http://www.math.sinica.edu.tw/chengsj/lie_2010.htm
12/20(一) 12/21(二) 12/22(三) 12/23(四)
0900 畢勒費大學(德) 中國科學院 九州大學
Ringel 席南華 Wakimoto
1000 Cluster algebra Weyl Group Poisson W-algebra
1010 麻省理工 哥倫比亞大學 北卡州立大學
Vogan Deng Lauda 景乃桓
1110 Unitary Repns KLR algebra VOA:Jack poly.
1130 北京理工大學 香港科技大學 倫敦城市大學 中研院
胡峻 朱永昌 Chuang 林正洪
1230 Hecke algebra Loop Groups 對稱群表現 VOA:holo. frame
1400 巴黎第七大學 北京理工大學 維吉尼亞科大
單芃 萬金奎 Li
1430 Hecke algebras sym. algebra 量子群
1440 巴黎第七大學 麻省理工 雪梨大學
Vasserot Lusztig Molev
1540 Hecke algebras Weyl Group 超李代數組合
1610 渥太華大學 維吉尼亞科大 喬治亞大學
Savage Shimozono Nakano
1710 Hecke algebras 代數群 超李代數同倫
http://www.math.sinica.edu.tw/chengsj/lie_2010.htm
以下是各演講摘要:
Joe Chuang (City University London)
Title:
A dual approach to representations of symmetric groups
Abstract:
I will describe how the representation theory of symmetric groups
could be organized around Ariki-Koike algebras arising as algebras of
extensions. This approach is inspired by an interpretation of level-rank
duality as Koszul duality. This is joint work with Hyohe Miyachi.
Jun Hu (Beijing Institute of Technology)
Title:
Morita equivalences of cyclotomic Hecke algebras of type G(r,p,n)
Abstract:
In this talk, we consider the modular representation theory of the
cyclotomic Hecke algebras of type G(r,p,n) when the parameters are (ε,q)
-separated. We set up a Morita equivalence of the cyclotomic Hecke algebras
of type G(r,p,n) and develop a Specht modules theory in that case. We show
that the decomposition numbers of these algebras are completely determined
by the Schur elements and the decomposition numbers of some related smaller
Ariki-Koike Hecke algebras. This is a joint work with Andrew Mathas.
Naihuan Jing (North Carolina State University)
Title:
On vertex operator realization of Jack polynomials
Abstract:
Mimachi and Yamada showed that Jack polynomials of rectangular shapes
are singular vectors for the Virasoro algebra associated with the lattice
vertex algebra of rank one using difference equation. We will discuss vertex
operator realization of Jack polynomials and directly prove that the product
of certain vertex operators are Jack polynomials and then generalize this
result to marked regular shapes. This is joint work with Wuxing Cai.
林正洪 (Academia Sinica)
Title:
Towards the classification of holomorphic framed vertex operator
algebras of central charge 24
Abstract:
In this talk, we discuss our recent work on classification of
holomorphic framed vertex operator algebras of central charge 24 using some
quadratic spaces.
Aaron Lauda (Columbia University)
Title:
Inventing the KLR algebras
Abstract:
I will explain the origins of the diagrammatic KLR algebras that
categorify one half of the quantum enveloping algebra associated to a
symmetrizable Kac-Moody algebra. I will show how generators for these
algebras arise by considering a certain bilinear form on the quantum group.
Relations in the KLR algebra arise from the bilinear form and from
computations involving partial flag varieties. This is joint with Mikhail
Khovanov.
Yiqiang Li (Virginia Tech)
Title:
Geometric realizations of quantum groups
Abstract:
I'll present a geometric construction of quantum groups by using the
geometry of double framed representation varieties of quivers.
George Lusztig (Massachusetts Institute of Technology)
Title:
Unipotent classes and conjugacy classes in the Weyl group
Abstract:
Let G be a connected reductive group over an algebraically closed
field. In this lecture I will describe a map from the set of conjugacy
classes in the Weyl group of G to the set of unipotent classes of G and also
a one sided inverse of this map. The definition of the map involves the study
of intersections of conjugacy classes with Bruhat cells.
Alexander Molev (University of Sydney)
Title:
Combinatorial bases for representations of the Lie superalgebra gl(m|n)
Abstract:
Covariant tensor representations of gl(m|n) occur as irreducible
components of tensor powers of the natural (m+n)-dimensional representation.
We construct a basis of each covariant representation which has the property
that the natural Lie subalgebras gl(m) and gl(n) act in this basis by the
classical Gelfand-Tsetlin formulas. The main role in the construction is
played by the fact that the subspace of gl(m)-highest vectors in any finite-
dimensional irreducible representation of gl(m|n) carries a structure of an
irreducible module over the Yangian Y(gl(n)). One consequence is a new proof
of the character formula for the covariant representations first found by
Berele and Regev and by Sergeev.
Dan Nakano (University of Georgia)
Title:
Cohomological Detection for Lie Superalgebras with Applications to
Support Varieties
Abstract:
In earlier work Boe, Kujawa, and the speaker investigated the
relative cohomology for classical Lie superalgebras and constructed support
varieties for modules over these algebras. The striking feature of these
support varieties was that for irreducible representations over basic
classical Lie superalgebras, the dimensions of the support varieties recover
the combinatorial notions of defect and atypicality defined by Kac and
Wakimoto. In this talk we use invariant theory to develop the notion of
cohomological detection for Type I classical Lie superalgebras. In
particular we show that the cohomology with coefficients in an arbitrary
module can be detected on smaller subalgebras. These results are used later
to affirmatively answer questions, which were originally posed by Bagci, Boe,
Kujawa and the speaker, about realizing support varieties for Lie
superalgebras via rank varieties constructed for the smaller detecting
subalgebras. This is joint work with Gus Lehrer and Ruibin Zhang.
Claus Michael Ringel (Universitt Bielefeld)
Title:
Cluster-concealed algebras
Abstract:
The cluster-tilted algebras have been introduced by Buan, Marsh and
Reiten, they are the endomorphism rings of cluster-tilting objects T in
cluster categories; we call such an algebra cluster-concealed in case T is
obtained from a preprojective tilting module. For example, all representation-
finite cluster-tilted algebras are cluster-concealed. If C is a
representation-finite cluster-tilted algebra, then the indecomposable
C-modules are shown to be determined by their dimension vectors. For a
general cluster-tilted algebra C, we are going to describe the dimension
vectors of the indecomposable C-modules in terms of the root system of a
quadratic form. The roots may have both positive and negative coordinates
and we have to take absolute values.
Alistair Savage (University of Ottawa)
Title:
Hecke algebras and a categorification of the Heisenberg algebra
Abstract:
In this talk, we will present a graphical category in terms of
certain planar braid-like diagrams. The definition of this category is
inspired by the representation theory of Hecke algebras of type A. The
Heisenberg algebra (in infinitely many generators), which plays an
important role in the description of certain quantum mechanical systems,
injects into the Grothendieck group of our category, yielding a
"categorification" of this algebra. We will also see that our graphical
category acts on the category of modules of Hecke algebras and of general
linear groups over finite fields. Additionally, other algebraic structures,
such as the affine Hecke algebra, appear naturally. This is joint work with
Anthony Licata and inspired by work of Mikhail Khovanov.
Peng Shan (Universit de Paris VII)
Title:
Fock spaces and cyclotomic rational double affine Hecke algebras
Abstract:
Recently, Pavel Etingof has given some conjectures on the filtration
of the Grothendieck group of the category O of cyclotomic rational double
affine Hecke algebras by the support of the modules. It yields in particular
the number of finite dimensional modules in O. I will report a proof of his
conjecture for some rational parameters. It uses a categorification of the
action of the affine Lie algebra of glm on the Fock space. This is a joint
work with Eric Vasserot.
Mark Shimozono (Virginia Tech)
Title:
Equivariant homology of the affine Grassmannian and Gromov-Witten
invariants of G/B
Abstract:
We give an explicit rule for the product of the degree two class with
an arbitrary Schubert class in the torus-equivariant homology ring (under
Pontryagin product) of the affine Grassmannian of a simple algebraic group G.
For classical type we also give a rule for the Schubert expansion of the
product of a special Schubert class (the special classes being generators)
with an arbitrary one. For G=SLn the formula is explicit and positive. By a
result of Peterson we obtain explicit formulae for equivariant Gromov-Witten
invariants for flag varieties G/B. This is joint with Thomas Lam.
Eric Vasserot (Universit de Paris VII)
Title:
On affine Hecke algebras of classical types
Abstract:
We shall explain a new combinatorial approach to the representation
theory of affine Hecke algebras of classical types, following a conjecture
of Enomoto-Kashiwara and Kashiwara-Miemietz. The proof uses new analogues of
quiver-Hecke algebras. This is a joint work with Varagnolo and (partially)
with Shan.
David Vogan (Massachusetts Institute of Technology)
Title:
Signatures of Hermitian forms and unitary representations
Abstract:
Suppose G is a real reductive Lie group. According to Gelfand's
philosophy of abstract harmonic analysis, the most basic question in the
representation theory of G is the classification of irreducible unitary
representations: the "unitary dual of G." One way to approach this question
is first to classify the irreducible hermitian representations: those
admitting a nondegenerate but possibly indefinite invariant hermitian form.
Knapp and Zuckerman gave a complete classification of irreducible hermitian
representations in a 1977 paper. What remains is to decide which of these
representations are unitary. I will describe joint work with Jeffrey Adams,
Marc van Leeuwen, Peter Trapa, and Wai Ling Yee on formulating an algorithm
to calculate the signature of the invariant Hermitian form on any irreducible
Hermitian representation. The algorithm uses the Kazhdan-Lusztig conjectures
for real reductive groups, in the refined form proved by Beilinson and
Bernstein ("Jantzen conjecture"). Using this algorithm, one can decide by a
finite calculation whether any particular irreducible representation is
unitary. As I will explain, it follows that the unitary dual for a fixed
reductive group G can be found by a finite calculation. I will also explain
work in progress by Marc van Leeuwen to implement this algorithm on a
computer.
Minoru Wakimoto (Kyushu University)
Title:
Bi-Hamiltonian systems of minimal Poisson W-algebras
Abstract:
As is well known, Poisson vertex algebras provide useful methods
for the study of integrable systems. Very simple examples are the KdV
equation and the Harry-Dym equation, both of which are integrable systems
associated to the Virasoro Poisson vertex algebra. In this talk, we discuss
the Hamiltonian structure on Poisson W-algebras and show that there exist
bi-Hamiltonian systems of KdV-type and of HD-type for any Poisson W-algebras
of nilpotent orbits.
Jinkui Wan (Beijing Institute of Technology)
Title:
Beyond Steinberg module multiplicity in the symmetric algebra
over GLn(q)
Abstract:
We determine the graded composition multiplicity in the symmetric
algebra S(V) of the natural GLn(q)-module V, or equivalently in the
coinvariant algebra of V, for a large class of irreduciblemodules around the
Steinberg module. This was built on a computation, via connections to
algebraic groups, of the Steinberg module multiplicity in a tensor product
of S(V) with other tensor spaces of fundamental weight modules. This is a
joint work with Weiqiang Wang.
Nanhua Xi (Chinese Academy of Sciences)
Title:
Isomorphisms between the group algebra of an affine Weyl group and
and its Hecke algebras
Abstract:
It is known that over complex field the group algebra of a Weyl
group is isomorphic to its generic Hecke algebra. In this talk we will show
that for affine Weyl groups, the conclusion is different. Part of the work
is done jointly with T.Shoji.
Yongchang Zhu (Hong Kong University of Science and Technology)
Title:
Automorphic forms of loop groups
Abstract:
After giving definitions of automorphic forms of loop groups, we
will give examples including Eisenstein series and Theta functions. And we
will discuss the relationship between arithmetic surfaces and automorphic
forms of loop groups.
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