the corresponding transform pair on the quantum SU(1,1) group. A weight on the C^*-algebra of continuous functions vanishing at infinity on the quantum SU(1,1) group is studied, which is left and right invariant in a weak sense with respect to a product defined using Wall functions. The Haar weight restricted to certain subalgebras are explicitly determined in terms of an infinitely supported Jackson integral and in terms of an infinitely supported Askey-Wilson type measure. For the evaluation the spectral analysis of explicit unbounded doubly infinite Jacobi matrices and some new summation formulas for basic hypergeometric series are needed. The spherical functions are calculated in terms of Askey-Wilson functions and big q-Jacobi functions. The corresponding spherical Fourier transforms are identified with special cases of the big q-Jacobi function transform and of the Askey-Wilson function transform.

math.QA↗