2019年度年会(於:東京工業大学)

SUMMARY: In 2012, Gamayun, Iorgov, and Lisovyy discovered that the tau function of the sixth Painlevé equation is a Fourier expansion of the 4-pt conformal block of the two dimensional conformal field theory with the central charge \(c=1\). I will explain about extensions of their construction to the other Painlevé equations and why conformal blocks appear here.

SUMMARY: We discuss recent developments of representation theory of affine Yangians. The main focus is on the Fock space representation which originates from the study of the spin Calogero–Sutherland model. We also show a relation between Yangians and quantized Coulomb branches associated with quiver representations.

SUMMARY: The rear mutations are transformations in cluster algebra theory. They are the transformations between rational expressions of cluster variables in terms of the initial cluster under the change of the initial cluster. In this talk, we discuss the duality between the (usual) mutations and the rear mutations through the \(C\)-matrices, the \(G\)-matrices, and the \(F\)-polynomials. In particular, we introduce the \(F\)-matrices, which is the maximal degree matrices of the \(F\)-polynomials, and show that they have the self-duality which is analogous to the duality between the \(C\)- and \(G\)-matrices by Nakanishi and Zelevinsky. This is a joint work with Shogo Fujiwara.

SUMMARY: Let \(X_r\) be a finite type Dynkin diagram, and \(\ell \) be a positive integer greater than or equal to two. The Y-system of type \(X_r\) with level \(\ell \) is a system of difference equations whose solutions have been proved to have periodicity. For a pair \((X_r, \ell )\), we define integer sequence called exponents using formulation of the Y-system by cluster algebras. We give a conjecture that express these numbers by the root system of type \(X_r\). We prove the conjecture for \((A_1 ,\ell )\) and \((A_r, 2)\) cases.

SUMMARY: The higher order \(q\)-Painlevé system \(q\)-\(P_{(n+1,n+1)}\) was proposed as a similarity reduction of the \(q\)-Drinfeld–Sokolov hierarchy of type \(A_{2n+1}^{(1)}\). It is a generalization of Jimbo–Sakai’s \(q\)-\(P_{\rm VI}\) for the basic hypergeometric function, is derived from the compatibility condition of a Lax pair and admits an affine Weyl group symmetry. In this talk, we show that Sakai’s \(q\)-Garnier system is obtained as a Schlesinger transformation for \(q\)-\(P_{(n+1,n+1)}\).

SUMMARY: In this talk we formulate a group of birational transformations which is isomorphic to an extended affine Weyl group of type \((A_{2n+1}+A_1+A_1)^{(1)}\) with the aid of mutations and permutations to a mutation-periodic quiver on a torus. This group provides four types of generalizations of Jimbo–Sakai’s \(q\)-Painlevé VI equation as translations of the affine Weyl group. Then the known three systems are obtained again; the \(q\)-Garnier system, a similarity reduction of the lattice \(q\)-UC hierarchy and a similarity reduction of the \(q\)-Drinfeld–Sokolov hierarchy.

SUMMARY: We propose a systematic way to get birational realization of Weyl groups in terms of cluster algebras.

SUMMARY: In recent years, research on 4D Painlevé systems have progressed mainly from the viewpoint of isomonodromy deformation of linear equations.In this talk we study the geometric aspects of 4D Painlevé systems by investigating the space of initial conditions in Okamoto–Sakai’s sense, which was a powerful tool in the original 2D case. Specifically, starting from known discrete symmetries, we construct the space of initial conditions for some 4D Painlevé systems, and using the Néron–Severi bilattice, clarify the whole group of their discrete symmetries.

SUMMARY: New analytic functions describing the stable and unstable manifolds at saddle fixed points of Hénon maps are discussed. These functions are obtained by using Borel–Laplace transform, and represented by asymptotic expansions that are convergent in common domains of some half plane and some neighborhood of infinity.

SUMMARY: We found a new analytic function to describe the stable and unstable manifolds of nonlinear 2D dynamical systems. An algorithm to construct this function and its concrete form is given by using the Borel–Laplace transform. To obtain the function, the bifurcation structure of a certain Riemann surface is investigated in detail, and it is shown that it is a global solution to the dynamical systems. We prove that the obtained function differs from any existing special functions.

SUMMARY: Buchstaber and Mikhailov introduced the polynomial Hamiltonian systems in \(\mathbb {C}^4\) with two polynomial integrals on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. In this talk, for the case of genus 3, we derive a relationship between these systems and KdV-hierarchy. More precisely, we construct two parametric deformation of the KdV-hierarchy by using the systems. This new system is integrated in the hyperelliptic sigma functions of genus 3.

SUMMARY: In this talk, I introduce a new method to prove fundamental theorems about combinatorics of Young tableaux. The main technique is a realization of jeu de taquin by means of the tropical KP equation.

SUMMARY: The shifted (interpolation) Jack polynomials are a multivariate analogue of the falling factorials. We obtain Pieri type formulas for the shifted Jack polynomials.

SUMMARY: We give explicit formulas for the Kostka polynomials with one column diagrams of type \(B_n\), \(C_n\) and \(D_n\).