## Infinite Analysis Seminar Tokyo

Seminar information archive ～08/08｜Next seminar｜Future seminars 08/09～

Date, time & place | Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
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**Seminar information archive**

### 2015/04/23

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Parabolic analogue of periodic Kazhdan-Lusztig polynomials (JAPANESE)

**Hideya Watanabe**(Department of Mathematics, Tokyo Institute of Technology, Graduate school of science and Engineering)Parabolic analogue of periodic Kazhdan-Lusztig polynomials (JAPANESE)

[ Abstract ]

We construct a parabolic analogue of so-called periodic modules, which are modules over the Hecke algebra

associated with an affine Weyl group.

These modules have a basis similar to Kazhdan-Lusztig basis.

Our construction enables us to see the relation between (ordinary)periodic KL-polynomials and parabolic ones.

We construct a parabolic analogue of so-called periodic modules, which are modules over the Hecke algebra

associated with an affine Weyl group.

These modules have a basis similar to Kazhdan-Lusztig basis.

Our construction enables us to see the relation between (ordinary)periodic KL-polynomials and parabolic ones.

### 2015/02/19

13:30-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Skein algebra and mapping class group (JAPANESE)

An extension of the LMO functor (JAPANESE)

**Shunsuke Tsuji**(Graduate School of Mathematical Sciences, the University of Tokyo) 13:30-15:00Skein algebra and mapping class group (JAPANESE)

[ Abstract ]

We define some filtrations of skein modules and the skein algebra on an oriented surface, and define the completed skein modules and

the completed skein algebra of the surface with respect to these filtration. We give an explicit formula for the action of the Dehn

twists on the completed skein modules in terms of the action of the completed skein algebra of the surface. As an application, we describe the action of the Johnson kernel on the completed skein modules.

We define some filtrations of skein modules and the skein algebra on an oriented surface, and define the completed skein modules and

the completed skein algebra of the surface with respect to these filtration. We give an explicit formula for the action of the Dehn

twists on the completed skein modules in terms of the action of the completed skein algebra of the surface. As an application, we describe the action of the Johnson kernel on the completed skein modules.

**Yuta Nozaki**(Graduate School of Mathematical Sciences, the University of Tokyo) 15:30-17:00An extension of the LMO functor (JAPANESE)

[ Abstract ]

Cheptea, Habiro and Massuyeau introduced the LMO functor as an extension of the LMO invariant of closed 3-manifolds. The LMO functor is “the monoidal category of Lagrangian cobordisms between surfaces with at most one boundary component” to “the monoidal category of top-substantial Jacobi diagrams”. In this talk, we extend the LMO functor to the case of any number of boundary components. Moreover, we explain that the internal degree d part of the extension is a finite-type invariant of degree d.

Cheptea, Habiro and Massuyeau introduced the LMO functor as an extension of the LMO invariant of closed 3-manifolds. The LMO functor is “the monoidal category of Lagrangian cobordisms between surfaces with at most one boundary component” to “the monoidal category of top-substantial Jacobi diagrams”. In this talk, we extend the LMO functor to the case of any number of boundary components. Moreover, we explain that the internal degree d part of the extension is a finite-type invariant of degree d.

### 2015/01/22

13:00-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

A construction of dynamical Yang-Baxter map with dynamical brace (JAPANESE)

Construction of Hopf algebroids by means of dynamical Yang-Baxter maps (JAPANESE)

**Diogo Kendy Matsumoto**(Faculty of Science and Engineerring, Waseda University) 13:00-14:30A construction of dynamical Yang-Baxter map with dynamical brace (JAPANESE)

[ Abstract ]

Brace is an algebraic system defined as a generalization of the radical ring. The radical ring means a ring $(R,+,¥cdot)$, which has a group structure with respect to $a*b:=ab+a+b$. By using brace, Rump constructs the non-degenerate Yang-Baxter map with unitary condition.

In this talk I will propose the dynamical brace, which is a generalization of the brace, and give a way to construct the dynamical Yang-Baxter map by using the dynamical brace. A dynamical Yang-Baxter map is a set-theoretical solution of the dynamical Yang-Baxter equation. Moreover, I will discuss algebraic and combinatorial properties of the dynamical brace.

Brace is an algebraic system defined as a generalization of the radical ring. The radical ring means a ring $(R,+,¥cdot)$, which has a group structure with respect to $a*b:=ab+a+b$. By using brace, Rump constructs the non-degenerate Yang-Baxter map with unitary condition.

In this talk I will propose the dynamical brace, which is a generalization of the brace, and give a way to construct the dynamical Yang-Baxter map by using the dynamical brace. A dynamical Yang-Baxter map is a set-theoretical solution of the dynamical Yang-Baxter equation. Moreover, I will discuss algebraic and combinatorial properties of the dynamical brace.

**Youichi Shibukawa**(Department of Mathematics, Hokkaido University) 15:00-16:30Construction of Hopf algebroids by means of dynamical Yang-Baxter maps (JAPANESE)

[ Abstract ]

A generalization of the Hopf algebra is a Hopf algebroid. Felder and Etingof-Varchenko constructed Hopf algebroids from the dynamical R-matrices, solutions to the quantum dynamical Yang-Baxter equation (QDYBE for short). This QDYBE was generalized, and several solutions called dynamical Yang-Baxter maps to this generalized equation were constructed. The purpose of this talk is to introduce construction of Hopf algebroids by means of dynamical Yang-Baxter maps. If time permits, I will explain that the tensor category of finite-dimensional L-operators associated with the suitable dynamical Yang-Baxter map is rigid. This tensor category is isomorphic to that consisting of finite-dimensional (dynamical) representations of the corresponding Hopf algebroid.

A generalization of the Hopf algebra is a Hopf algebroid. Felder and Etingof-Varchenko constructed Hopf algebroids from the dynamical R-matrices, solutions to the quantum dynamical Yang-Baxter equation (QDYBE for short). This QDYBE was generalized, and several solutions called dynamical Yang-Baxter maps to this generalized equation were constructed. The purpose of this talk is to introduce construction of Hopf algebroids by means of dynamical Yang-Baxter maps. If time permits, I will explain that the tensor category of finite-dimensional L-operators associated with the suitable dynamical Yang-Baxter map is rigid. This tensor category is isomorphic to that consisting of finite-dimensional (dynamical) representations of the corresponding Hopf algebroid.

### 2015/01/15

15:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

On Gram matrices of the Shapovalov form of a basic representation of a

quantum affine group (ENGLISH)

Continuous and Infinitesimal Hecke algebras (ENGLISH)

**Shunsuke Tsuchioka**(Graduate School of Mathematical Sciences, the University of Tokyo) 15:00-16:30On Gram matrices of the Shapovalov form of a basic representation of a

quantum affine group (ENGLISH)

[ Abstract ]

We consider Gram matrices of the Shapovalov form of a basic

representation

of a quantum affine group. We present a conjecture predicting the

invariant

factors of these matrices and proving that it gives the correct

invariants

when one specializes or localizes the ring $\mathbb{Z}[v,v^{-1}]$ in

certain ways.

This generalizes Evseev's theorem which settled affirmatively

the K\"{u}lshammer-Olsson-Robinson conjecture that predicts

the generalized Cartan invariants of the symmetric groups.

This is a joint work with Anton Evseev.

We consider Gram matrices of the Shapovalov form of a basic

representation

of a quantum affine group. We present a conjecture predicting the

invariant

factors of these matrices and proving that it gives the correct

invariants

when one specializes or localizes the ring $\mathbb{Z}[v,v^{-1}]$ in

certain ways.

This generalizes Evseev's theorem which settled affirmatively

the K\"{u}lshammer-Olsson-Robinson conjecture that predicts

the generalized Cartan invariants of the symmetric groups.

This is a joint work with Anton Evseev.

**Alexander Tsymbaliuk**(SCGP (Simons Center for Geometry and Physics)) 17:00-18:30Continuous and Infinitesimal Hecke algebras (ENGLISH)

[ Abstract ]

In the late 80's V. Drinfeld introduced the notion of the

degenerate affine Hecke algebras. The particular class of those, called

symplectic reflection algebras, has been rediscovered 15 years later by

[Etingof and Ginzburg]. The theory of those algebras (which include also

the rational Cherednik algebras) has attracted a lot of attention in the

last 15 years.

In this talk we will discuss their continuous and infinitesimal versions,

introduced by [Etingof, Gan, and Ginzburg]. Our key result relates those

classical algebras to the simplest 1-block finite W-algebras.

In the late 80's V. Drinfeld introduced the notion of the

degenerate affine Hecke algebras. The particular class of those, called

symplectic reflection algebras, has been rediscovered 15 years later by

[Etingof and Ginzburg]. The theory of those algebras (which include also

the rational Cherednik algebras) has attracted a lot of attention in the

last 15 years.

In this talk we will discuss their continuous and infinitesimal versions,

introduced by [Etingof, Gan, and Ginzburg]. Our key result relates those

classical algebras to the simplest 1-block finite W-algebras.

### 2014/12/11

15:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Renormalization group method for many-electron systems (JAPANESE)

Unitary transformations and multivariate special

orthogonal polynomials (JAPANESE)

**Yohei Kashima**(Graduate School of Mathematical Sciences, the University of Tokyo) 15:00-16:30Renormalization group method for many-electron systems (JAPANESE)

[ Abstract ]

We consider quantum many-body systems of electrons

hopping and interacting on a lattice at positive temperature.

As it is possible to write down each order term rigorously in

principle, the perturbation series expansion with the coupling

constant between electrons is thought as a valid method to

compute physical quantities. By directly estimating each term,

one can prove that the perturbation series is convergent if the

coupling constant is less than some power of temperature. This is,

however, a serious constraint for models of interacting electrons

in low temperature. In order to ensure the analyticity of physical

quantities of many-electron systems with the coupling constant in

low temperature, renormalization group methods have been developed

in recent years. As one progress in this direction, we construct a

renormalization group method for the half-filled Hubbard model on a

square lattice, which is a typical model of many-electron, and prove

the following. If the system contains the magnetic flux pi (mod 2 pi)

per plaquette, the free energy density of the system is analytic with

the coupling constant in a neighborhood of the origin and it uniformly

converges to the infinite-volume, zero-temperature limit. It is known

that the flux pi condition is sufficient for the free energy density

to be minimum. Thus, it follows that the same analyticity and the

convergent property hold for the minimum free energy density of the

system.

We consider quantum many-body systems of electrons

hopping and interacting on a lattice at positive temperature.

As it is possible to write down each order term rigorously in

principle, the perturbation series expansion with the coupling

constant between electrons is thought as a valid method to

compute physical quantities. By directly estimating each term,

one can prove that the perturbation series is convergent if the

coupling constant is less than some power of temperature. This is,

however, a serious constraint for models of interacting electrons

in low temperature. In order to ensure the analyticity of physical

quantities of many-electron systems with the coupling constant in

low temperature, renormalization group methods have been developed

in recent years. As one progress in this direction, we construct a

renormalization group method for the half-filled Hubbard model on a

square lattice, which is a typical model of many-electron, and prove

the following. If the system contains the magnetic flux pi (mod 2 pi)

per plaquette, the free energy density of the system is analytic with

the coupling constant in a neighborhood of the origin and it uniformly

converges to the infinite-volume, zero-temperature limit. It is known

that the flux pi condition is sufficient for the free energy density

to be minimum. Thus, it follows that the same analyticity and the

convergent property hold for the minimum free energy density of the

system.

**Genki Shibukawa**(Institute of Mathematics for Industory, Kyushu University) 17:00-18:30Unitary transformations and multivariate special

orthogonal polynomials (JAPANESE)

[ Abstract ]

Investigations into special orthogonal function systems by

using unitary transformations have a long history.

This is, by calculating an image of some unitary transform (e.g. Fourier

trans.) of a known orthogonal system, we derive a new orthogonal system

and obtain its fundamental properties.

This basic concept and technique have been known since ancient times for

a single variable case, and recently these multivariate analogue has

been studied by Davidson, Olafsson, Zhang, Faraut, Wakayama et.al..

In our talk, we introduce the unitary picture for the circular Jacobi

polynomials obtained by Shen, further give a multivariate analogue of

the results of Shen.

These polynomials, which we call multivariate circular Jacobi (MCJ)

polynomials, are generalizations (2-parameter deformation) of the

spherical (zonal) polynomials that are different from the Jack or

Macdonald polynomials, which are well known as an extension of spherical

polynomials.

We also remark that the weight function of their orthogonality relation

coincides with the circular Jacobi ensemble defined by Bourgade,et,al.,

and the modified Cayley transform of the MCJ polynomials satisfy with

some quasi differential equation.

In addition, we can give a generalization of MCJ polynomials as

including the Jack polynomials.

For this generalized MCJ polynomials, we would like to present some

conjectures and problems.

If we have time, we also describe a unitary picture of Meixner,

Charlier and Krawtchouk polynomials which are typical examples of

discrete orthogonal systems, and mention their multivariate analogue.

Investigations into special orthogonal function systems by

using unitary transformations have a long history.

This is, by calculating an image of some unitary transform (e.g. Fourier

trans.) of a known orthogonal system, we derive a new orthogonal system

and obtain its fundamental properties.

This basic concept and technique have been known since ancient times for

a single variable case, and recently these multivariate analogue has

been studied by Davidson, Olafsson, Zhang, Faraut, Wakayama et.al..

In our talk, we introduce the unitary picture for the circular Jacobi

polynomials obtained by Shen, further give a multivariate analogue of

the results of Shen.

These polynomials, which we call multivariate circular Jacobi (MCJ)

polynomials, are generalizations (2-parameter deformation) of the

spherical (zonal) polynomials that are different from the Jack or

Macdonald polynomials, which are well known as an extension of spherical

polynomials.

We also remark that the weight function of their orthogonality relation

coincides with the circular Jacobi ensemble defined by Bourgade,et,al.,

and the modified Cayley transform of the MCJ polynomials satisfy with

some quasi differential equation.

In addition, we can give a generalization of MCJ polynomials as

including the Jack polynomials.

For this generalized MCJ polynomials, we would like to present some

conjectures and problems.

If we have time, we also describe a unitary picture of Meixner,

Charlier and Krawtchouk polynomials which are typical examples of

discrete orthogonal systems, and mention their multivariate analogue.

### 2014/11/20

15:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

An explicit formula for the specialization of nonsymmetric

Macdonald polynomials at $t = \infty$ (JAPANESE)

divisor function and strict partition (JAPANESE)

**Fumihiko Nomoto**(Department of Mathematics, Tokyo Institute of Technology, Graduate school of science and Engineering) 15:00-16:30An explicit formula for the specialization of nonsymmetric

Macdonald polynomials at $t = \infty$ (JAPANESE)

[ Abstract ]

Orr-Shimozono obtained an explicit formula for nonsymmetric Macdonald polynomials with Hecke parameter $t$ set to $\infty$, which is described in terms of an affine root system

and an affine Weyl group. On the basis of this work, we give another explicit formula for the specialization above, which is described in terms of the quantum Bruhat graph associated with a finite root system and a finite Weyl group.

More precisely, we interpret the specialization above as the graded character of an explicitly specified set of quantum Lakshmibai-Seshadri (LS) paths. Here we note that the set of quantum LS paths (of a given shape) provides an explicit realization of the crystal basis of a quantum Weyl module over the quantum affine algebra.

In this talk, I will explain our explicit formula

by exhibiting a few examples.

Also, I will give an outline of the proof.

Orr-Shimozono obtained an explicit formula for nonsymmetric Macdonald polynomials with Hecke parameter $t$ set to $\infty$, which is described in terms of an affine root system

and an affine Weyl group. On the basis of this work, we give another explicit formula for the specialization above, which is described in terms of the quantum Bruhat graph associated with a finite root system and a finite Weyl group.

More precisely, we interpret the specialization above as the graded character of an explicitly specified set of quantum Lakshmibai-Seshadri (LS) paths. Here we note that the set of quantum LS paths (of a given shape) provides an explicit realization of the crystal basis of a quantum Weyl module over the quantum affine algebra.

In this talk, I will explain our explicit formula

by exhibiting a few examples.

Also, I will give an outline of the proof.

**Masanori Ando**(Wakkanai Hokusei Gakuen University) 17:00-18:30divisor function and strict partition (JAPANESE)

[ Abstract ]

We know that the q-series identity of Uchimura-type is related with the divisor function.

It is obtained also as a specialization of basic hypergeometric series.

In this seminar, we interprete this identity from the point of view of combinatorics of partitions of integers.

We give its proof by using the mock involution map.

We know that the q-series identity of Uchimura-type is related with the divisor function.

It is obtained also as a specialization of basic hypergeometric series.

In this seminar, we interprete this identity from the point of view of combinatorics of partitions of integers.

We give its proof by using the mock involution map.

### 2014/10/03

13:30-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

KPZ equation and Macdonald process (JAPANESE)

Entanglement spectra in topological phases and coupled Tomonaga-Luttinger liquids (JAPANESE)

**Tomohiro SASAMOTO**(Department of Physics, Tokyo Institute of Technology) 13:30-15:00KPZ equation and Macdonald process (JAPANESE)

**Shunsuke FURUKAWA**(Department of Physics, the Tokyo University) 15:30-17:00Entanglement spectra in topological phases and coupled Tomonaga-Luttinger liquids (JAPANESE)

[ Abstract ]

The entanglement spectrum (ES) has been found to provide useful probes of topological phases of matter and other exotic strongly correlated states. For the system's ground state, the ES is defined as the full eigenvalue spectrum of the reduced density matrix obtained by tracing out the degrees of freedom in part of the system. A key result observed in various topological phases and other gapped systems has been the remarkable correspondence between the ES and the edge-state spectrum. While this correspondence has been analytically proven for some topological phases, it is interesting to ask what systems show this correspondence more generally and how the ES changes when the bulk energy gap closes.

We here study the ES in two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. In addition to having direct applications to ladder systems, this problem is closely related to the entanglement properties of two-dimensional topological phases. Based on the calculation for coupled chiral TLLs, we provide a simple physical proof for the correspondence between edge states and the ES in quantum Hall systems consistent with previous numerical and analytical studies. We also discuss violations of this correspondence in gapped and gapless phases of coupled non-chiral TLLs.

Reference: R. Lundgren, Y. Fuji, SF, and M. Oshikawa, Phys. Rev. B 88, 245137 (2013).

The entanglement spectrum (ES) has been found to provide useful probes of topological phases of matter and other exotic strongly correlated states. For the system's ground state, the ES is defined as the full eigenvalue spectrum of the reduced density matrix obtained by tracing out the degrees of freedom in part of the system. A key result observed in various topological phases and other gapped systems has been the remarkable correspondence between the ES and the edge-state spectrum. While this correspondence has been analytically proven for some topological phases, it is interesting to ask what systems show this correspondence more generally and how the ES changes when the bulk energy gap closes.

We here study the ES in two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. In addition to having direct applications to ladder systems, this problem is closely related to the entanglement properties of two-dimensional topological phases. Based on the calculation for coupled chiral TLLs, we provide a simple physical proof for the correspondence between edge states and the ES in quantum Hall systems consistent with previous numerical and analytical studies. We also discuss violations of this correspondence in gapped and gapless phases of coupled non-chiral TLLs.

Reference: R. Lundgren, Y. Fuji, SF, and M. Oshikawa, Phys. Rev. B 88, 245137 (2013).

### 2014/09/22

13:30-16:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Colored HOMFLY homology of knots and links (ENGLISH)

**Satoshi Nawata**(Theoretical Physics at NIKHEF)Colored HOMFLY homology of knots and links (ENGLISH)

[ Abstract ]

In this talk I will present structural properties of colored HOMFLY homology of knots and links. These rich properties of the categorification of the colored HOMFLY polynomial are obtained by using various methods: physics insights, representation theory of Lie super-algebras, double affine Hecke algebras, etc. This in turn enables computation of colored HOMFLY homology for various classes of knots and links and consequent computation of super-A-polynomial - the deformation of the classical A-polynomial. I will also explain recent results and special additional properties for colored Kauffman homology as well as the case of links. Although I will try to give a talk accessible to mathematicians, there is no proof and rigorousness in this talk.

In this talk I will present structural properties of colored HOMFLY homology of knots and links. These rich properties of the categorification of the colored HOMFLY polynomial are obtained by using various methods: physics insights, representation theory of Lie super-algebras, double affine Hecke algebras, etc. This in turn enables computation of colored HOMFLY homology for various classes of knots and links and consequent computation of super-A-polynomial - the deformation of the classical A-polynomial. I will also explain recent results and special additional properties for colored Kauffman homology as well as the case of links. Although I will try to give a talk accessible to mathematicians, there is no proof and rigorousness in this talk.

### 2014/07/16

10:30-12:00 Room #002 (Graduate School of Math. Sci. Bldg.)

From the Hilbert scheme to m/n Pieri rules (ENGLISH)

**Andrei Negut**(Columbia University, Department of Mathematics)From the Hilbert scheme to m/n Pieri rules (ENGLISH)

[ Abstract ]

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

### 2014/07/15

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

From the shuffle algebra to the Hilbert scheme (ENGLISH)

**Andrei Negut**(Columbia University, Department of Mathematics)From the shuffle algebra to the Hilbert scheme (ENGLISH)

[ Abstract ]

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

### 2014/07/13

14:00-15:00 Room #002 (Graduate School of Math. Sci. Bldg.)

From vertex operators to the shuffle algebra (ENGLISH)

**Andrei Negut**(Columbia University, Department of Mathematics)From vertex operators to the shuffle algebra (ENGLISH)

[ Abstract ]

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

In this series of talks, we will discuss several occurrences of shuffle

algebras: in representation theory, in geometry of moduli spaces, and in

the combinatorics of symmetric functions. All the connections will be

explained in detail.

### 2013/05/11

10:30-12:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Saga of Dunkl elements (ENGLISH)

**Anatol Kirillov**(RIMS Kyoto Univ.)Saga of Dunkl elements (ENGLISH)

[ Abstract ]

The Dunkl operators has been introduced by C. Dunkl in the middle of

80's of the last century as a powerful mean in the study of orthogonal

polynomials related with finite Coxeter groups. Later it was discovered

a deep connection of the the Dunkl operators with the theory of

Integrable systems and Invariant Theory.

In my talk I introduce and study a certain class of nonhomogeneous

quadratic algebras together with the distinguish set of mutually

commuting elements inside of each, the so-called universal Dunkl elements.

The main problem I would like to discuss is : What is the algebra

generated by universal Dunkl elements in a different representations of

the quadratic algebra introduced ?

I'm planning to present partial answers on that problem related with

classical and quantum Schubert and Grothendieck Calculi as well as the

theory of elliptic series.

Also some interesting algebraic properties of the quadratic algebra(s)

in question will be described.

The Dunkl operators has been introduced by C. Dunkl in the middle of

80's of the last century as a powerful mean in the study of orthogonal

polynomials related with finite Coxeter groups. Later it was discovered

a deep connection of the the Dunkl operators with the theory of

Integrable systems and Invariant Theory.

In my talk I introduce and study a certain class of nonhomogeneous

quadratic algebras together with the distinguish set of mutually

commuting elements inside of each, the so-called universal Dunkl elements.

The main problem I would like to discuss is : What is the algebra

generated by universal Dunkl elements in a different representations of

the quadratic algebra introduced ?

I'm planning to present partial answers on that problem related with

classical and quantum Schubert and Grothendieck Calculi as well as the

theory of elliptic series.

Also some interesting algebraic properties of the quadratic algebra(s)

in question will be described.

### 2013/03/30

13:30-15:30 Room #002 (Graduate School of Math. Sci. Bldg.)

On the extended algebra of type sl_2 at positive rational level (ENGLISH)

**Simon Wood**(Kavli IPMU)On the extended algebra of type sl_2 at positive rational level (ENGLISH)

[ Abstract ]

I will be presenting my recent work with Akihiro Tsuchiya

(arXiv:1302.6435).

I will explain how to construct a certain VOA called the "extended

algebra of type sl_2 at positive rational level"

as a subVOA of a lattice VOA, by means of screening operators. I will

then show that this VOA carries a kind of exterior sl_2 action and then

show how one can compute the structure Zhu's algebra and the Poisson

algebra as well as classify all simple modules by using the screening

operators and the sl_2 action. Important concepts such as screening

operators or Zhu's algebra and the Poisson algebra of a VOA will be

reviewed in the talk.

I will be presenting my recent work with Akihiro Tsuchiya

(arXiv:1302.6435).

I will explain how to construct a certain VOA called the "extended

algebra of type sl_2 at positive rational level"

as a subVOA of a lattice VOA, by means of screening operators. I will

then show that this VOA carries a kind of exterior sl_2 action and then

show how one can compute the structure Zhu's algebra and the Poisson

algebra as well as classify all simple modules by using the screening

operators and the sl_2 action. Important concepts such as screening

operators or Zhu's algebra and the Poisson algebra of a VOA will be

reviewed in the talk.

### 2013/02/16

13:30-15:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Generalized Calogero-Moser type systems and Cherednik Algebras (ENGLISH)

**Alexey Silantyev**(Tokyo. Univ.)Generalized Calogero-Moser type systems and Cherednik Algebras (ENGLISH)

[ Abstract ]

Calogero-Moser systems can be obtained using Dunkl operators, which

define the polynomial representation of the corresponding rational

Cherednik algebra. Parabolic ideals invariant under the action of the

Dunkl operators give submodules of Cherednik algebra. Considering the

corresponding quotient-modules one yields the generalized (or deformed)

Calogero-Moser systems. In the same way we construct the generalized

elliptic Calogero-Moser systems using the elliptic Dunkl operators

obtained by Buchstaber, Felder and Veselov. The Macdonald-Ruijsenaars

systems (difference (relativistic) Calogero-Moser type systems) can be

considered in terms of Double Affine Hecke Algebra (DAHA). We construct

appropriate submodules in the polynomial representation of DAHA, which

were obtained by Kasatani for some affine root systems. Considering the

corresponding quotient representation we derive the generalized

(deformed) Macdonald-Ruijsenaars systems for any affine root system,

which where obtained by Sergeev and Veselov for the A series. This is

joint work with Misha Feigin.

Calogero-Moser systems can be obtained using Dunkl operators, which

define the polynomial representation of the corresponding rational

Cherednik algebra. Parabolic ideals invariant under the action of the

Dunkl operators give submodules of Cherednik algebra. Considering the

corresponding quotient-modules one yields the generalized (or deformed)

Calogero-Moser systems. In the same way we construct the generalized

elliptic Calogero-Moser systems using the elliptic Dunkl operators

obtained by Buchstaber, Felder and Veselov. The Macdonald-Ruijsenaars

systems (difference (relativistic) Calogero-Moser type systems) can be

considered in terms of Double Affine Hecke Algebra (DAHA). We construct

appropriate submodules in the polynomial representation of DAHA, which

were obtained by Kasatani for some affine root systems. Considering the

corresponding quotient representation we derive the generalized

(deformed) Macdonald-Ruijsenaars systems for any affine root system,

which where obtained by Sergeev and Veselov for the A series. This is

joint work with Misha Feigin.

### 2012/12/15

13:30-15:00 Room #117 (Graduate School of Math. Sci. Bldg.)

current and integrability (ENGLISH)

**Vincent Pasquier**(CEA, Saclay, France)current and integrability (ENGLISH)

[ Abstract ]

I will describe some problems related to currents in XXZ chains:

Drude conductivity, Linbladt equation, tasep, matrix ansatz,

in particular the relation of permanent currents with integrability.

If time permits I will also discuss a nonrelated subject:

deformation of fusion rules in minimal models and Macdonald polynomials.

I will describe some problems related to currents in XXZ chains:

Drude conductivity, Linbladt equation, tasep, matrix ansatz,

in particular the relation of permanent currents with integrability.

If time permits I will also discuss a nonrelated subject:

deformation of fusion rules in minimal models and Macdonald polynomials.

### 2012/12/01

13:30-15:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Manin matrices and quantum integrable systems (ENGLISH)

**Alexey Silantyev**(Univ. Tokyo)Manin matrices and quantum integrable systems (ENGLISH)

[ Abstract ]

Manin matrices (known also as right quantum matrices) is a class of

matrices with non-commutative entries. The natural generalization of the

usual determinant for these matrices is so-called column determinant.

Manin matrices, their determinants and minors have the most part of the

properties possessed by the usual number matrices. Manin matrices arise

from the RLL-relations and help to find quantum analogues of Poisson

commuting traces of powers of Lax operators and to establish relations

between different types of quantum commuting families. The RLL-relations

also give us q-analogues of Manin matrices in the case of trigonometric

R-matrix (which define commutation relations for the quantum affine

algebra).

Manin matrices (known also as right quantum matrices) is a class of

matrices with non-commutative entries. The natural generalization of the

usual determinant for these matrices is so-called column determinant.

Manin matrices, their determinants and minors have the most part of the

properties possessed by the usual number matrices. Manin matrices arise

from the RLL-relations and help to find quantum analogues of Poisson

commuting traces of powers of Lax operators and to establish relations

between different types of quantum commuting families. The RLL-relations

also give us q-analogues of Manin matrices in the case of trigonometric

R-matrix (which define commutation relations for the quantum affine

algebra).

### 2012/03/09

13:30-14:30 Room #002 (Graduate School of Math. Sci. Bldg.)

On Hall algebra of complexes (JAPANESE)

**Shintarou Yanagida**(Kobe Univ.)On Hall algebra of complexes (JAPANESE)

[ Abstract ]

The topic of my talk is the Hall algebra of complexes,

which is recently introduced by T. Bridgeland.

I will discuss its properties and relation to

auto-equivalences of derived category.

If I have enough time,

I will also discuss the relation

of this Hall algebra to the so-called Ding-Iohara algebra.

The topic of my talk is the Hall algebra of complexes,

which is recently introduced by T. Bridgeland.

I will discuss its properties and relation to

auto-equivalences of derived category.

If I have enough time,

I will also discuss the relation

of this Hall algebra to the so-called Ding-Iohara algebra.

### 2011/10/29

11:00-14:30 Room #117 (Graduate School of Math. Sci. Bldg.)

CKP Hierarchy, Bosonic Tau Function, Bosonization Formulae and Orthogonal Polynomials both in Odd and Even Variables

(based on a joint work with Johan van de Leur and Takahiro Shiota) (ENGLISH)

Kernel function identities associated with van Diejen's $q$-difference operators

and transformation formulas for multiple $q$-hypergeometric series (JAPANESE)

**Alexander Orlov**(Nonlinear Wave Processes Laboratory, Oceanology Institute (Moscow)) 11:00-12:00CKP Hierarchy, Bosonic Tau Function, Bosonization Formulae and Orthogonal Polynomials both in Odd and Even Variables

(based on a joint work with Johan van de Leur and Takahiro Shiota) (ENGLISH)

[ Abstract ]

We develop the theory of CKP hierarchy introduced in the papers of Kyoto school where the CKP tau function is written as a vacuum expectation value in terms of free bosons. We show that a sort of odd currents naturaly appear. We consider bosonization formulae which relate bosonic Fock vectors to polynomials in even and odd Grassmannian variables, where both sets play a role of higher times.

We develop the theory of CKP hierarchy introduced in the papers of Kyoto school where the CKP tau function is written as a vacuum expectation value in terms of free bosons. We show that a sort of odd currents naturaly appear. We consider bosonization formulae which relate bosonic Fock vectors to polynomials in even and odd Grassmannian variables, where both sets play a role of higher times.

**Yasuho Masuda**(Kobe Univ. ) 13:30-14:30Kernel function identities associated with van Diejen's $q$-difference operators

and transformation formulas for multiple $q$-hypergeometric series (JAPANESE)

### 2011/10/22

13:30-16:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Department of Mathematics) 13:30-14:30

Quantization of Quasimaps' Spaces (joint work with M. Finkelberg) (ENGLISH)

Instituteof Biochemical Physics) 15:00-16:00

Quantum integrable models with elliptic R-matrices

and elliptic hypergeometric series (ENGLISH)

**Leonid Rybnikov**(IITP, and State University Higher School of Economics,Department of Mathematics) 13:30-14:30

Quantization of Quasimaps' Spaces (joint work with M. Finkelberg) (ENGLISH)

[ Abstract ]

Quasimaps' space Z_d (also known as Drinfeld's Zastava space) is a

remarkable compactification of the space of based degree d maps from

the projective line to the flag variety of type A. The space Z_d has a

natural Poisson structure,

which goes back to Atiyah and Hitchin. We describe

the Quasimaps' space as some quiver variety, and define the

Atiyah-Hitchin Poisson structure in quiver terms.

This gives a natural way to quantize this Poisson structure.

The quantization of the coordinate ring of the Quasimaps' space turns

to be some natural subquotient of the Yangian of type A.

I will also discuss some generalization of this result to the BCD types.

Quasimaps' space Z_d (also known as Drinfeld's Zastava space) is a

remarkable compactification of the space of based degree d maps from

the projective line to the flag variety of type A. The space Z_d has a

natural Poisson structure,

which goes back to Atiyah and Hitchin. We describe

the Quasimaps' space as some quiver variety, and define the

Atiyah-Hitchin Poisson structure in quiver terms.

This gives a natural way to quantize this Poisson structure.

The quantization of the coordinate ring of the Quasimaps' space turns

to be some natural subquotient of the Yangian of type A.

I will also discuss some generalization of this result to the BCD types.

**Anton Zabrodin**(Instituteof Biochemical Physics) 15:00-16:00

Quantum integrable models with elliptic R-matrices

and elliptic hypergeometric series (ENGLISH)

[ Abstract ]

Intertwining operators for infinite-dimensional representations of the

Sklyanin algebra with spins l and -l-1 are constructed using the technique of

intertwining vectors for elliptic L-operator. They are expressed in

terms of

elliptic hypergeometric series with operator argument. The intertwining

operators obtained (W-operators) serve as building blocks for the

elliptic R-matrix

which intertwines tensor product of two L-operators taken in

infinite-dimensional

representations of the Sklyanin algebra with arbitrary spin. The

Yang-Baxter equation

for this R-matrix follows from simpler equations of the star-triangle

type for the

W-operators. A natural graphic representation of the objects and

equations involved

in the construction is used.

Intertwining operators for infinite-dimensional representations of the

Sklyanin algebra with spins l and -l-1 are constructed using the technique of

intertwining vectors for elliptic L-operator. They are expressed in

terms of

elliptic hypergeometric series with operator argument. The intertwining

operators obtained (W-operators) serve as building blocks for the

elliptic R-matrix

which intertwines tensor product of two L-operators taken in

infinite-dimensional

representations of the Sklyanin algebra with arbitrary spin. The

Yang-Baxter equation

for this R-matrix follows from simpler equations of the star-triangle

type for the

W-operators. A natural graphic representation of the objects and

equations involved

in the construction is used.

### 2011/06/02

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On the module category of $¥overline{U}_q(¥mathfrak{sl}_2)$ (JAPANESE)

**Yoshihisa Saito**(Graduate School of Mathematical Sciences, Univ. of Tokyo)On the module category of $¥overline{U}_q(¥mathfrak{sl}_2)$ (JAPANESE)

[ Abstract ]

In the representation theory of quantum groups at roots of unity, it is

often assumed that the parameter $q$ is a primitive $n$-th root of unity

where $n$ is a odd prime number. However, there has recently been

increasing interest in the the cases where $n$ is an even integer ---

for example, in the study of logarithmic conformal field theories, or in

knot invariants. In this talk,

we work out a fairly detailed study on the category of finite

dimensional

modules of the restricted quantum $¥overline{U}_q(¥mathfrak{sl}_2)$

where

$q$ is a $2p$-th root of unity, $p¥ge2$.

In the representation theory of quantum groups at roots of unity, it is

often assumed that the parameter $q$ is a primitive $n$-th root of unity

where $n$ is a odd prime number. However, there has recently been

increasing interest in the the cases where $n$ is an even integer ---

for example, in the study of logarithmic conformal field theories, or in

knot invariants. In this talk,

we work out a fairly detailed study on the category of finite

dimensional

modules of the restricted quantum $¥overline{U}_q(¥mathfrak{sl}_2)$

where

$q$ is a $2p$-th root of unity, $p¥ge2$.

### 2010/09/14

10:30-14:00 Room #117 (Graduate School of Math. Sci. Bldg.)

AGT conjectures and recursion formulas (JAPANESE)

classification of solutions to the reflection equation associated to

trigonometrical $R$-matrix of Belavin (JAPANESE)

**Shintarou Yanagida**(Kobe Univ.) 10:30-11:30AGT conjectures and recursion formulas (JAPANESE)

**Yuji Yamada**(Rikkyo Univ.) 13:00-14:00classification of solutions to the reflection equation associated to

trigonometrical $R$-matrix of Belavin (JAPANESE)

### 2010/09/13

10:30-15:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Polynomial representations of DAHA of type $C^¥vee C$ and boundary qKZ equations (JAPANESE)

CFT, Isomonodromy deformations and Nekrasov functions (JAPANESE)

Twisted de Rham theory---resonances and the non-resonance (JAPANESE)

**Masahiro Kasatani**(Tokyo Univ.) 10:30-11:30Polynomial representations of DAHA of type $C^¥vee C$ and boundary qKZ equations (JAPANESE)

[ Abstract ]

First I will review basic facts about

the double affine Hecke algebra of type $C^¥vee C$

and its polynomial representation.

Next I will intrduce a boundary qKZ equation

and construct its solution in terms of the polynomial representation.

First I will review basic facts about

the double affine Hecke algebra of type $C^¥vee C$

and its polynomial representation.

Next I will intrduce a boundary qKZ equation

and construct its solution in terms of the polynomial representation.

**Yasuhiko Yamada**(Kobe Univ.) 13:00-14:00CFT, Isomonodromy deformations and Nekrasov functions (JAPANESE)

[ Abstract ]

This talk is an introduction to the relation between conformal filed

theories

and super symmetric gauge theories (Alday-Gaiotto-Tachikawa conjecture)

from the point of view of differential equations (in particular

isomonodromy

deformations).

This talk is an introduction to the relation between conformal filed

theories

and super symmetric gauge theories (Alday-Gaiotto-Tachikawa conjecture)

from the point of view of differential equations (in particular

isomonodromy

deformations).

**Katsuhisa Mimachi**(Tokyo Institute of Technology) 14:30-15:30Twisted de Rham theory---resonances and the non-resonance (JAPANESE)

### 2010/09/12

10:30-17:00 Room #117 (Graduate School of Math. Sci. Bldg.)

A factorization formula for Macdonald polynomials at roots of unity (JAPANESE)

W algebras and symmetric polynomials (JAPANESE)

Quantizing the difference Painlev¥'e VI equation (JAPANESE)

On a bijective proof of a factorization formula for Macdonald

polynomials at roots of unity (JAPANESE)

**Hideaki Morita**(Muroran Institute of Technology) 10:30-11:30A factorization formula for Macdonald polynomials at roots of unity (JAPANESE)

[ Abstract ]

We consider a combinatorial property of Macdonald polynomials at roots

of unity.

If we made some plethystic substitution to the variables,

Macdonald polynomials are subjected to a certain decomposition rule

when a parameter is specialized at roots of unity.

We review the result and give an outline of the proof.

This talk is based on a joint work with F. Descouens.

We consider a combinatorial property of Macdonald polynomials at roots

of unity.

If we made some plethystic substitution to the variables,

Macdonald polynomials are subjected to a certain decomposition rule

when a parameter is specialized at roots of unity.

We review the result and give an outline of the proof.

This talk is based on a joint work with F. Descouens.

**Junichi Shiraishi**(Tokyo Univ.) 13:00-14:00W algebras and symmetric polynomials (JAPANESE)

[ Abstract ]

It is well known that we have the factorization property of the Macdonald polynomials under the principal specialization $x=(1,t,t^2,t^3,¥cdots)$. We try to better understand this situation in terms of the Ding-Iohara algebra or the deformend $W$-algebra. Some conjectures are presented in the case of $N$-fold tensor representation of the Fock modules.

It is well known that we have the factorization property of the Macdonald polynomials under the principal specialization $x=(1,t,t^2,t^3,¥cdots)$. We try to better understand this situation in terms of the Ding-Iohara algebra or the deformend $W$-algebra. Some conjectures are presented in the case of $N$-fold tensor representation of the Fock modules.

**Koji Hasegawa**(Tohoku Univ.) 14:30-15:30Quantizing the difference Painlev¥'e VI equation (JAPANESE)

[ Abstract ]

I will review two constructions for quantum (=non-commutative) version of

q-difference Painleve VI equation.

I will review two constructions for quantum (=non-commutative) version of

q-difference Painleve VI equation.

**Yasuhide Numata**(Graduate School of Information Science and Technology, Tokyo Univ.) 16:00-17:00On a bijective proof of a factorization formula for Macdonald

polynomials at roots of unity (JAPANESE)

[ Abstract ]

The subject of this talk is a factorization formula for the special

values of modied Macdonald polynomials at roots of unity.

We give a combinatorial proof of the formula, via a result by

Haglund--Haiman--Leohr, for some special classes of partitions,

including two column partitions.

(This talk is based on a joint work with F. Descouens and H. Morita.)

The subject of this talk is a factorization formula for the special

values of modied Macdonald polynomials at roots of unity.

We give a combinatorial proof of the formula, via a result by

Haglund--Haiman--Leohr, for some special classes of partitions,

including two column partitions.

(This talk is based on a joint work with F. Descouens and H. Morita.)

### 2010/09/11

13:00-17:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Three-term recurrence relations for a $BC_n$-type basic hypergeometric function and their application (JAPANESE)

TBA (JAPANESE)

AGT conjecture and geometric engineering (JAPANESE)

**Masahiko Ito**(School of Science and Technology for Future Life, Tokyo Denki University) 13:00-14:00Three-term recurrence relations for a $BC_n$-type basic hypergeometric function and their application (JAPANESE)

[ Abstract ]

$BC_n$-type basic hypergeometric series are a certain $q$-analogue

of an integral representation for the Gauss hypergeometric function.

They are defined as multiple $q$-series satisfying Weyl group symmetry of type $C_n$,

and they are a multi-sum generalization of the basic hypergeometric series

in a class of what is called (very-)well-poised. In my talk I will explain

an explicit expression for the $q$-difference system of rank $n+1$

satisfied by a $BC_n$-type basic hypergeometric series with 6+1 parameters

as first order simultaneous $q$-difference equations with a concrete basis.

For this purpose I introduce two types of symmetric Laurent polynomials

which I call the $BC$-type interpolation polynomials. The polynomials satisfy

three-term relations like a contiguous relation for the Gauss hypergeometric

function. As an application, I will show another proof for the product formula

of the $q$-integral introduced by Gustafson.

$BC_n$-type basic hypergeometric series are a certain $q$-analogue

of an integral representation for the Gauss hypergeometric function.

They are defined as multiple $q$-series satisfying Weyl group symmetry of type $C_n$,

and they are a multi-sum generalization of the basic hypergeometric series

in a class of what is called (very-)well-poised. In my talk I will explain

an explicit expression for the $q$-difference system of rank $n+1$

satisfied by a $BC_n$-type basic hypergeometric series with 6+1 parameters

as first order simultaneous $q$-difference equations with a concrete basis.

For this purpose I introduce two types of symmetric Laurent polynomials

which I call the $BC$-type interpolation polynomials. The polynomials satisfy

three-term relations like a contiguous relation for the Gauss hypergeometric

function. As an application, I will show another proof for the product formula

of the $q$-integral introduced by Gustafson.

**Masatoshi Noumi**(Kobe Univ.) 14:30-15:30TBA (JAPANESE)

**Masato Taki**(YITP Kyoto Univ.) 16:00-17:00AGT conjecture and geometric engineering (JAPANESE)

### 2009/12/22

10:00-14:00 Room #056 (Graduate School of Math. Sci. Bldg.)

離散周期KP方程式の簡約と、初期値問題の解の構成

Laplacian on graphs: Examples from physics

**岩尾 慎介**(東大数理) 10:00-11:00離散周期KP方程式の簡約と、初期値問題の解の構成

[ Abstract ]

様々に簡約された離散周期KP方程式に対して、スペクトル曲線を用いた逆散乱解法を考える。 このとき、簡約の種類によっては、超楕円とは限らない代数曲線が多数あらわれてくる。 本講演では、簡約周期KP方程式の初期値問題の解を構成する方法を紹介する。この方法はFayの恒等式を用いない構成的なもので、わかりやすいものである。

様々に簡約された離散周期KP方程式に対して、スペクトル曲線を用いた逆散乱解法を考える。 このとき、簡約の種類によっては、超楕円とは限らない代数曲線が多数あらわれてくる。 本講演では、簡約周期KP方程式の初期値問題の解を構成する方法を紹介する。この方法はFayの恒等式を用いない構成的なもので、わかりやすいものである。

**Y. Avishai**(Ben Gurion University) 13:00-14:00Laplacian on graphs: Examples from physics

[ Abstract ]

When the Laplacian operator is written as a second order difference operator the physicists refer to it as a tight-binding model. In two dimensions the eigenvalue problem connects the function at a given point to the sum of its values on its nearest neighbors. In numerous physical problems, some of the coefficients are multiplied by phase factors. This problem is amazingly rich and the pattern of eigenvalues E(φ) has a fractal nature known as the Hofstadter butterfly.

I will discuss some of these models and especially concentrate on two problems, which I solved recently, where the vertices of the graphs are located on the sphere S2. The first one corresponds to the famous problem of the Dirac magnetic monopole, while in the second one, the eigenfunctions are two component vectors and the phase factors are replaced by unitary 2x2 matrices. This is relevant to the spin-orbit problem in Physics. In both cases the solutions can be obtained in closed form, and exhibit a beautiful symmetry pattern. Their elucidation requires some special techniques in graph theory. Quite surprisingly, the spectra of the two systems coincide at one symmetry point.

When the Laplacian operator is written as a second order difference operator the physicists refer to it as a tight-binding model. In two dimensions the eigenvalue problem connects the function at a given point to the sum of its values on its nearest neighbors. In numerous physical problems, some of the coefficients are multiplied by phase factors. This problem is amazingly rich and the pattern of eigenvalues E(φ) has a fractal nature known as the Hofstadter butterfly.

I will discuss some of these models and especially concentrate on two problems, which I solved recently, where the vertices of the graphs are located on the sphere S2. The first one corresponds to the famous problem of the Dirac magnetic monopole, while in the second one, the eigenfunctions are two component vectors and the phase factors are replaced by unitary 2x2 matrices. This is relevant to the spin-orbit problem in Physics. In both cases the solutions can be obtained in closed form, and exhibit a beautiful symmetry pattern. Their elucidation requires some special techniques in graph theory. Quite surprisingly, the spectra of the two systems coincide at one symmetry point.