Double Affine Hecke Algebra Research Articles

Generalized double affine Hecke algebra for double torus

We propose a generalization of the double affine Hecke algebra of type-C∨C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^\\vee C_1$$\\end{document} at specific parameters by introducing a “Heegaard dual” of the Hecke operators. Shown is a relationship with the skein algebra on double torus. We give automorphisms of the algebra associated with the Dehn twists on the double torus.

Higher rank (𝑞,𝑡)-Catalan polynomials, affine Springer fibers, and a finite rational shuffle theorem

We introduce the higher rank ( q , t ) (q,t) -Catalan polynomials and prove they equal truncations of the Hikita polynomial to a finite number of variables. Using affine compositions and a certain standardization map, we define a d i n v \mathtt {dinv} statistic on rank r r semistandard ( m , n ) (m,n) -parking functions and prove c o d i n v \mathtt {codinv} counts the dimension of an affine space in an affine paving of a parabolic affine Springer fiber. Combining these results, we give a finite analogue of the Rational Shuffle Theorem in the context of double affine Hecke algebras. Lastly, we also give a Bizley-type formula for the higher rank Catalan numbers in the non-coprime case.

Symmetry and Pieri rules for the bisymmetric Macdonald polynomials

Bisymmetric Macdonald polynomials can be obtained through a process of antisymmetrization and t-symmetrization of non-symmetric Macdonald polynomials. Using the double affine Hecke algebra, we show that the evaluation of the bisymmetric Macdonald polynomials satisfies a symmetry property generalizing that satisfied by the usual Macdonald polynomials. We then obtain Pieri rules for the bisymmetric Macdonald polynomials where the sums are over certain vertical strips.

On some double Nahm sums of Zagier

Zagier provided eleven conjectural rank two examples for Nahm's problem. All of them have been proved in the literature except for the fifth example, and there is no q-series proof for the tenth example. We prove that the fifth and the tenth examples are in fact equivalent. Then we give a q-series proof for the fifth example, which confirms a recent conjecture of Wang. This also serves as the first q-series proof for the tenth example, whose explicit form was conjectured by Vlasenko and Zwegers in 2011 and whose modularity was proved by Cherednik and Feigin in 2013 via nilpotent double affine Hecke algebras.

Schur-Weyl duality for quantum toroidal superalgebras

We establish the Schur-Weyl type duality between double affine Hecke algebras and quantum toroidal superalgebras, generalizing the well known result of Vasserot-Varagnolo [26] to the super case.

Type A DAHA and doubly periodic tableaux

Analogously to the construction of Suzuki and Vazirani, we construct representations of the GLm-type Double Affine Hecke Algebra at roots of unity. These representations are graded and the weight spaces for the X-variables are parametrized by the combinatorial objects we call doubly periodic tableaux. We show that our representations exhaust all graded X-semisimple representations, and the direct sum of all our representations is faithful. Analogously to the construction of Jordan and Vazirani of rectangular DAHA representations, we show that our representations can be interpreted in terms of ribbon fusion categories associated to Uq(glN) at roots of unity. Combining the ribbon structure with faithfulness we deduce a conjecture of Morton and Samuelson about realization of DAHA as a skein algebra of the torus with base string modulo certain local relations.

On the basic representation of the double affine Hecke algebra at critical level

We construct the basic representation of the double affine Hecke algebra at critical level [Formula: see text] associated to an irreducible reduced affine root system [Formula: see text] with a reduced gradient root system. For [Formula: see text] of untwisted type such a representation was studied by Oblomkov [A. Oblomkov, Double affine Hecke algebras and Calogero–Moser spaces, Represent. Theory 8(2004) 243–266] and further detailed by Gehles [K. E. Gehles, Properties of Cherednik algebras and graded Hecke algebras, PhD thesis, University of Glasgow (2006)] in the presence of minuscule weights.

Knizhnik–Zamolodchikov functor for degenerate double affine Hecke algebras: algebraic theory

In this article, we define an algebraic version of the Knizhnik–Zamolodchikov (KZ) functor for the degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with the KZ monodromy functor constructed by Varagnolo–Vasserot. We prove the double centraliser property for our functor and give a characterisation of its kernel. We establish these results for a family of algebras, called quiver double Hecke algebras, which includes the degenerate double affine Hecke algebras as special cases.

A Littlewood–Richardson rule for Koornwinder polynomials

Koornwinder polynomials are q-orthogonal polynomials equipped with extra five parameters and the $$B C_n$$ -type Weyl group symmetry, which were introduced by Koornwinder (Contemp Math 138:189–204, 1992) as multivariate analogue of Askey–Wilson polynomials. They are now understood as the Macdonald polynomials associated with the affine root system of type $$(C^\vee _n,C_n)$$ via the Macdonald–Cherednik theory of double affine Hecke algebras. In this paper, we give explicit formulas of Littlewood–Richardson coefficients for Koornwinder polynomials, i.e., the structure constants of the product as invariant polynomials. Our formulas are natural $$(C^\vee _n,C_n)$$ -analogue of Yip’s alcove-walk formulas (Math Z 272:1259–1290, 2012) which were given in the case of reduced affine root systems.

On the genus two skein algebra

We study the skein algebra of the genus 2 surface and its action on the skein module of the genus 2 handlebody. We compute this action explicitly, and we describe how the module decomposes over certain subalgebras in terms of polynomial representations of double affine Hecke algebras. Finally, we show that this algebra is isomorphic to the $t=q$ specialisation of the genus two spherical double affine Hecke algebra recently defined by Arthamonov and Shakirov.

Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

We construct a family of representations of affine Hecke algebras, which depend on a number of auxiliary parameters g_i, and which we refer to as metaplectic representations. We realize these representations as quotients of certain parabolically induced modules, and we apply the method of Baxterization (localization) to obtain actions of corresponding Weyl groups on rational functions on the torus. Our construction both generalizes and provides a conceptual proof of earlier results of Chinta, Gunnells, and Puskas, which had depended on a crucial computer verification. A key motivation is that when the parameters g_i are specialized to certain Gauss sums, the resulting representation and its localization arise naturally in the consideration of p-parts of Weyl group multiple Dirichlet series. In this special case, similar results have been previously obtained in the literature by the study of Iwahori Whittaker functions for principal series of metaplectic covers of reductive p-adic groups. However this technique is not available for generic parameters g_i. It turns out that the metaplectic representations can be extended to the double affine Hecke algebra, where they share many important properties with Cherednik’s basic polynomial representation, which they generalize. This allows us to introduce families of metaplectic polynomials, which depend on the g_i, and which generalize Macdonald polynomials. In this paper we discuss in some detail the situation for type A, which is of considerable interest in algebraic combinatorics. We postpone some of the proofs, as well as a discussion of other types, to the sequel.

DAHAs and Skein Theory

We give a skein-theoretic realization of the $\mathfrak{gl}_n$ double affine Hecke algebra of Cherednik using braids and tangles in the punctured torus. We use this to provide evidence of a relationship we conjecture between the classical skein algebra of the punctured torus and the elliptic Hall algebra of Burban and Schiffmann.

A note on discrete dynamical systems in theories of class S

In this note we consider the set of line operators in theories of class S. We show that this set carries the action of a natural discrete dynamical system associated with the BPS spectrum. We discuss several applications of this perspective; the relation with global properties of the theory; the set of constraints imposed on the spectrum generator, in particular for the case of SU(2) mathcal{N} = 2*; and the relation between line defects and certain spherical Double Affine Hecke Algebras.

Cyclotomic expansion of generalized Jones polynomials

In (Compos. Math. 152(7): 1333–1384, 2016), Berest and Samuelson proposed a conjecture that the Kauffman bracket skein module of any knot in \(S^3\) carries a natural action of a rank 1 double-affine Hecke algebra \(SH_{q,t_1, t_2}\) depending on 3 parameters \(q, t_1, t_2\). As a consequence, for a knot K satisfying this conjecture, we defined a three-variable polynomial invariant \(J^K_n(q,t_1,t_2)\) generalizing the classical coloured Jones polynomials \(J^K_n(q)\). In this paper, we give explicit formulas and provide a quantum group interpretation for the polynomials \(J^K_n(q,t_1,t_2)\). Our formulas generalize the so-called cyclotomic expansion of the classical Jones polynomials constructed by Habiro (Invent. Math. 171(1): 1–81, 2008) : as in the classical case, they imply the integrality of \(J^K_n(q,t_1,t_2)\) and, in fact, make sense for an arbitrary knot K independent of whether or not it satisfies the conjecture of Berest and Samuelson (Compos. Math. 152(7): 1333–1384, 2016). When one of the Hecke deformation parameters is set to be 1, we show that the coefficients of the (generalized) cyclotomic expansion of \(J^K_n(q,t_1)\) are expressed in terms of Macdonald orthogonal polynomials.

InverseK-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type

AbstractWe prove an explicit inverse Chevalley formula in the equivariantK-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a$\mathbb {Z}\left [q^{\pm 1}\right ]$-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars$e^{\lambda }$, where$\lambda $is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type$E_8$. The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetricq-Toda operators for minuscule weights in ADE type.

Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type (C1∨,C1)

Assume that F is an algebraically closed field with characteristic zero. The universal Racah algebra ℜ is a unital associative F -algebra defined by generators and relations. The generators are A , B , C , D and the relations state that [ A , B ] = [ B , C ] = [ C , A ] = 2 D and each of [ A , D ] + A C − B A , [ B , D ] + B A − C B , [ C , D ] + C B − A C is central in ℜ. The universal additive DAHA (double affine Hecke algebra) H of type ( C 1 ∨ , C 1 ) is a unital associative F -algebra generated by t 0 , t 1 , t 0 ∨ , t 1 ∨ and the relations state that t 0 + t 1 + t 0 ∨ + t 1 ∨ = − 1 and each of t 0 2 , t 1 2 , t 0 ∨ 2 , t 1 ∨ 2 is central in H . Each H -module is an ℜ-module by pulling back via the algebra homomorphism ℜ → H given by A ↦ ( t 1 ∨ + t 0 ∨ ) ( t 1 ∨ + t 0 ∨ + 2 ) 4 , B ↦ ( t 1 + t 1 ∨ ) ( t 1 + t 1 ∨ + 2 ) 4 , C ↦ ( t 0 ∨ + t 1 ) ( t 0 ∨ + t 1 + 2 ) 4 . Let V denote any finite-dimensional irreducible H -module. The set of ℜ-submodules of V forms a lattice under the inclusion partial order. We classify the lattices that arise by this construction. As a consequence, the ℜ-module V is completely reducible if and only if t 0 is diagonalizable on V .

Kauffman skein algebras and quantum Teichmüller spaces via factorization homology

We compute the factorization homology of the four-punctured sphere and punctured torus over the quantum group [Formula: see text] explicitly as categories of equivariant modules using the framework developed by Ben-Zvi et al. We identify the algebra of [Formula: see text]-invariants (quantum global sections) with the spherical double affine Hecke algebra of type [Formula: see text], in the four-punctured sphere case, and with the “cyclic deformation” of [Formula: see text] in the punctured torus case. In both cases, we give an identification with the corresponding quantum Teichmüller space as proposed by Teschner and Vartanov as a quantization of the moduli space of flat connections.

Elliptic Double Affine Hecke Algebras

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the C<sub>n</sub> version of the construction to construct a flat noncommutative deformation of the nth symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.

Double Affine Hecke Algebras and Congruence Groups

The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA always admit a faithful action by automorphisms of a finite index subgroup of the Artin group of type $A_{2}$, which descends to a faithful outer action of a congruence subgroup of $SL(2,\mathbb{Z})$ or $PSL(2,\mathbb{Z})$. This was previously known only in some special cases and, to the best of our knowledge, not even conjectured to hold in full generality. The structural intricacies of DAAG/DAHA are captured by the underlying semisimple data and, to a large extent, by adjoint data; we prove our main result by reduction to the adjoint case. Adjoint DAAG/DAHA correspond in a natural way to affine Lie algebras, or more precisely to their affinized Weyl groups, which are the semi-direct products $W\ltimes Q^{\vee}$ of the Weyl group $W$ with the coroot lattice $Q^{\vee}$. We now describe our results for the adjoint case in greater detail. We first give a new Coxeter-type presentation for adjoint DAAG as quotients of the Coxeter braid groups associated to certain crystallographic diagrams that we call double affine Coxeter diagrams. As a consequence we show that the rank two Artin groups of type $A_{2},B_{2},G_{2}$ act by automorphisms on the adjoint DAAG/DAHA associated to affine Lie algebras of twist $r=1,2,3$, respectively. This extends a fundamental result of Cherednik for $r=1$. We show further that the above rank two Artin group action descends to an outer action of congruence subgroup $\Gamma_{1}(r)$. In particular $\Gamma_{1}( r) $ acts naturally on the set of isomorphism classes of representations of an adjoint DAAG/DAHA of twist type $r$, giving rise to a projective representation of $\Gamma_{1}( r) $ on the space of a $\Gamma_{1}( r) $-stable representation.

Finite-dimensional irreducible modules of the universal DAHA of type

Assume that is an algebraically closed field and let q denote a nonzero scalar in that is not a root of unity. The universal DAHA (double affine Hecke algebra) of type is an unital associative -algebra defined by generators and relations. The generators are and the relations assert that In this paper we describe the finite-dimensional irreducible -modules from many viewpoints and classify the finite-dimensional irreducible -modules up to isomorphism. The proofs are carried out in the language of linear algebra.