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Infinite Analysis Seminar Tokyo
Seminar information archive ～03/23｜Next seminar｜Future seminars 03/24～
| Date, time & place | Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
|---|
Seminar information archive
2016/02/08
13:30-15:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Vincent Pasquier (IPhT Saclay)
q-Bosons, Toda lattice and Baxter Q-Operator (ENGLISH)
Vincent Pasquier (IPhT Saclay)
q-Bosons, Toda lattice and Baxter Q-Operator (ENGLISH)
[ Abstract ]
I will use the Pieri rules of the Hall Littlewood polynomials to construct some
lattice models, namely the q-Boson model and the Toda Lattice Q matrix.
I will identify the semi infinite chain transfer matrix of these models with well known
half vertex operators. Finally, I will explain how the Gaudin determinant appears in the evaluation
of the semi infine chain scalar products for an arbitrary spin and is related to the Macdonald polynomials.
I will use the Pieri rules of the Hall Littlewood polynomials to construct some
lattice models, namely the q-Boson model and the Toda Lattice Q matrix.
I will identify the semi infinite chain transfer matrix of these models with well known
half vertex operators. Finally, I will explain how the Gaudin determinant appears in the evaluation
of the semi infine chain scalar products for an arbitrary spin and is related to the Macdonald polynomials.
2015/09/17
14:00-15:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Simon Wood (The Australian National University)
Classifying simple modules at admissible levels through
symmetric polynomials (ENGLISH)
Simon Wood (The Australian National University)
Classifying simple modules at admissible levels through
symmetric polynomials (ENGLISH)
[ Abstract ]
From infinite dimensional Lie algebras such as the Virasoro
algebra or affine Lie (super)algebras one can construct universal
vertex operator algebras. These vertex operator algebras are simple at
generic central charges or levels and only contain proper ideals at so
called admissible levels. The simple quotient vertex operator algebras
at these admissible levels are called minimal model algebras. In this
talk I will present free field realisations of the universal vertex
operator algebras and show how they allow one to elegantly classify
the simple modules over the simple quotient vertex operator algebras
by using a deep connection to symmetric polynomials.
From infinite dimensional Lie algebras such as the Virasoro
algebra or affine Lie (super)algebras one can construct universal
vertex operator algebras. These vertex operator algebras are simple at
generic central charges or levels and only contain proper ideals at so
called admissible levels. The simple quotient vertex operator algebras
at these admissible levels are called minimal model algebras. In this
talk I will present free field realisations of the universal vertex
operator algebras and show how they allow one to elegantly classify
the simple modules over the simple quotient vertex operator algebras
by using a deep connection to symmetric polynomials.
2015/07/17
14:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Simon Wood (The Australian National University)
Classifying simple modules at admissible levels through symmetric polynomials (ENGLISH)
Simon Wood (The Australian National University)
Classifying simple modules at admissible levels through symmetric polynomials (ENGLISH)
[ Abstract ]
From infinite dimensional Lie algebras such as the Virasoro
algebra or affine Lie (super)algebras one can construct universal
vertex operator algebras. These vertex operator algebras are simple at
generic central charges or levels and only contain proper ideals at so
called admissible levels. The simple quotient vertex operator algebras
at these admissible levels are called minimal model algebras. In this
talk I will present free field realisations of the universal vertex
operator algebras and show how they allow one to elegantly classify
the simple modules over the simple quotient vertex operator algebras
by using a deep connection to symmetric polynomials.
From infinite dimensional Lie algebras such as the Virasoro
algebra or affine Lie (super)algebras one can construct universal
vertex operator algebras. These vertex operator algebras are simple at
generic central charges or levels and only contain proper ideals at so
called admissible levels. The simple quotient vertex operator algebras
at these admissible levels are called minimal model algebras. In this
talk I will present free field realisations of the universal vertex
operator algebras and show how they allow one to elegantly classify
the simple modules over the simple quotient vertex operator algebras
by using a deep connection to symmetric polynomials.
2015/07/09
15:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Yuta Nozaki (Graduate School of Mathematical Sciences, the University of Tokyo) 15:00-16:30
An extension of the LMO functor and formal Gaussian integrals (JAPANESE)
On the relative number of ends of higher dimensional Thompson groups (JAPANESE)
Yuta Nozaki (Graduate School of Mathematical Sciences, the University of Tokyo) 15:00-16:30
An extension of the LMO functor and formal Gaussian integrals (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 certain Jacobi diagrams”.
In this talk, we extend the LMO functor to the case of any number of
boundary components.
In particular, we focus on a formal Gaussian integral, that is an
essential tool to construct the LMO functor.
Motoko Kato (Graduate School of Mathematical Sciences, the University of Tokyo) 17:00-18:30Cheptea, 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 certain Jacobi diagrams”.
In this talk, we extend the LMO functor to the case of any number of
boundary components.
In particular, we focus on a formal Gaussian integral, that is an
essential tool to construct the LMO functor.
On the relative number of ends of higher dimensional Thompson groups (JAPANESE)
[ Abstract ]
In 2004, Brin defined n−dimensional Thompson group nV for every natural number n ≥ 1. nV is a generalization of the Thompson group V . The Thompson group V can be described as a subgroup of the homeomorphism group of the Cantor set C. In this point of view, nV is a subgroup of the homeomorphism group of Cn. We prove that the number of ends of nV is equal to 1 and there is a subgroup of nV such that the relative number of ends is ∞. As a corollary of the second result, for each n, nV has Haagerup property and it can not be the fundamental group of a compact K ̈ahler manifold. These results are the generalizations of the corresponding results of Farley, who studied the Thompson group V .
In 2004, Brin defined n−dimensional Thompson group nV for every natural number n ≥ 1. nV is a generalization of the Thompson group V . The Thompson group V can be described as a subgroup of the homeomorphism group of the Cantor set C. In this point of view, nV is a subgroup of the homeomorphism group of Cn. We prove that the number of ends of nV is equal to 1 and there is a subgroup of nV such that the relative number of ends is ∞. As a corollary of the second result, for each n, nV has Haagerup property and it can not be the fundamental group of a compact K ̈ahler manifold. These results are the generalizations of the corresponding results of Farley, who studied the Thompson group V .
2015/06/25
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Akane Nakamura (Tokyo University, Graduate School of Mathematical Sciences)
Autonomous limit of the 4-dimensional Painlev¥.FN"e-type equations and degeneration of curves of genus two (JAPANESE)
Akane Nakamura (Tokyo University, Graduate School of Mathematical Sciences)
Autonomous limit of the 4-dimensional Painlev¥.FN"e-type equations and degeneration of curves of genus two (JAPANESE)
[ Abstract ]
The Painlev¥.FN"e equations have been generalized from various aspects. Recently, the 4-dimensional Painlev¥N"e-type equations were classified by corresponding linear equations(Sakai, Kawakami-N.-Sakai, Kawakami). In this talk, I explain an attempt to characterize the 40 types of integrable systems obtained as the autonomous limit of the 4-dimensional Painlev¥N"e-type equations, by inspecting the degenerations of their spectral curves, which are curves of genus two.
The Painlev¥.FN"e equations have been generalized from various aspects. Recently, the 4-dimensional Painlev¥N"e-type equations were classified by corresponding linear equations(Sakai, Kawakami-N.-Sakai, Kawakami). In this talk, I explain an attempt to characterize the 40 types of integrable systems obtained as the autonomous limit of the 4-dimensional Painlev¥N"e-type equations, by inspecting the degenerations of their spectral curves, which are curves of genus two.
2015/05/28
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Yuki Arano (Graduate School of Mathematical Sciences, the University of Tokyo)
Unitary spherical representations of Drinfeld doubles (JAPANESE)
Yuki Arano (Graduate School of Mathematical Sciences, the University of Tokyo)
Unitary spherical representations of Drinfeld doubles (JAPANESE)
[ Abstract ]
It is known that the Drinfeld double of the quantized
enveloping algebra of a semisimple Lie algebra looks similar to the
quantized enveloping algebra of the complexification of the Lie algebra.
In this talk, we investigate the unitary representation theory of such
Drinfeld double via its analogy to that of the complex Lie group.
We also talk on an application to operator algebras.
It is known that the Drinfeld double of the quantized
enveloping algebra of a semisimple Lie algebra looks similar to the
quantized enveloping algebra of the complexification of the Lie algebra.
In this talk, we investigate the unitary representation theory of such
Drinfeld double via its analogy to that of the complex Lie group.
We also talk on an application to operator algebras.
2015/04/23
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Hideya Watanabe (Department of Mathematics, Tokyo Institute of Technology, Graduate school of science and Engineering)
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.)
Shunsuke Tsuji (Graduate School of Mathematical Sciences, the University of Tokyo) 13:30-15:00
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) 13:30-15:00
Skein 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.
Yuta Nozaki (Graduate School of Mathematical Sciences, the University of Tokyo) 15:30-17:00We 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.
An 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.)
Diogo Kendy Matsumoto (Faculty of Science and Engineerring, Waseda University) 13:00-14:30
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) 13:00-14:30
A 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.
Youichi Shibukawa (Department of Mathematics, Hokkaido University) 15:00-16:30Brace 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.
Construction 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.)
Shunsuke Tsuchioka (Graduate School of Mathematical Sciences, the University of Tokyo) 15:00-16:30
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) 15:00-16:30
On 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.
Alexander Tsymbaliuk (SCGP (Simons Center for Geometry and Physics)) 17:00-18:30We 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.
Continuous 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.)
Yohei Kashima (Graduate School of Mathematical Sciences, the University of Tokyo) 15:00-16:30
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) 15:00-16:30
Renormalization 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.
Genki Shibukawa (Institute of Mathematics for Industory, Kyushu University) 17:00-18:30We 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.
Unitary 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.)
Fumihiko Nomoto (Department of Mathematics, Tokyo Institute of Technology, Graduate school of science and Engineering) 15:00-16:30
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) 15:00-16:30
An 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.
Masanori Ando (Wakkanai Hokusei Gakuen University) 17:00-18:30Orr-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.
divisor 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.)
Tomohiro SASAMOTO (Department of Physics, Tokyo Institute of Technology) 13:30-15:00
KPZ equation and Macdonald process (JAPANESE)
Shunsuke FURUKAWA (Department of Physics, the Tokyo University) 15:30-17:00
Entanglement spectra in topological phases and coupled Tomonaga-Luttinger liquids (JAPANESE)
Tomohiro SASAMOTO (Department of Physics, Tokyo Institute of Technology) 13:30-15:00
KPZ equation and Macdonald process (JAPANESE)
Shunsuke FURUKAWA (Department of Physics, the Tokyo University) 15:30-17:00
Entanglement 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.)
Satoshi Nawata (Theoretical Physics at NIKHEF)
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.)
Andrei Negut (Columbia University, Department of Mathematics)
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.)
Andrei Negut (Columbia University, Department of Mathematics)
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.)
Andrei Negut (Columbia University, Department of Mathematics)
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.)
Anatol Kirillov (RIMS Kyoto Univ.)
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.)
Simon Wood (Kavli IPMU)
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.)
Alexey Silantyev (Tokyo. Univ.)
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.)
Vincent Pasquier (CEA, Saclay, France)
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.)
Alexey Silantyev (Univ. Tokyo)
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.)
Shintarou Yanagida (Kobe Univ.)
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.)
Alexander Orlov (Nonlinear Wave Processes Laboratory, Oceanology Institute (Moscow)) 11:00-12:00
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)) 11:00-12:00
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)
[ 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.
Yasuho Masuda (Kobe Univ. ) 13:30-14:30We 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.
Kernel 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.)
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)
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.
Anton Zabrodin ( 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.
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.
