We investigate the structure of $\tau$-functions for the elliptic difference Painlevé equation of type $E_8$. Introducing the notion of ORG $\tau$-functions for the $E_8$ lattice, we construct some particular solutions which are expressed in terms of elliptic hypergeometric integrals. Also, we discuss how this construction is related to the framework of lattice $\tau$-functions associated with the configuration of generic nine points in the projective plane.

We will give several descriptions of some basic examples of wild character varieties, including a discussion of links to work of Sibuya, Calabi and Euler, amongst others.

We construct a hyperbolic modular double – an algebra lying in between the Faddeev modular double for $U_q(sl_2)$ and the elliptic modular double. The intertwining operator for this algebra leads to an integral operator solution of the Yang-Baxter equation associated with a generalized Faddeev-Volkov lattice model introduced by the second author. We describe also the L-operator and finite-dimensional R-matrices for this model.

Starting from a recently found branching rule for the six-parameter family of symmetric Macdonald-Koornwinder polynomials, we arrive by degeneration at corresponding branching formulas for symmetric hypergeometric orthogonal polynomials of Wilson, continuous Hahn, Jacobi, Laguerre, and Hermite type.

We propose a new method to compute connection matrices of quantum Knizhnik-Zamolodchikov equations associated to integrable vertex models with super algebra and Hecke algebra symmetries. The scheme relies on decomposing the underlying spin representation of the affine Hecke algebra in principal series modules and invoking the known solution of the connection problem for quantum affine Knizhnik-Zamolodchikov equations associated to principal series modules. We apply the method to the spin representation underlying the $\mathcal{U}_q\bigl(\widehat{\mathfrak{gl}}(2|1)\bigr)$ Perk-Schultz model. We show that the corresponding connection matrices are described by an elliptic solution of the dynamical quantum Yang-Baxter equation with spectral parameter.

The conical function and its relativistic generalization can be viewed as eigenfunctions of the reduced 2-particle Hamiltonians of the hyperbolic Calogero-Moser system and its relativistic generalization. We prove new product formulas for these functions. As a consequence, we arrive at explicit diagonalizations of integral operators that commute with the 2-particle Hamiltonians and reduced versions thereof. The kernels of the integral operators are expressed as integrals over products of the eigenfunctions and explicit weight functions. The nonrelativistic limits are controlled by invoking novel uniform limit estimates for the hyperbolic gamma function.

We propose a unified approach to transformation and summation formulas for multiple basic hypergeometric series of type $A$ on the basis of balanced duality transformations. We study two classes of transformations, one between an $n$-ple sum and an $m$-ple sum, and the other between two $n$-ple sums. Though the latter are not simple special cases of the former, we can still derive them from the former in a systematic way. Our derivation utilizes the fact that some multiple basic hypergeometric series have hidden symmetry which originates from the relation to basic hypergeometric series in one variable. We also give remarks on the related summation formulas.

Arakawa, Suzuki and Tsuchiya defined a correspondence between certain modules of the trigonometric Cherednik algebra $\mathfrak{C}_N$ depending on a parameter $\kappa\in\mathbb{C}\,$, and certain modules of the affine Lie algebra $\widehat{\mathfrak{sl}}_m$ of level $\kappa-m\,$. We give a detailed proof of this correspondence by working with the affine Lie algebra $\widehat{\mathfrak{gl}}_m$ alongside of $\,\widehat{\mathfrak{sl}}_m\,$. We also relate this construction to a correspondence between certain modules of the degenerate affine Hecke algebra $\mathfrak{H}_N$ and all modules of $\mathfrak{sl}_m$ or $\mathfrak{gl}_m\,$. The latter correspondence was constructed earlier by Cherednik.

We will look at the Catalan numbers from the Rigged Configurations point of view originated [10] from an combinatorial analysis of the Bethe Ansatz Equations associated with the higher spin anisotropic Heisenberg models. Our strategy is to take a combinatorial interpretation of the Catalan number $C_n$ as the number of standard Young tableaux of rectangular shape $(n^2)$, or equivalently, as the Kostka number $K_{(n^2),1^{2n}}$, as the starting point of our research. We observe that the rectangular (or multidimensional) Catalan numbers $ C(m,n)$, introduced and studied by P. MacMahon [23], [34], see also [35], can be identified with the corresponding Kostka numbers $K_{(n^m),1^{mn}}$, and therefore can be treated by the Rigged Configurations technique. Based on this technique we study the stretched Kostka numbers and polynomials, and give a proof of a strong rationality of the stretched Kostka polynomials. This result implies a polynomiality property of the stretched Kostka and stretched Littlewood–Richardson coefficients [8], [28], [17]. Finally, we give a brief introduction to a rigged configuration version of the Robinson–Schensted–Knuth correspondence.

Another application of the Rigged Configuration technique presented, is a new family of counterexamples to Okounkov's log-concavity conjecture [27].

Finally, we apply Rigged Configurations technique to give a combinatorial proof of the unimodality of the principal specialization of the internal product of Schur functions. In fact we prove a combinatorial (fermionic) formula for generalized $q$-Gaussian polynomials which is a far generalization of the so-called $KOH$-identity [26], as well as it manifests the unimodality property of the $q$-Gaussian polynomials.

We reformulate a central extension of Felder's elliptic quantum group in the FRST formulation as a topological algebra $E_{q,p}(\widehat{\mathfrak{gl}}_N)$ over the ring of formal power series in $p$. We then discuss the isomorphism between $E_{q,p}(\widehat{\mathfrak{gl}}_N)$ and the elliptic algebra $U_{q,p}(\widehat{\mathfrak{gl}}_N)$ of the Drinfeld realization. An evaluation $H$-algebra homomorphism from $U_{q,p}(\widehat{\mathfrak{gl}}_N)$ to a dynamical extension of the quantum affine algebra $U_{q}(\widehat{\mathfrak{gl}}_N)$ resolves the problem into the one discussed by Ding and Frenkel in the trigonometric case. We also provide some useful formulas for the elliptic quantum determinants.

We discuss quadratic transformations for orthogonal polynomials in one and two variables. In the one-variable case we list many (or all) quadratic transformations between families in the Askey scheme or $q$-Askey scheme. In the two-variable case we focus, after some generalities, on the polynomials associated with root system $BC_2$, i.e., $BC_2$-type Jacobi polynomials if $q=1$ and Koornwinder polynomials in two variables in the $q$-case.

In this paper we show how the Cherednik algebra of type $\check{C_1}C_1$ appears naturally as quantisation of the group algebra of the monodromy group associated to the sixth Painlevé equation. This fact naturally leads to an embedding of the Cherednik algebra of type $\check{C_1}C_1$ into $Mat(2,\mathbb T_q)$, i.e. $2\times 2$ matrices with entries in the quantum torus. For $q=1$ this result is equivalent to say that the Cherednik algebra of type $\check{C_1}C_1$ is Azumaya of degree 2 [31]. By quantising the action of the braid group and of the Okamoto transformations on the monodromy group associated to the sixth Painlevé equation we study the automorphisms of the Cherednik algebra of type $\check{C_1}C_1$ and conjecture the existence of a new automorphism. Inspired by the confluences of the Painlevé equations, we produce similar embeddings for the confluent Cherednik algebras $\mathcal H_V,\mathcal H_{IV},\mathcal H_{III},\mathcal H_{II}$ and $\mathcal H_{I},$ defined in [27].

We consider a complex version of the Gauss hypergeometric integral from the view point of the twisted de Rham theory. In particular, we give a formula to express the complex hypergeometric integral in terms of the hermitian form of the ordinary Gauss hypergeometric integrals.

In a previous article [11] a novel class of elliptic Lax pairs for integrable lattice equations was introduced. The present article proposes a de-autonomisation of those Lax pairs leading to a class of elliptic discrete isomonodromic deformation problems. We analyse the systems of compatibility conditions using some (possibly novel) higher order elliptic identities.

Inspired by the new type of Capelli identity associated with the regular representation of quaternions obtained by An Huang [6], we present a more basic type of Capelli identities for reducible modules of the matrix algebras. As a related subject in view of reducible modules, we also discuss the Capelli identities associated with group determinants, for which a precise formulation and an answer are given.