2017年度秋季総合分科会(於:山形大学)

SUMMARY: Reverse plane partitions are combinatorial objects of tableau-type. One of the most important topics in reverse (and ordinary) plane partitions is the exploration of generating functions and partition functions which can be nicely factored. In this talk we show a close connection between reverse plane partitions and an integrable dynamical system called the discrete two-dimensional (2D) Toda molecule. We consider reverse plane partitions of arbitrary shape and of bounded size of parts and find that each non-vanishing solution to the discrete 2D Toda molecule gives a partition function which can be nicely factored. For the result a combinatorial interpretation of the discrete 2D Toda molecule in terms of lattice paths is crucial. As an instance from a specific solution we obtain a new partition function which generalizes both MacMahon’s generating function for boxed plane partitions and the trace generating function for reverse plane partitions of unbounded size of parts.

SUMMARY: We will discuss on the recent development of the representation theory of the quantum toroidal algebra of type \(\mathfrak {gl}(1)\) i.e. the Ding–Iohara–Miki algebra.

SUMMARY: We propose several solvable chaotic systems that have pseudo-invariants. The solution of the systems can be represented by elliptic functions associated to elliptic curves with complex multiplication. We also discuss invariant measures of the systems.

SUMMARY: The speaker will present a result of linear ordinary differential equations which are parametrized on Hermitian symmetric space and invariant under the action of symmetric groups. They are generalization s of the classical Lame equation. We will see a relation between such differential equations and automorphic forms for symplectic groups.

SUMMARY: If the tau function does not vanish at the origin, it is known that the coefficients are given by Giambelli formula and that it characterizes solutions of the KP hierarchy. In this talk, we deal with a generalization of Giambelli formula to the case when the tau function vanishes at the origin.

SUMMARY: In the talk, we give criteria for algebraic independence of solutions to a certain system of first order algebraic difference equations. First, as a background of our results, Ostrowski gave a criterion for algebraic independence of integrals of given functions. Inspired by Ostrowski’s result, Kolchin gave a criterion for algebraic independence of exponentials of integrals of given functions. Hardouin gave difference analogues of Ostrowski’s result and Kolchin’s result, that is, criteria for algebraic independence of functions satisfying first order linear difference equations. We generalize the analogues to more general systems of algebraic difference equations. As an application of our results, we show algebraic independence of multiple gamma functions and derivatives of the gamma function. As another application, we show algebraic independence of the logarithmic function, \(q\)-polylogarithm functions and \(q\)-exponential functions.

SUMMARY: Aomoto introduces the global \(q\)-difference de Rham complex and the Čech complex. These two complex should be isomorphic to each other in general case. To prove this, we introduce a sheaf-theoretic version of these complexes. We introduce the sheaf \(\mathscr {D} _{{}_q \! \Bbb {P} ^1 _\bullet } ^\bullet \) of \(q\)-difference operators, which is a \(q\)-analogue of the sheaf of differential operators on \(\Bbb {P} ^1\). We define the sheaf \(\mathscr {D} _{{}_q \! \Bbb {P} ^1 _\bullet } ^\bullet \) on some simplicial space \({}_q \! \Bbb {P} ^1 _\bullet \) (which we also define). For some \(\mathscr {D} _{{}_q \! \Bbb {P} ^1 _\bullet }\)-module \(\mathscr {M} ^\bullet \) we consider \(\Bbb {R} \mathscr {H}om _{\mathscr {D} _{{}_q \! \Bbb {P} ^1 _\bullet } ^\bullet } \left ( \mathscr {O} ^\bullet , \mathscr {M} ^\bullet \right )\), where \(\mathscr {O} ^\bullet \) is the structure sheaf on \({}_q \! \Bbb {P} ^1 _\bullet \). Both the (global) \(q\)-difference de Rham complex and the Čech complex are quasi-isomrphic to \(\Bbb {R} \Gamma \Bbb {R} \mathscr {H}om _{\mathscr {D} _{{}_q \! \Bbb {P} ^1 _\bullet } ^\bullet } \left ( \mathscr {O} ^\bullet , \mathscr {M} ^\bullet \right )\).

SUMMARY: We give a proof of functional equations of Nekrasov partition functions for \(A_{1}\)-singularity, suggested by Ito–Maruyoshi–Okuda. We follow method by Nakajima–Yoshioka based on the theory of wall-crossing formula developed by Mochizuki.

SUMMARY: We extend Dubrovin’s almost duality of Frobenius structures to Saito structures without metric. Then we formulate and study the existence and uniqueness problem of the natural Saito structure on the orbit spaces of finite complex reflection groups from the viewpoint of the almost duality. We give a complete answer to the problem for the irreducible groups.

SUMMARY: Schur \(Q\)-functions are a family of symmetric functions introduced by Schur in his study of projective representations of symmetric groups. They are obtained by putting \(t=-1\) in the Hall–Littlewood functions associated to the root system of type \(A\). (Schur functions are the \(t=0\) specialization.) This talk concerns symplectic \(Q\)-functions, which are obtained by putting \(t=-1\) in the Hall–Littlewood functions associated to the root system of type \(C\). We discuss several Pfaffian identities as well as a combinatorial description for them. Also we present some positivity conjectures.

SUMMARY: We will present a determinant formula for the \(BC_n\) elliptic hypergeometric integrals of Selberg type. The formula means that the determinant whose entries are the \(BC_n\) type elliptic Selberg integrals is expressed explicitly as a product of elliptic gamma functions. For this purpose we make use of the elliptic Lagrange interpolation functions, which play important roles in both definition of the determinant and proof of the product formula.

SUMMARY: It is known that the Painlevé VI is obtained by connection preserving deformation of some linear differential equations, and the Heun equation is obtained by a specialization of the linear differential equations. We inverstigate degenerations of the Ruijsenaars-van Diejen difference opearators and show difference analogues of the Painlevé–Heun correspondence.

SUMMARY: The Sasano system is a higher order generalization of the sixth Painlevé equations. The Sasano system of the \(2N\)-th order admits the affine Weyl group symmetry of type \(D_{2N+2}^{(1)}\) as the group of Bäcklund transformations. We propose, in this talk, some \(q\)-analogues of the Sasano system of the fourth order. Each of our systems also admits the affine Weyl group symmetry.

SUMMARY: The \(q\)-Painlevé VI equation (\(q\)-\(P_{\rm {VI}}\)) was introduced by Jimbo and Sakai. Recently, from a viewpoint of the Heine’s basic hypergeometric function \({}_{n+1}\phi _n\), we proposed a higher order generalization of \(q\)-\(P_{\rm {VI}}\), which admits an extended affine Weyl group symmetry of type \(A_{2n+1}^{(1)}\). In this talk, we formulate \(\tau \)-functions for the generalized \(q\)-\(P_{\rm {VI}}\) on the root lattice \(Q(A_{2n+1})\) and show that they satisfy Hirota–Miwa type bilinear relations.

SUMMARY: We construct cluster algebras the coefficients of which satisfy several \(q\)-discrete Painlevé equations. These cluster algebras are obtained from quivers with the mutation-period property. We will also show that degeneration of \(q\)-discrete Painlevé equations corresponds to confluence of quivers.

SUMMARY: When we geometrically construct the Painlevé systems by certain rational surfaces called the spaces of initial values, we can variously select deformation directions and singularity configurations in \({\mathbb P}^1\times {\mathbb P}^1\). However, in order to obtain a simple evolution equation, it is necessary to select the deformation direction and coordinates (singularity configuration) suitably. In this talk, we give some examples of simple evolution equations for the case of \(q\)-\(E_ 7^{(1)}\) type, and we discuss a relation among them and their special solutions.

SUMMARY: In this talk we establish a way of performing the deautonomization for a pair of an autonomous mapping and an anti-canonical divisor. Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painlevé equations using this deautonomization procedure. We also introduce a technique which allows us to obtain factorized expressions of discrete Painlevé equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai’s classification. (Based on joint work with A. S. Carstea and A. Dzhamay.)

SUMMARY: We analyze the wavefunctions of the six-vertex models by extending the celebrated Izergin–Korepin analysis which was originally invented by Korepin and Izergin to study the domain wall boundary partition functions. We show that the method can give a systematic way to study the relation between the wavefunctions and symmetric functions. We mainly illustrate the method by taking the basic wavefunctions of the XXZ-type six-vertex models as an example. Next, we show the result for the case of triangular boundary conditions, Felderhof models and their generalizations such as the elliptic analogue.

SUMMARY: We consider the two dimensional Ising model by using the CTM (corner transfer matirx) method. We derive the corner Hamiltonian for the Ising model in a magnetic field by purturbatively inserting the magnetic field. This is just a purturbative approximation because the existence of a real magnetic field violates some assumptions for the validity of the CTM method. Thus we find that the CTM method goes well at a particular pure imaginary magnetic filed, and obtain the explicit expression of the spontaneous magnetization.

SUMMARY: A bosonization of the quantum affine superalgebra \(U_q(\widehat {sl}(M|N))\) is presented for an arbitrary level \(k \in {\bf C}\). Screening operators that commute with \(U_q(\widehat {sl}(M|N))\) are presented for the level \(k \neq -M+N\). This talk is based on arXiv.1701.03645 to appear in Commun. Math. Phys.

SUMMARY: Nakatsu–Takasaki have studied partition functions of five-dimensional gauge theories and topological string theories by using an algebra of fermions. Recently, it has been realized that partition functions of six-dimensional gauge theories are represented by elliptic functions. As an attempt to apply the method due to Nakatsu–Takasaki to the six-dimensional cases, the author has introduced an elliptic analog of an algebra of fermions. An elliptic analog of the quantum torus algebra is also constructed.

SUMMARY: In this talk, I explain that singular vectors of a certain algebra arising from the level \(N\) representation of a Hopf algebra called Ding–Iohara–Miki algebra correspond to generalized Macdonald functions (also called AFLT basis or fixed point basis). This correspondence is a generalization of coincidence between singular vectors of the deformed W algebra and ordinary Macdonald functions. Moreover, we also obtain a formula for Kac determinant of the level \(N\) representation of DIM algebra.

SUMMARY: Recently a cluster realization of Drinfeld–Jimbo quantum groups \(\mathcal {U}_q(\mathfrak {g})\) has been found via the positive representations, where an embedding into a quantum torus algebra \(\mathcal {X}_\mathfrak {g}\) is described by certain quiver diagram. Using this new realization, we discuss its application towards the proof of the tensor product decomposition of the positive representations of split real quantum groups restricted to the Borel part, as well as the proof by Schrader–Shapiro on the full decomposition in type \(A_n\), thus solving part of the long-standing conjecture of the closure of positive representations under taking the tensor product.

SUMMARY: By using representation theory of the elliptic quantum group \(U_{q,p}(\widehat {\mathfrak {sl}}_N)\), we present a systematic method of deriving the weight functions. The resultant \(\mathfrak {sl}_N\) type elliptic weight functions are new, and give elliptic and dynamical analogue of those obtained by Mimachi and Tarasov–Varchenko in the trigonometric case. We then discuss some basic properties of the elliptic weight functions. We also present an explicit formula for formal elliptic hypergeometric integral solution to the elliptic \(q\)-KZ equation.

SUMMARY: We construct a finite-dimensional representation of the elliptic quantum group \(U_{q,p}(\widehat {\mathfrak {sl}}_N)\) on the Gelfand–Tsetlin basis. The resultant representation is described in terms of the partitions of \([1,n]\) in a combinatorial way, and gives an elliptic and dynamical analogue of the geometric representation of \(U_q(\widehat {\mathfrak {sl}}_N)\) on the equivariant \(K\)-theory constructed by Vasserot and Nakajima. By comparing the elliptic weight functions presented in the previous talk and the elliptic stable envelopes proposed by Aganagic–Okounkov, we conjecture that our finite-dimensional representation provides a geometric representation of \(U_{q,p}(\widehat {\mathfrak {sl}}_N)\) on the equivariant elliptic cohomology.