Infinite Analysis Seminar
Infinite Analysis Seminar | Date: | March 25, 2006, 11:00-12:30 |
| Speaker: | Alain Lascoux ( C.N.R.S., Institut Gaspard Monge Université de Marne-la-Vallee, FRANCE ) |
| Title: |
q-Demazure characters (joint work with F. Descouens)
Schur functions (i.e. irreducible characters of the linear group) have been extended in different directions : non symmetric characters (Demazure), or symmetric functions with a parameter $q$ (Hall-Littlewood polynomials). I shall give a linear basis of the ring of polynomials which specializes to Demazure characters for $q=0$ and contains as a subfamily the Hall-Littlewood polynomials. The Yang-Baxter operators play an esential role in this construction. |
| Room: | RIMS, Room 206 |
| Date: | January 14, 2006, 11:00-12:30 |
| Speaker: | Yosihisa Saito ( University of Tokyo ) |
| Title: |
On Hecke algebras associated with elliptic root systems
In my lecture I will talk on a q-analogue of the Weyl group of the 2-toroidal Lie algebras. There are two different ways to define a q-analogue of them. The first one is ``double affine Hecke algebra'' due to Cherednik. The second one is ``elliptic Hecke algebra'' which is the main topic on this talk. First I will give a definition of elliptic Hecke algebras attaching to elliptic root systems due to Kyoji Saito. After that, I will discuss on the relationship between double affine Hecke algebras and elliptic Hecke algebras. |
| Room: | RIMS, Room 206 |
| Date: | December 17, 2005, 11:00-12:30 |
| Speaker: | Oleksandr Khomenko ( Max Planck Institute, Bonn, Germany ) |
| Title: |
On tilting modules over algebraic groups
Determining the characters of tilting modules over algebraic groups in prime characteristic is a very difficult problem. At the moment there is no general conjecture about what the answer can be. After a motivational discussion in this talk we present a partial description of the characters of tilting modules. The result is formulated by comparing them to the (known) characters of tilting modules over quantum groups. |
| Room: | RIMS, Room 206 |
| Date: | October 12, 19, November 2, 16, 30 10:30-12:00 |
| Speaker: | Mark Shimozono 氏（Virginia) |
| 内容は，math.AG/0308142 Knutson, Miller and Shimozono, Four positive formulae for type A quiver polynomials の解説です． | |
| Room: | RIMS, Room 102 |
| Date: | September 24, Saturday, 11:00-12:30 |
| Speaker: | Ivan Cherednik ( UNC Chapel Hill, USA) |
| Title: |
Skew Young diagrams and Hecke algebras,affine and double
affine.
I will begin with the description of the semisimple representations of affine Hecke algebras of type GL in terms of the skew Young diagrams. The affine Hecke algebra technique changed a great deal our understanding of the skew diagrams and their combinatorial applications, the theory of Young projections, and the Frobenius character formula. However the semisimple representations do not play any significant role in the theory of affine Hecke algebras and p-adic groups. In the theory of double affine Hecke algebras of type GL(n), the semisimple representation are described via periodic infinite skew diagrams. These representations are of fundamental importance for the theory of dahas mainly because of the applications, known and expected, in the harmonic analysis. |
| Room: | RIMS, Room 206 |
| Date: | September 24, Saturday, 13:45-14:45, |
| Speaker: | Anatol N.Kirillov (RIMS. Kyoto University) |
| Title: |
On some remarkable quadratic algebras, Schubert
calculus and beyond.
I will describe certain quadratic algebras together with commutative subalgebras therein. Depending on the situation, the latter happened to be isomorphic to the (quantum) cohomology or Grothendieck rings of flag varieties and so on. |
| Room: | RIMS, Room 206 |
| Date: | December 9, Thursday, 12:00-13:00 |
| Speaker: | Takeshi Suzuki (RIMS) |
| Title: |
Double affine Hecke algebras and conformal field theory
Motivated by conformal field theory on Riemann sphere with $n+1$ points ($n$ moving points + $1$ fixed point), we introduce a certain space of coinvariants obtained from tensor product of representations of the affine Lie algebra ${\mathfrak g}$ of type $A$. On this space, we define an action of the rational double affine Hecke algebra (or the rational Cherednik algebra) $\mathcal H$ through Knizhnik-Zamolodchikov connections. This construction gives a functor from the category of highest weight modules over ${\mathfrak g}$ to the category of $\mathcal H$-modules. We show that integrable ${\mathfrak g}$-modules correspond by this functor to irreducible $\mathcal H$-modules whose structure is described combinatorially. We also present character formulas for these irreducible $H$-modules, which are described by level restricted standard tableaux on skew Young diagrams and the $H$-functions in RSOS model. By similar argument based on conformal field theory on Riemann sphere with $n+2$ points, we obtain a correspondence between highest weight modules over ${\mathfrak g}$ and modules over the degenerate affine Hecke algebra, and parallel results for integrable modules. |
| Room: | RIMS, Room 402 |
| Date: | November 20, 2004, 11:00-12:00 |
| Speaker: | Mikhail Olshanetsky (ITEP, Moscow) |
| Title: |
Hitchin Systems - Hecke correspondence and two-dimensional version
We describe two types of Hitchin systems - the first one is the Elliptic Calogero-Moser system (ECMS) related to the holomorphic vector bundles of degree zero over an elliptic curve and Elliptic Top (ET) related to the bundles of degree one.There exists a map from ECMS to ET (symplectic Hecke correspondence). To describe an analogues of the Hitchin systems in dimension 1+1 we consider infinite rank holomorphic bundles. The main example is the Calogero-Moser field theory and the Landau-Lifshitz equation. The symplectic Hecke correspondence is generalized to this case as well. |
| Room: | RIMS, Room 206 |
| Date: | November 20, 2004, 13:30-14:30 |
| Speaker: | Vladimir Mangazeev (Australian National Univ.) |
| Title: |
Eight-vertex model and non-stationary Lame equation
We study the ground state eigenvalues of Baxter's Q-operator for the eight-vertex model in a special case when it describes the off-critical deformation of the $\Delta=-1/2$ six-vertex model. We show that these eigenvalues satisfy a "non-stationary Lame equation" with the time-dependent potential given by Weierstrass elliptic function where the modular parameter $\tau$ plays the role of time.In the scaling limit the equation transforms into a "non-stationary Mathieu equation" which is closely related to the theory of dilute polymers on a cylinder, massive sine-Gordon model and the Painleve III equation. |
| Room: | RIMS, Room 206 |
| Date: | October 23, 2004, 11:00-12:00 |
| Speaker: | Eric Opdam ( Korteveg de Vries Institute for Mathematics, Amsterdam, The Netherlands ) |
| Title: |
Category O for rational Cherednik algebras and representations
of Hecke algebras at roots of unity
The category of representations Mod_H of a finite dimensional Hecke algebra H becomes very complicated when the parameters are specialized to certain bad roots of unity. In this talk we will first discuss the properties of a larger, more simple category O of lowest weight modules over the rational Cherednik algebra. This category shares many structural similarities with the category of highest weight modules over a simple Lie algebra. Then we show, using Dunkl operators, that Mod_H is actually a quotient of O via a localization functor. It is remarkable that this construction works for all complex reflection groups whose braid group has a Coxeter-like presentation.
Literature: |
| Room: | RIMS, Room 206 |
| Date: | October 23, 2004, 13:30-14:30 |
| Speaker: | Sergey Oblezin ( ITEF, Moscow, Russia ) |
| Title: |
On a class of representations of the Yangian and moduli space of
monopoles
A new class of infinite dimensional representations of the Yangians $Y(\frak{g})$ and $Y(\frak{b})$ corresponding to a complex semisimple algebra $\frak{g}$ and its Borel subalgebra $\frak{b}\subset\frak{g}$ is constructed. It is based on the generalization of the Drinfeld realization of $Y(\frak{g})$, $\frak{g}=\frak{gl}(N)$ in terms of quantum minors to the case of an arbitrary semisimple Lie algebra $\frak{g}$. The Poisson geometry associated with the constructed representations is described. In particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of $G$-monopoles defined as the components of the space of based maps of $\mathbb{P}^1$ into the generalized flag manifold $X=G/B$. Thus the constructed representations of the Yangian may be considered as a quantization of the moduli space of the monopoles. ( Joint with A. Gerasimov, S. Kharchev, D. Lebedev ) |
| Room: | RIMS, Room 206 |
| Date: | October 23, 2004, 11:00-12:00 |
| Speaker: | Eric Opdam ( Korteveg de Vries Institute for Mathematics, Amsterdam, The Netherlands ) |
| Title: |
Category O for rational Cherednik algebras and representations
of Hecke algebras at roots of unity
The category of representations Mod_H of a finite dimensional Hecke algebra H becomes very complicated when the parameters are specialized to certain bad roots of unity. In this talk we will first discuss the properties of a larger, more simple category O of lowest weight modules over the rational Cherednik algebra. This category shares many structural similarities with the category of highest weight modules over a simple Lie algebra. Then we show, using Dunkl operators, that Mod_H is actually a quotient of O via a localization functor. It is remarkable that this construction works for all complex reflection groups whose braid group has a Coxeter-like presentation.
Literature: |
| Room: | RIMS, Room 206 |
| Date: | October 23, 2004, 13:30-14:30 |
| Speaker: | Sergey Oblezin ( ITEF, Moscow, Russia ) |
| Title: |
On a class of representations of the Yangian and moduli space of
monopoles
A new class of infinite dimensional representations of the Yangians $Y(\frak{g})$ and $Y(\frak{b})$ corresponding to a complex semisimple algebra $\frak{g}$ and its Borel subalgebra $\frak{b}\subset\frak{g}$ is constructed. It is based on the generalization of the Drinfeld realization of $Y(\frak{g})$, $\frak{g}=\frak{gl}(N)$ in terms of quantum minors to the case of an arbitrary semisimple Lie algebra $\frak{g}$. The Poisson geometry associated with the constructed representations is described. In particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of $G$-monopoles defined as the components of the space of based maps of $\mathbb{P}^1$ into the generalized flag manifold $X=G/B$. Thus the constructed representations of the Yangian may be considered as a quantization of the moduli space of the monopoles. ( Joint with A. Gerasimov, S. Kharchev, D. Lebedev ) |
| Room: | RIMS, Room 206 |
| Date: | October 21 (Thu), 15:00-16:30 |
| Speaker: | Daniel Barlet (Univ. Nancy) |
| Title: |
Integration of meromorphic cohomology classes and applications
I shall present tools introduced in sevral papers in collaboration with Jon Magnusson (Island), namely integration of meromorphic cohomology classes. They are used to describe how positivity conditions on the normal bundle of a compact complex submanifold Y of codimension n+1 in a complex manifold Z can be transfert into positivity conditions for a Cartier divisor in the space of compact n-cycles in Z. As an application of these tools we prove that the following problem has a positive answer in many cases : Let Z be a compact connected complex manifold of dimension n+p. Let Y a submanifold in Z of dimension p-1 whose normal bundle is positive. Assume that Z can be covered by an analytic family of compact n-cycles parametrized par a compact normal space S. Is the algebraic dimension of Z at least p ? |
| Room: | RIMS, Room 402 |
| organizer: | Masaki Kashiwara |
| Date: | September 25, 2004, 11:00-12:30 |
| Speaker: | Vladimir Rubtsov ( Universite d'Angers, France ) |
| Title: |
Commuting families in skew fields: Integrable systems
associated with elliptic algebras
We propose a simple algebraic construction of commutative subsets in skew fields, which can be thought as an analog of Separation of Variables for Integrable Systems ( both in quantum and classical cases ). The applications include some known examples of Integrable Systems associated with Poisson surfaces ( Beauville-Mukai systems ) as well as some new families of com- muting elements in Sklyanin-Feigin-Odesskii elliptic algebras. |
| Room: | RIMS, Room 206 |
| Date: | September 25, 2004, 13:30-14:30 |
| Speaker: | Hjalmar Rosengren ( Chalmers University of Technology, Sweden ) |
| Title: |
Harmonic analysis on the Sklyanin algebra
This talk will be a progress report on recent work linking representation theory of elliptic quantum groups with elliptic hypergeometric series. We will argue that Takebe's intertwining vectors (related to the vertex-IRF transformation for the 8-vertex model) are a natural starting-point for harmonic analysis on the Sklyanin algebra. These are bases solving a generalized eigenvalue problem in a Sklyanin algebra representation. The transition coefficients between different such bases are given by (analytically continued) elliptic 6j-symbols. As an application, we can prove Sklyanin's 1983 conjecture concerning Sklyanin algebra invariant integration on the torus. |
| Room: | RIMS, Room 206 |
| Date: | May 29, 2004, 11:00-12:15 |
| Speaker: | Rei Inoue Yamazaki ( RIMS, Kyoto University ) |
| Title: |
Matrix realization of affine Jacobi varieties and
the extended Lotka-Volterra lattice
Consider a gauge equivalence class M of polynomial matrices of a spectral parameter, and let X be the algebraic curve given by the common characteristic equation for M. It is known that M is isomorphic to the affine part of the Jacobi variety, Jaff(X). In the context of classical integrable Hamiltonian systems, an orbit in M corresponds to the level set of some Lax matrix, and Jaff(X) is the invariant manifold of the system. We construct a family of M and corresponded representatives explicitly by starting with certain types of Lax matrices, and study the isomorphic map from the representative to Jaff(X). Further we apply these to discuss the algebraic complete integrability of the extended Lotka-Volterra lattice with a periodic boundary condition. This work corresponds to a partial extension of the result by Smirnov and Zeitlin. |
| Room: | RIMS, Room 206 |
| Date: | May 29, 2004, 13:30-14:30 |
| Speaker: | Masahiro Kasatani ( Kyoto University ) |
| Title: |
The vanishing ideal on double shifted diagonal and Jack
and Macdonald polynomials
We construct a ideal defined by certain zero condition on the double shifted diagonal. We give a basis by Jack and Macdonald polynomials and its linear combination with specialized parameters. A character formula (Hilbert-Poincaré series) is given. Joint work with T.Miwa, A.N.Sergeev, A.P.Veselov |
| Room: | RIMS, Room 206 |
| Date: | February 28, 2004, 11:00-12:30 |
| Speaker: | Edward Frenkel ( University of California, Berkeley, USA ) |
| Title: |
Opers, flag varieties and Bethe Ansatz
We consider the problem of diagonalization of the hamiltonians of the Gaudin model, which is a quantum chain model associated to a simple Lie algebra. The hamiltonians of this model act on the tensor product of finite-dimensional representations of this Lie algebra. We show that the eigenvalues of the Gaudin hamiltonians are encoded by the so-called "opers" on the projective line, associated to the Langlands dual Lie algebra. These opers have regular singularities at the marked points with prescribed residues and trivial monodromy representation. The Bethe Ansatz is a procedure to construct explicitly the eigenvectors of the generalized Gaudin hamiltonians. We show that each solution of the Bethe Ansatz equations defines what we call a "Miura oper" on the projective line. Moreover, we show that the space of Miura opers is a union of copies of the flag manifold (of the dual group), one for each oper. This allows us to prove that all solutions of the Bethe Ansatz equations, corresponding to a fixed oper, are in one-to-one correspondence with the points of an open dense subset of the flag manifold. For the Lie algebras of types A,B,C similar results were obtained by other methods by I.Scherbak and A.Varchenko and by E.Mukhin and A.Varchenko. |
| Room: | RIMS, Room 206 |
| Date: | December 13, 2003, 11:00-12:15 |
| Speaker: | Michael Kleber ( Brandeis University,Boston, USA ) |
| Title: |
Symmetric functions and representations of quantum groups
Consider the following bizarre operation: let $V(\lambda)$ be an irreducible finite-dimensional representation of $gl(n)$, and look at its restriction to the orthogonal subalgebra $o(n)$. Decompose this as a direct sum of irreducibles, and then replace each $o(n)$ module with the $sp(n)$ module with the same highest weight. The resulting (reducible) symplectic representation is, remarkably, the restriction of an irreducible finite-dimensional representation of the Yangian $Y(sp(n))$. This unnatural-seeming operation is inspired by a theorem about automorphisms of the ring of symmetric functions, which in turn was inspired by work of Kirillov and Reshetikhin which predicted the above decompositions in the special case where $\lambda$ is a multiple of a fundamental weight. This is joint work with Ian Grojnowski (Cambridge). [pdf] |
| Room: | RIMS, Room 206 |
| Date: | September 6, 2003, 11:00--12:15 |
| Speaker: | Vladimir Bazhanov ( Australian National University ) |
| Title: |
High Level Eigenvalues of Local Integrals of Motion in Conformal
Field Theory
We analyse the structure of eigenvalues of the CFT local integrals of motion for their high level eigenstates in the irreducible Virasoro module with the help of Bethe Ansatz and functional relations. The previously known relation between one-dimensional Schroedinger equation and the vacuum eigenvalues is extended to the higher-level case. |
| Room: | RIMS, Room 206 |
| Date: | September 6, 2003, 13:30--15:00 |
| Speaker: | Sergey Loktev ( Moscow State University ) |
| Title: |
Multidimensional Weyl Modules
We consider Weyl modules (in the sense of V.Chari and A.Pressley) over multidimensional currents. Using the result of M. Haiman we calculate the dimension for the two dimensional currents over $sl_2$ and in some cases $sl_n$. If some time remains we discuss the connection with multidimension fusion procedure and geometry of Hilbert schemes. Ref: math.QA/0212001 |
| Room: | RIMS, Room 206 |
| Date: | May 31, 2003, 11:00--12:15 |
| Speaker: | Vladimir Bazhanov ( Australian National University ) |
| Title: |
Spectral Theory of Schr\"odinger Equation
and Conformal Field Theory
The conformal field theory (CFT) can be understood as a completely integrable quantum field theory as it possesses an infinite set of mutually commuting local integrals of motion. Therefore the simultaneous diagonalization of these integrals of motion is a fundamental spectral problem of the CFT. In this talk we discuss a remarkable connection of this problem with the spectral theory of the one-dimensional Schr\"odinger equation. Some recent applications of this connection, particularly the analytical results for the quantum impurity problem with arbitrary magnetic fields (the Coqblin-Schrieffer model) related with $WA_{n-1}$-algebra will be briefly discussed. We will also reveal some celebrated (but well forgotten) results of J.L.Lagrange on resolution of the general algebraic equations of an arbitrary degree obtained in 1768. |
| Room: | RIMS, Room 206 |
| Date: | May 31, 2003, 13:30--15:00 |
| Speaker: | Maxim Nazarov ( University of York ) |
| Title: |
Mixed Hook-Length Formula for Affine Hecke Algebras
Let H(l) be the affine Hecke algebra corresponding to the group GL(l) over a p-adic field with the residue field of cardinality q. Regard H(l) as an associative algebra over the field C(q) of rational functions in q with complex coefficients. Consider the H(l+m)-module W induced from the tensor product of evaluation modules over the algebras H(l) and H(m). The module W depends on two partitions (of l and of m) and on two non-zero elements of the field C(q). There is a canonical operator J acting in W, it corresponds to the trigonometric R-matrix. The algebra H(l+m) contains the finite dimensional Hecke algebra (the q-analogue of the symmetric group ring CS(l+m)) as a subalgebra, and the operator J commutes with the action of this subalgebra on W. Under this action, W decomposes into irreducible subspaces according to the Littlewood-Richardson rule. We compute the eigenvalues of J, corresponding to certain multiplicity-free irreducible components of W. In particular, we give a formula for the ratio of two eigenvalues of J, corresponding to the ``highest'' and the ``lowest'' components. We also give irreducibily criterion for W as H(l+m)-module. |
| Room: | RIMS, Room 206 |
| Date: | April 26 (Sat), 2003, 11:00-12:15 |
| Speaker: | Yoshihiro TAKEYAMA ( Kyoto University ) |
| Title: |
Counting minimal form factors of the restricted sine-Gordon model
We revisit the issue of counting all local fields of the restricted sine-Gordon model, in the case corresponding to a perturbation of minimal unitary conformal field theory. The problem amounts to the study of a quotient of certain space of polynomials which enter the integral representation for form factors. This space may be viewed as a q-analog of the space of conformal coinvariants associated with U_q(sl_{2}2) with q=\sqrt{-1}. We prove that its character is given by the restricted Kostka polynomial multiplied by a simple factor. As a result, we obtain a formula for the truncated character of the total space of local fields in terms of the Virasoro characters. This is a joint work with M. Jimbo and T. Miwa. |
| Room: | RIMS, Room 206 |
| Date: | April 26(Sat), 2003, 13:30-15:00 |
| Speaker: | Kazuhiko HIKAMI ( Tokyo University ) |
| Title: |
Zagier's q-series identity and knot invariant
We define a generalization of Zagier's q-series which is relate to the Rogers-Ramanujan type q-series, and study asymptotic behavior. These q-series originate from recent studies of "Volume Conjecture", and they coincide with knot invariant for torus knots when q is root of unity. |
| Room: | RIMS, Room 206 |
| Date: | February 22 (Sat), 2003, 11:00-12:15 |
| Speaker: | Nikolai KITANINE (Graduate School 0f Mathematics, University of Tokyo) |
| Title: |
Asymptotic analysis of one correlation function of
the XXZ Heisenberg spin chain
The asymptotic behavior of the emptiness formation probability is considered for the masseless regime of the XXZ Heisenberg spin chain. The leading term of the asymptotic can be obtained from the multiple integral representation using the saddle point method. This result can be compared with known formulae for some particular cases. |
| Room: | RIMS, Room 206 |
| Date: | February 22 (Sat), 2003, 13:30-15:00 |
| Speaker: | J.F. VAN DIEJEN (Institute of Mathematics and Physics, University of Talca, Talca, Chile) |
| Title: |
Explicit construction of the Macdonald polynomials
Macdonald polynomials are usually defined as orthogonal polynomials with respect to a q-Selberg type integration measure. Theoretically these polynomials can be constructed via a Gram-Schmidt process, but in practice this method is not very attractive from a computational point of view. Algebraic tools such as Pieri formulas, raising operators, and shift operators also turn out to be quite limited from a computational point of view. Still, in applications in mathematical physics and combinatorics it is often desired to have very explicit information regarding the monomial expansion of the Macdonald polynomials. In this talk I want to present an explicit formula for the coefficients of the Macdonald polynomials aimed at efficient computations. The formula in question is valid for arbitrary crystallographic root systems. The talk is based on joint work with L. Lapointe and J. Morse. |
| Room: | RIMS, Room 206 |
| Date: | January 25, 2003, 11:00-12:00 |
| Speaker: | Teruhisa TSUDA (Graduate School of Mathematics, University of Tokyo) |
| Title: |
Universal characters and an extension of the KP hierarchy
The universal character is a polynomial attached to a pair of partitions, which is a generalization of the Schur polynomial. We define the vertex operators, which play roles of the raising operators for the universal character. By using the vertex operators, we obtain a hierarchy of non- linear partial differential equations of infinite order, which can be regarded as an extension of the KP hierarchy (termed the UC hierarchy). We investigate solutions of the UC hierarchy; the totality of the space of solutions forms a direct product of two infinite dimensional Grassmann manifolds, and its infinitesimal transformations are described by an infinite dimensional Lie algebra  .
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| Room: | RIMS, Room 206 |
| Date: | January 25, 2003, 13:30-15:00 |
| Speaker: | Toshiki NAKASHIMA (Department of Mathematics, Sophia University, Tokyo) |
| Title: |
Geometric crystal on Schubert varieties
We develop the theory of geometric crystals and unipotent crystals in Kac-Moody setting, which is introduced by Berenstein and Kazhdan for reductive cases. Next, we define an unipotent crystal structure on finite Schubert cells/varieties associated with Kac-Moody groups and induce the geometric crystal structure on them. Considering a "positive structure" on them and show that the geometric crystal structure on finite Schubert cells is a kind of "tropicalization" of certain Kashiwara's crystal. |
| Room: | RIMS, Room 206 |
| Date: | December 7(Sat), 2002, 11:00-12:00 |
| Speaker: | Masato OKADO (Graduate School 0f Engineering Sciences, Osaka University) |
| Title: | Geometric crystal and tropical R |
| Room: | RIMS, Room 206 |
| Date: | December 7(Sat), 2002, 13:10-14:40 |
| Speaker: | Tomoyuki ARAKAWA (Graduate School of Mathematical Sciences, Nagoya University) |
| Title: |
On vanishing conjecture of Frenkel-Kac-Wakimoto
We (partially) prove the vanishing conjecture of Frenkel-Kac-Wakimoto related to the minimal series representations of $W$-algebra. |
| Room: | RIMS, Room 206 |
| Date: | November 2(Sat), 2002, 11:00-12:00 |
| Speaker: | Yuji HARA (Graduate School of Mathematical Sciences, University of Tokyo) |
| Title: |
Affine Analogue of Jack's Polynomials for $\hat{sl_2}$ Affine analogue of Jack's polynomials corresponding to $\hat{sl_2}$ was introduced and studied by Etingof and Kirillov Jr. Using the Wakimoto representation, we give an integral formula of elliptic Selberg type for the affine Jack polynomials. From this integral formula, an action of the modular group $SL_2(Z)$ on the space of affine Jack polynomials is computed. For simple cases, we write down affine Jack polynomials in terms of some modular and elliptic functions. The talk is based on the preprint math.QA/0210236. |
| Room: | RIMS, Room 206 |
| Date: | November 2, 13:30-14:30 |
| Speaker: | Sayaka HAMADA (Graduate School of Mathematical Sciences, Kyushu University) |
| Title: |
Proof of Baker-Forrester's constant term conjecture
for the cases $N_{1}=2, 3$ The Baker-Forrester constant term conjecture is an extension of the $q$-Morris constant term identity. We give a proof of the conjecture in the cases $N_{1}=2, 3$ by using the $q$-integral representation for Macdonald polynomials. |
| Room: | RIMS, Room 206 |
| Date: | July 27(Sat), 2002, 11:00-12:30, |
| Speaker: | Roman BEZRUKAVNIKOV (University of Chicago, USA, and RIMS) |
| Title: |
Local geometric Langlands program, and
representations of simple Lie algebras
in prime characteristic In the first part of the talk I will give a brief introduction to the geometric Langlands program of Beilinson and Drinfeld; and describe my results (partly joint with S. Arkhipov) which can be viewed as a partial progress in the (local version of the) program. In the second part I will describe my joint results with Mirkovic and Rumynin, which adapt the D-module technique to the theory of representations of simple Lie algebras in prime characteristic. A (somewhat mysterious) relation between the two constructions is expected to yield a proof of Lusztig's conjectures on characters of irreducible representations of simple Lie algebras in positive characteristic. |
| Room: | RIMS, Room 206 |
| Date: | July 27(Sat), 2002, 14:00- 15:30 |
| Speaker: | Takayuki HIBI (Osaka University, Graduate School of Science) |
| Title: |
Counting lattice points and constructing
triangulations of convex polytopes Counting lattice points and constructing triangulations of convex polytopes are traditional topics in combinatorics on convex polytopes. In my talk algebraic aspects on these topics with emphasizing the close connections with Gr"obner bases will be discussed. No special knowledge on combinatorics and algebra will be required. |
| Room: | RIMS, Room 206 |
| Date: | June 29(Sat), 2002, 11:00-12:00 |
| Speaker: | David NADLER (University of Chicago, USA, and RIMS) |
| Title: |
Perverse sheaves on real affine Grassmannians It is known that a certain category of perverse sheaves on the affine Grassmannian Gr of a reductive complex algebraic group G is a tensor category equivalent to the category of finite-dimensional representations of the dual group G^ of G. The result can be thought of as a geometric version of the Satake isomorphism. It is of fundamental importance in the Geometric Langlands Program. It turns out that a similar statement is true for real groups: a certain category of perverse sheaves on the affine Grassmannian Gr_R of a reductive real algebraic group G_R is equivalent to the category of finite-dimensional representations of a reductive complex algebraic subgroup H of the dual group G^. The root system of H is closely related to the restricted root system of G_R. The fact that $H$ is reductive implies that an interesting family of real algebraic maps satisfies the conclusion of the Decomposition Theorem of Beilinson-Bernstein-Deligne. |
| Room: | RIMS, Room 206 |
| Date: | June 29(Sat), 2002, 14:00-15:30 |
| Speaker: | Nikolai KITANINE (Tokyo University, and RIMS) |
| Title: |
Correlation functions of the XXZ spin chain The correlation functions of the XXZ model are studied for different regimes using the algebraic Bethe ansatz technique. Different correlation functions such as emptiness formation probability, generation functional of the third components of spins correlation functions and two point functions can be obtained as multiple integrals of a very particular structure. The integrals can be calculated explicitly in some particular points, such as the free fermion point, reproducing well known results, but also for $\Delta=1/2$ giving the first explicit asymptotic result for the correlation functions of the XXZ model beyond the free fermion point. It permits also to establish a connection between the correlation functions and alternating sign matrices. |
| Room: | RIMS, Room 206 |
| Date: | May 25(Sat), 2002, 11:00-12:00 |
| Speaker: | Susumu ARIKI (RIMS, Kyoto University) |
| Title: |
Tameness of Hecke algebras of type B,
and Green correspondence Let Hn(q) be the Hecke algebra of type B, q be a primitive eth root of unity. In this talk, we show that Hn(q) has tame representation type if and only if q=-1 and n=2. We conjecture that if e > 2 then there is no tame block. Toward this goal, we formulate and prove the Green correspondence for block algebras of the Hecke algebras. |
| Room: | RIMS, Room 206 |
| Date: | May 25(Sat), 2002, 14:00-15:00 |
| Speaker: | Boris FEIGIN (RIMS, Kyoto University) |
| Title: |
Deformation theory, Lie algebra cohomology construction
of (formal) moduli space and formality theorem In the talk we present a certain approach which, probably, helps to understand better the formality theorem by M. Kontsevich, and to find a more simple proof. |
| Room: | RIMS, Room 206 |
| Date: | February 16(Sat), 2002, 11:00-12:30 |
| Speaker: | Alexander BELAVIN (L.D. Landau Institute of Theoretical Physics, Moscow, Russia, and RIMS, Kyoto University) |
| Title: |
New relations in the algebra of the Baxter Q-operators We consider irreducible cyclic representations of the Quantum Group connected with the R-matrix of the six-vertex model. At roots of unity the Baxter Q-operator can be represented as trace of a tensor product of the cyclic representation of the L-operators and satisfies the Baxter T-Q-equation. We find some new algebraic relations between the cyclic L-operators and, as a consequence, between the Q-operators. |
| Room: | RIMS, Room 206 |
| Date: | February 16(Sat), 2002, 14:30-16:00 |
| Speaker: | Kazuo HABIRO (RIMS, Kyoto University) |
| Title: |
On the Witten-Reshetikhin-Turaev invariant of integral homology 3-spheres
We study the sl_2 Witten-Reshetikhin-Turaev invariants of integral homology 3-spheres. We define an invariant of integral homology spheres with values in a completion of a polynomial ring which unifies all the sl_2 Witten-Reshetikhin-Turaev invariants at various roots of unity. |
| Room: | RIMS, Room 206 |
| Date: | January 12(Sat), 2002, 11:00-12:00 |
| Speaker: | Yoshihiro TAKEYAMA (RIMS, Kyoto University) |
| Title: |
Form factors of SU(N) invariant Thirring model We obtain a new integral formula for solutions of the rational quantum Knizhnik-Zamolodchikov equation associated with Lie algebra sl(N) at level zero. Our formula contains the integral representation of form factors of SU(N) invariant Thirring model constructed by F. Smirnov. We give recurrence relations for our solutions to be the form factors, and check that the recurrence relations hold for the form factors of the energy momentum tensor. |
| Room: | RIMS, Room 206 |
| Date: | January 12(Sat), 2002, 14:00-15:00 |
| Speaker: | Takeshi SUZUKI (RIMS, Kyoto University) |
| Title: |
On representations of double affine Iwahori-Hecke
algebras of type A We study a class of representations of (degenerate/nondegenerate) double affine Iwahori-Hecke algebras of type A. Under certain conditions on parameters, we show that any irreducible module of this class is given as a unique irreducible quotient of some parabolically induced module. |
| Room: | RIMS, Room 206 |
| Date: | December 18 (Tue), 2001, 15:00- |
| Speaker: | Nicholas WITTE (Department of Mathematics & Statistics, University of Melbourne, Parkville, Victoria 3010, Australia) |
| Title: | Random Matrix Theory & Integrable Dynamical Systems |
| Room: | RIMS, Room 202 |
| Date: | December 1 (Sat), 2001, 11:00-12:00 |
| Speaker: | Alexei Zamolodchikov (LPM, Univ. Montpellier II, Montpellier, France) |
| Title: |
Generalized Mathieu equation and Liouville TBA Four apparently different structures will be discussed together with unexpected relations between them. The first is the relation between the so-called Q-operator as constructed by V.Bazhanov, S.Lukyanov and A.Zamolodchikov (hep-th/9412229, 9604044, 9805008) and standard eigen-value problem for radial Schrodinger equation in homogeneous attractive potential, first observed by P.Dorey and R.Tateo in 1999. The second relation is proposed between the spectral problem for certain ordinary differential equation, which I call the generalized Mathieu equation, and Liouville thermodynamic Bethe ansatz (TBA) equation (a non-linear integral equation which arizes in attempt to treat the Liouville field theory as completely integrable system). A solution to the last equation can be ``explicitely constructed'' in terms of formal analytic continuation of the Q-operators to the region where the Virasoro central charge c > 25. |
| Room: | RIMS, Room 206 |
| Date: | October 27 (Sat), 2001 14:00-15:15 |
| Speaker: | Toshiki NAKASHIMA (Sophia University, Tokyo) |
| Title: |
Piecewise-linear Combinatorics of Crystal Bases We give a general way of representing the crystal (base) corresponding to the nilpotent part of quantized Kac-Moody algebras and their integrable highest weight modules, which is called polyhedral realizations. Those crystal bases are realized as the set of integer solutions of a system of linear inequalities (in infinitely many variables) under certain conditions. We produce the system of linear inequalities by some piecewise-linear operators on linear functions. We also describe braid-type isomorphisms in terms of piecewise-linear functions. The polyhedral realizations are applied to describe explicitly the crystal bases for arbitrary rank 2 Kac-Moody algebra cases, the classical $A_n$-case |
| Room: | RIMS, Room 206 |
| Date: | October 27 (Sat), 2001 15:30-16:45 |
| Speaker: | Kouichi TAKEMURA (Yokohama City University) |
| Title: |
The Heun equation and the Calogero-Moser-Sutherland model We propose and develop the Bethe Ansatz method for the Heun's equation. As an application, we justify the holomorphic perturbation for the 1-particle Inozemtsev model from the trigonometric model. We will give comments on the relationships to the finite gap property and the expression of the monodromy of the Heun's equation in terms of the hyper-elliptic integral. |
| Room: | RIMS, Room 206 |
| Date: | October 27 (Sat), 2001, 17:00-18:00 |
| Speaker: | Sergei Loktev ( Institute of Theoretical and Experemental Physics (ITEP) and Independent University, Moscow, Russia ) |
| Title: |
Coinvariants of one-dimensional lattice VOAs
Two character formulas for the spaces of coinvariants are proposed. First of them is a formula of widely used fermionic type. The other (bosonic) one is a q-analog of the Verlinde formula for dimensions. Along the way a fermionic formula for supernomial coefficients is obtained. Based on joint work with B.L. Feigin and I.Yu.Tipunin |
| Room: | RIMS, Room 206 |
| Date: | June 30 (Sat), 2001, 11:00-12:00 |
| Speaker: | Yoshihiro Takeyama (RIMS) |
| Title: |
On form factors of SU(2) invariant Thirring model
Integral formulae for form factors of a large family of charged local operators in SU(2) invariant Thirring model are given extending Smirnov's construction of form factors of chargeless local operators in the sine-Gordon model. New abelian symmetry acting on this family of local operators is found. It creates Lukyanov's operators which are not in the above family of local operators in general. This is a joint work with Atsushi Nakayashiki. |
| Room: | RIMS, Room 102 |
| Date: | June 30 (Sat), 2001, 14:00-15:00 |
| Speaker: | Boris FEIGIN (RIMS) |
| Title: | Coinvariants for 1-dimensional lattice VOA and the $sl_2$-multinomial coefficents |
| Room: | RIMS, Room 102 |
| Date: | June 30 (Sat), 2001, 15:30-16:30 |
| Speaker: | Martin GUEST (Tokyo Metropolitan University) |
| Title: |
Introduction to quantum cohomology and related integrable systems
This will be an introduction to quantum cohomology, mainly through concrete examples. Relations with systems of pde will be discussed, including for example the observation of Givental and Kim that the quantum cohomology of the flag manifold is related to the quantum Toda lattice. |
| Room: | RIMS, Room 102 |
| Date: | December 5 (Tue), 2000, 10:30-11:30 |
| Speaker: | V. E. ADLER (Institute of Mathematics, Ufa, Russia) |
| Title: | On the discretizations of the Landau-Lifshitz equation |
| Room: | RIMS, Room 402 |
| Date: | December 4 (Mon), 2000, 10:30-11:30 |
| Speaker: | V. E. ADLER (Institute of Mathematics, Ufa, Russia) |
| Title: | Unified approach to the relativistic Toda type lattices and their discrete analogs |
| Room: | RIMS, Room 009 |
| Date: | November 17 (Wed), 2000, 12:00-13:00 |
| Speaker: | A. A. BELAVIN (Landau Institute for Theoretical Phisics) |
| Title: |
Correspondence between the XXZ model in roots of unity and
the one-dimensional quantum Ising chain with different boundary conditions
We consider the integrable XXZ model with special open boundary conditions that renders its Hamiltonian ${SU(2)}_q$ symmetric, and the one-dimensional quantum Ising model with four different boundary conditions. We show that for each boundary condition the Ising quantum chain is exactly given by the Minimal Model of integrable lattice theory $LM(3, \, 4)$. This last theory is obtained as the result of the quantum group reduction of the XXZ model at anisotropy $\Delta=(q + q^{-1})/2=\sqrt{2}/2$, with a number of sites in the latter defined by the type of boundary conditions. |
| Room: | RIMS, Room 102 |
| Date: | November 16 (Thu), 2000, 14:00-15:00 |
| Speaker: | A. A. BELAVIN (Landau Institute for Theoretical Phisics) |
| Title: |
Truncation of functional relations in the XXZ model
The integrable XXZ model with a special open boundary condition is considered. We study Sklyanin transfer matrices after quantum group reduction in roots of unity. In this case Sklyanin transfer matrices satisfy a closed system of truncated functional equations. The algebraic reason for the truncation is found. The important role in proving of the result is performed by Zamolodchikov algebra . |
| Room: | RIMS, Room 202 |
| Date: | June 22 (Thu), 2000, 10:30-11:30 |
| Speaker: | Evgeni MUKHIN (Berkeley) |
| Title: | Combinatorics of q-characters |
| Room: | Dept. of Math., Kyoto Univ., Room 123 |
| Date: | June 22 (Thu), 2000, 11:45-12:45 |
| Speaker: | Hiraku NAKAJIMA (Kyoto) |
| Title: | Finite dimensional representations of quantum affine algebras |
| Room: | Dept. of Math., Kyoto Univ., Room 123 |
| Date: | May 31 (Wed), 2000, 13:15-14:15 |
| Speaker: | T. BAKER (RIMS) |
| Title: | Rigged configurations for $C^{(1)}_n$ crystals |
| Room: | RIMS, Room 102 |
| Date: | May 31 (Wed), 2000, 14:30-15:30 |
| Speaker: | B. TSYGAN (Pensilvania State University) |
| Title: | Index theorems and formality |
| Room: | RIMS, Room 102 |
| Date: | May 31 (Wed), 2000, 15:45-16:45 |
| Speaker: | B. FEIGIN (RIMS) |
| Title: | Semiinfinite cohomologies for sl2^ and Langlands-type D-modules |
| Room: | RIMS, Room 102 |
| Date: | May 30 (Tue), 2000, 13:15-14:15 |
| Speaker: | Y. PUGAI (RIMS) |
| Title: | Comment on vertex operators |
| Room: | RIMS, Room 202 |
| Date: | May 30 (Tue), 2000, 14:30-15:30 |
| Speaker: | T. UMEDA (Math. Department) |
| Title: | Pfaffian and determinant |
| Room: | RIMS, Room 202 |
| Date: | May 30 (Tue), 2000, 15:45-16:45 |
| Speaker: | T. MIWA (Math. Department) |
| Title: | Combinatorics of coinvariants |
| Room: | RIMS, Room 202 |
| Date: | February 18 (Fri), 2000, 12:00-13:00 |
| Speaker: | Yuji HARA (University of Tokyo) |
| Title: | Correlation functions of the XXZ model with a boundary |
| Room: | RIMS, Room 102 |
| Date: | January 14 (Fri), 2000, 12:00-13:00 |
| Speaker: | Hidetaka SAKAI (Kyoto University) |
| Title: | Rational surfaces and geometry of the Painlev\'e equations |
| Room: | RIMS, Room 102 |