2019年度秋季総合分科会(於:金沢大学)

SUMMARY: The classical Schur–Weyl duality produces a strong representation-theoretic connection between the complex simple Lie algebra \(\mathfrak {sl}_{n}\) and the symmetric group \(\mathfrak {S}_{d}\). Its natural quantum affine analogue, called the quantum affine Schur–Weyl duality, is played by their quantum affinizations: the quantum affine algebra \(U_{q}(\widehat {\mathfrak {sl}}_{n})\) and the affine Hecke algebra of \(GL_d\). It induces a functor with several good properties between the categories of finite-dimensional modules. Moreover it has a beautiful geometric interpretation via the equivariant \(K\)-theory of flag varieties. The main topic of this talk is a further variant of the quantum affine Schur–Weyl duality associated with a Dynkin quiver, which was originally introduced by Kang–Kashiwara–Kim as a special case of their general construction. Here the players are replaced by the quantum affine algebra and the quiver Hecke algebra (also known as Khovanov–Lauda–Rouquier algebra) of the corresponding \(ADE\) type. We see that the induced functor enjoys good properties just like the usual case. Also we present its geometric interpretation via the equivariant \(K\)-theory of Nakajima’s graded quiver varieties.

SUMMARY: Cluster algebra is an algebraic structure generated by operations of a quiver called the mutations and their associated simple birational mappings, and it was introduced by Fomin and Zelevinsky in 2000. We present a systematic derivation of tropical (i.e., subtraction-free birational) realization of Weyl groups for various Dynkin diagrams. Our result is related with a class of tropical Weyl groups actions defined on certain rational varieties and also (higher-order) \(q\)-Painlevé equations. Key ingredients of the argument are the combinatorial aspects of reflections associated with \(n\)-cycles in the quiver. This talk is based on a joint work with Tetsu Masuda and Naoto Okubo.

SUMMARY: We present a conjecture for the asymptotically free eigenfunctions for the \(B_n\) \(q\)-Toda operator, which can be regarded as a brunching formula from the \(B_n\) \(q\)-Toda eigenfunction restricted to the \(A_{n-1}\) \(q\)-Toda eigenfunctions.

SUMMARY: In this talk, I will explain a duality formula for the matrix elements of 2N-valent intertwining operators of the Ding–Iohara–Miki algebra. This formula gives an algebraic description of 5D (K-theoretic) AGT correspondence and shows a spectral duality with respect to the Ding–Iohara–Miki algebra arising from string theory.

SUMMARY: The Koornwinder operator is a multi-variable generalization of the Askey–Wilson operator. In this talk, we will talk about the realization of Koornwinder operator on the Fock space. We also briefly discuss the relations between the currents to define the Koornwinder operator and the Drinfeld currents of the Ding–Iohara–Miki algebra.

SUMMARY: Dimofte, Gukov, Soibelman discovered remarkable identities for quantum dilogarithm functions on a non-commutative algebra as wall-crossing formulas, which are of the form “infinite product = finite product”. We derived one of the identities algebraically by using explicit product presentations of the universal R-matrix of the affine quantum group \(U_q(\widehat {\mathfrak {sl}_3})\). The presentations were constructed by K. Ito, which correspond to convex orders of positive roots. We calculated explicitly all the root vectors determined by certain convex orders, and obtained two different presentations of the universal R-matrix. Equating them and projecting both sides by a certain good representation yields an “infinite product = finite product” type identity. Specializing it gives the wall-crossing formula proposed by Dimofte et al.

SUMMARY: We study \(q\)-Stokes problems on basic hypergeometric equations with one regular singular points. We solve the \(q\)-Stokes problems of basic hypergeometric equations whose the Newton diagram has three segments at an irregular singular point.

SUMMARY: We introduce two variants of \(q\)-hypergeometric equation. We obtain several explicit solutions of variants of \(q\)-hypergeometric equation. We show that a variant of \(q\)-hypergeometric equation can be obtained by a restriction of \(q\)-Appell equation of two variables.

SUMMARY: We consider the minimal resolution of \(A_{1}\) singularity and the quotient stack of the plane by \(\lbrace \pm 1 \rbrace \), and moduli spaces of framed sheaves on them. We want to propose (-2) blow-up formula relating integrals over these moduli spaces for some cases.

SUMMARY: For a given marked surface \((S,M)\) and a fixed tagged triangulation \(T\) of \((S,M)\), we show that each tagged triangulation \(T'\) of \((S,M)\) is uniquely determined by the intersection numbers of tagged arcs of \(T\) and tagged arcs of \(T'\). As an application, each cluster in the cluster algebra \(\mathcal {A}(T)\) is uniquely determined by its \(F\)-matrix which is a new numerical invariant of the cluster introduced in Fujiwara and Gyoda.

SUMMARY: Birational rowmotion is a discrete dynamical system associated with a finite poset \(P\), which provides a birational lift of combinatorial rowmotion acting on order ideals of \(P\). If \(P\) is a product of two chains, then birational rowmotion has nice properties such as periodicity and homomesy. In this talk we extend these properties to minuscule posets. One of our results asserts that, if \(P\) is a minuscule poset arising from a simple Lie algebra \(\mathfrak {g}\), then the birational rowmotion map on \(P\) has order equal to the Coxeter number of \(\mathfrak {g}\). Also, as a generalization of promotion, we introduce birational Coxeter-motion on minuscule posets, and prove similar properties.

SUMMARY: We prove that dual factorial Schur \(P\)-functions provide solutions of BKP hierarchy. For the proof we used the criteria of Shigyo on the recursive relations of the coefficients of expansion in terms of Schur Q-functions.

SUMMARY: We introduce a multivariate analogue of Bernoulli polynomials and give their fundamental properties: difference and differential relations, symmetry, explicit formula, inversion formula, multiplication theorem, and binomial type formula.