Download Shanghai Conference on Representation Theory

Shanghai Conference on Representation Theory
þ°L«ØIS¬Æ
December 7 –December 11, 2015
2015c12 7F–12 11F
Organized by
Toshiaki Shoji (Bid²) (Chair, Tongji University)
Susumu Ariki (k7Ñ) (Osaka University)
Hiraku Nakajima (¥
) (RIMS, Kyoto University)
Nanhua Xi (RHu) (Chinese Academy of Sciences)
Hebing Rui (
ÚW) (Harbin Institute of Technology)
Yucai Su (€˜â) (Tongji University)
Qiang Fu (G r) (Tongji University)
Tian He Hotel, Chongming Island, Shanghai, China
¥Iþ°Â²UäŒËA
Speakers:
T. Deshpande (TIFR, Mumbai)
J. Du (Ú #) (U. of New South Wales, Sydney)
Z. Fan (•ëW) (Harbin Engineering U., Harbin)
M. Finkelberg (National Research U., Moscow)
P. Fiebig (U. of Erlangen, Erlangen)
Q. Fu (G r) (Tongji U., Shanghai)
M. Geck (U. of Stuttgart, Stuttgart)
J. Hu (
) (Beijing Institute of Technology, Beijing)
T. Ikeda (³X
) (Okayama U. of Science, Okayama)
O. Iyama (žì?) (Nagoya U., Nagoya)
C. Mautner (U. of California, Riverside)
H. Nakajima (¥
) (Kyoto U., Kyoto)
K. Nishiyama (Üì•) (Aoyama Gakuin U., Tokyo)
M. Okado (—
<) (Osaka City U., Osaka)
F. Qin (Ú ~) (U. Strasbourg, Strasbourg)
Y. Saito (àBÂÈ) (U. of Tokyo, Tokyo)
P. Shan (ü C) (U. Paris-Sud, Orsay)
L. Song (y
) (Shanghai Normal U., Shanghai)
M. Watanabe (ÞÜ ä) (U. of Tokyo, Tokyo)
D. Yang (
À) (Nanjing U., Nanjing)
P. Zhang (Ù â) (Shanghai Jiaotong U., Shanghai)
H. Zheng (x Ó) (Peking U.)
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Schedule
Lecture Room:
Talks from December 7 to 11 are given in Tian He Hotel, The fourth floor
Conference Room, Chongming Island, Shanghai, China.
Schedule of the talks:
Monday, 7 December
9:25–9:30
Opening
9:30–10:20
Meinolf Geck: Modular Harish-Chandra series.
10:20–10:40 Tea Break
10:40–11:30 Tanmay Deshpande: Character sheaves on solvable algebraic
groups.
11:30–14:00 Lunch Break
14:00–14:50 Masato Okado: Tetrahedron equation and generalized quantum
groups.
14:50–15:00 Tea Break
15:00–15:50 Qiang Fu: Quantum affine gln via Hecke algebras.
15:50–16:20 Tea Break
16:20–17:10 Linliang Song: Higher Schur-Weyl duality for cyclotomic walled
Brauer algebras.
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Tuesday, 8 December
9:30–10:20
Osamu Iyama: Lattice structure of preprojective algebras and
Weyl groups.
10:20–10:40 Tea Break
10:40–11:30 Dong Yang: Relative singularity category vs singularity category.
11:30–14:00 Lunch Break
14:00–14:50 Pu Zhang: Infinite ladders of triangulated categories.
14:50–15:00 Tea Break
15:00–15:50 Takeshi Ikeda: Degeneracy loci formulas in K-theory of the symplectic Grassmannian bundles.
15:50–16:20 Tea Break
16:20–17:10 Jun Hu: On involutions in Weyl groups.
Wendesday, 9 December
9:30–10:20
Michael Finkelberg: Kostka-Shoji polynomials.
10:20–10:40 Tea Break
10:40–11:30 Masaki Watanabe: Kraskiewicz-Pragacz modules and some positivity properties of Schubert polynomials.
11:30–13:00 Lunch Break
13:00–18:00 Excursion
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Thursday, 10 December
9:30–10:20
Hiraku Nakajima: Towards a mathematical definition of Coulomb
branches of 3-dimensional N = 4 gauge theories.
10:20–10:40 Tea Break
10:40–11:30 Fan Qin: Monoidal categorifications and triangular bases of quantum cluster algebras.
11:30–14:00 Lunch Break
14:00–14:50 Jie Du: Presenting and extending Hecke endomorphism algebras.
14:50–15:00 Tea Break
15:00–15:50 Carl Mautner : Hypertoric and Matroidal Schur algebras.
15:50–16:20 Tea Break
16:20–17:10 Hao Zheng: Functoriality of Drinfeld Centers.
Friday, 11 December
9:30–10:20
Peter Fiebig : Periodic patterns in representation theory and moment graph theory.
10:20–10:40 Tea Break
10:40–11:30 Peng Shan: Categorification of coideal algebras.
11:30–14:00 Lunch Break
14:00–14:50 Yoshihisa Saito: On Elliptic Artin Groups.
14:50–15:00 Tea Break
15:00–15:50 Zhaobin Fan: Positivity of canonical bases under comultiplication.
15:50–16:20 Tea Break
16:20–17:10 Kyo Nishiyama: Exotic Grassmannian and its nilpotent variety.
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Abstracts
Character sheaves on solvable algebraic groups
Tanmay Deshpande
Let G be an algebraic group over a finite field Fq . The theory of character sheaves is a
geometric approach to the character theory of the finite groups G(Fqn ). In the 1980’s Lusztig
developed a theory of character sheaves on reductive groups and more recently Boyarchenko
and Drinfeld have developed a theory of character sheaves on unipotent groups.
I will begin by introducing the main goals and some conjectures about the theory of
character sheaves on general algebraic groups. I will then describe the theory of character
sheaves on solvable groups and the relationship between character sheaves and irreducible
characters of such groups.
Presenting and extending Hecke endomorphism algebras
Jie Du (Ú #)
Hecke endomorphism algebras are a natural generalisation of q-Schur algebras from symmetric groups to arbitrary Coxeter groups W . Thus, they are certain endomorphism algebras of
some q-permutation modules associated with finite parabolic subgroups of W . In the first
half, we will discuss presentations of these algebras by generators and relations of Hecke
type. The presentation problem was raised by G. Williamson in his thesis and its solution
in the integral case has potential applications in the (singular) Soergel bimodule categorification. In the second half, we will restrict to finite Weyl groups W . By enlarging the
q-permutation module, we will prove that the extended Hecke endomorphism algebra has
nicer properties such as the quasi-heredity and a category equivalence over a local ring of
its module category with the category O for the rational Cherednik algebra associated to
W . Thus, we proved in the local case a conjecture of Du–Parshall–Scott 20 year ago. This
is a report on recent joint works with Jesen–Su and Parshall–Scott.
Positivity of canonical bases under comultiplication
Zhaobin Fan (•ëW)
We show the positivity of the canonical basis for a modified quantum affine sln under the
comultiplication. Moreover, we establish the positivity of the i-canonical basis defined by
Li and Wang with respect to the coideal subalgebra structure. This is a joint work with
Yiqiang Li.
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Kostka-Shoji polynomials
Michael Finkelberg
These polynomials were introduced and studied by Shoji; they are analogues of Kostka polynomials Kλ,µ when λ and µ are r-partitions. I will present a conjectural analogue of the
Lusztig-Kato formula for Kostka-Shoji polynomials and their related geometric interpretation as multiplicities in the space of sections of certain vector bundles over flag varieties. I
will also present another geometric interpretation of their stable version, as Poincare polynomials of IC stalks of a mirabolic version of zastava spaces (for r = 2).
Periodic patterns in representation theory
and moment graph theory
Peter Fiebig
We introduce a filtered version of the category of sheaves on an affine moment graph and
show that the multiplicities of the projective objects yield upper bounds on Jordan–Holder
multiplicities for baby Verma modules over a modular Lie algebra for all prime characteristics
above the Coxeter number. Conjecturally, the multiplicities agree in all cases. We also
discuss the connection to the representation theory of affine Kac-Moody algebras in positive
level, and to parity sheaves on the affine flag manifold. This is joint work with Martina
Lanini.
Quantum affine gln via Hecke algebras
Qiang Fu (G r)
In 1990, Beilinson–Lusztig–MacPherson discovered a realization for quantum gln via a geometric setting of quantum Schur algebras. We will use affine quantum Schur algebras to
realize quantum affine gln and discuss its application. This talk is based on joint works with
Jie Du.
Modular Harish-Chandra series
Meinolf Geck
We survey the current state of knowledge concerning the problem of describing the HarishChandra series of modular representations of finite groups of Lie type in non-describing
characteristic. One of the crucial ingredients is the theory of Hecke algebras with unequal
parameters.
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On involutions in Weyl groups
Jun Hu (
)
This talk is based on recent joint work with Zhang Jing. Let (W, S) be a Coxeter system and
∗ be an automorphism of W with order ≤ 2 such that s∗ ∈ S for any s ∈ S. Let I∗ be the
set of twisted involutions in W . We study the reduced I∗ -expressions of twisted involutions
and the braided I∗ -transformations between them. If W is the Weyl group of type An , Bn
andDn and ∗ = id, we determine all the basic braided I∗ -transformations, and show that
any two reduced I∗ -expressions for a given involution can be transformed into each other
through a series of basic braid I∗ -transformations. As an application, we verify a conjecture
of Lusztig in Type A.
Degeneracy loci formulas in K-theory
of the symplectic Grassmannian bundles
Takeshi Ikeda (³X )
Let E be a vector bundle of rank 2n over a smooth variety X equipped with a symplectic
structure. For 0 ≤ k ≤ n, let SGk (E) denote the Grassmannian bundle that parametrizes
isotropic subbundles S ⊂ E of rank n − k. Fix a flag of isotropic subbundles 0 = F n ⊂ · · · ⊂
F 0 , rank(F i ) = n − i. Let F −i (1 ≤ i ≤ n) be the complement of F i with respect to the
symplectic form. We consider a Schubert variety defined as the locus in SGk (E) where the
prescribed dimensions for the intersections of the fibers of the bundles S and F i . Our aim
is to describe the structure sheaf of the Schubert variety in K(SGk (E)), the K-ring of the
coherent sheaves on SGk (E). We obtained a formula that is written as a sum of Pfaffians
such that each entry is a K-theoretic (relative) Segre classes of the vector bundles involved.
For the Lagrangian case SG0 (E), the formula is reduced to a single Pfaffian generalizing
the Schur Q-function. This is a joint work with Thomas Hudson, Tomoo Matsumura, and
Hiroshi Naruse.
Lattice structure of preprojective algebras and Weyl groups
Osamu Iyama (žì?)
Tilting theory of the preprojective algebra A of an acyclic quiver Q categorifies the corresponding Coxeter group W . When Q is Dynkin, there exists an isomorphism of lattices
between W with the opposite weak order and torsion classes of A (that is, a full subcategory
of mod A closed under factor modules and extensions). This give bijections between joinirreducible elements in W and stones of A (that is, A-modules X satisfying EndA (X) = k
and Ext1A (X, X) = 0). As an application, for type A, we characterize the lattice quotients
of W coming from algebra quotients of A. This is a joint work with N. Reading, I. Reiten
and H. Thomas.
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Hypertoric and Matroidal Schur algebras
Carl Mautner
(joint with Tom Braden) Schur algebras are remarkable finite-dimensional algebras that
interpolate the representation theory of the symmetric groups and the general linear groups
and are most interesting in small characteristics. Motivated by Springer theory and a duality
for symplectic singularities, we introduce a new family of algebras associated to hypertoric
varieties (and more generally matroids). We show that these new algebras share a number
of the nice properties of Schur algebras - in particular, they are quasi-hereditary and behave
well under Ringel duality.
Towards a mathematical definition of Coulomb branches
of 3-dimensional N = 4 gauge theories
Hiraku Nakajima (¥
)
Consider the 3-dimensional N = 4 supersymmetric gauge theory associated with a compact
Lie group G and its quaternionic representation M. Physicists study its Coulomb branch,
which is a noncompact hyper-Kähler manifold, appearing geometric representation theory.
I propose a mathematical definition of the coordinate ring of the Coulomb branch. This is
partly based on a joint work with Braverman, Finkelberg.
Exotic Grassmannian and its nilpotent variety
Kyo Nishiyama (Üì•)
Let V = Cn be a complex vector space and W = V ⊕ V . We are interested in the Grassmannian Grn (W ) of n-dimensional spaces in W . Let us put X = Grn (W ) × V and consider
the diagonal action of K = GL(V ). We call X an exotic Grassmannian (or we may call it
”enhanced” Grassmannian instead). Our goal is to consider a moment map for K-variety
X; to define a conormal variety and an ”exotic” nilpotent variety EX ; and then to establish
a ”Steinberg-Springer” theory for them (cf. Henderson-Trapa (2012)).
In the talk, I will describe the correspondence between the K-orbits in X and exotic nilpotent orbits in EX . We can also play the same game for the ”exotic” Lagrangian
Grassmannian.
This is a joint work in progress with Lucas Fresse.
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Tetrahedron equation and generalized quantum groups
Masato Okado (—
<)
Tetrahedron equation is a 3D analogue of the Yang-Baxter equation. Starting from its known
solutions, R and L, and taking a trace with respect to one vector space, one obtains a large
family of solutions of the Yang-Baxter equation depending on the choice of R or L at each
site. They are then identified with the quantum R matrices for the Hopf algebras known as
generalized quantum groups, which contains an affinization of the quantum superalgebras
associated with sl(m|n). This is a joint work with Atsuo Kuniba and Sergey Sergeev.
Monoidal categorifications and triangular bases
of quantum cluster algebras
Fan Qin (Ú ~)
In this talk, I will give an introduction of the cluster algebras arising from the representation
theory of quantum affine algebras. I will construct a common triangular basis of such a
cluster algebra, which is parametrized by the tropical points. The construction implies
the monoidal categorification conjecture of Hernandez-Leclerc and, in this situation, the
Fock-Goncharov conjecture.
On Elliptic Artin Groups
Yoshihisa Saito (àBÂÈ)
Motivated in study of singularity theory, Kyoji Saito introduced the notion of elliptic root
systems, and studied the corresponding Weyl group (called the elliptic Weyl groups) with
Tadayoshi Takebayashi. Several years ago, the author and Midori Shiota considered the corresponding Hecke algebra (called the elliptic Hecke algebra), and showed they coincide with
Cherednik’s double affine Hecke algebras. In this talk, we try to consider the corresponding
Atrin groups (or braid group), called the elliptic Artin groups. Some basic properties of
these groups will be also discussed.
Categorification of coideal algebras
Peng Shan (ü C)
Certain coideal subalgebras of the quantized enveloping algebra of sln arise naturally as
Schur-Weyl dual to Hecke algebras of type B. They admit a theory of canonical bases.
In this talk, we explain how to categorify the coideal algebras using a 2-category analogue
to Khovanov-Lauda-Rouquier’s categorification of Uq (sln ), and some applications of this
2-category. This is a joint work with Huanchen Bao, Weiqiang Wang and Ben Webster.
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Higher Schur-Weyl duality for
cyclotomic walled Brauer algebras
Linliang Song (y
)
Rui and Su establish the mixed Schur-Weyl duality between general linear Lie super-algebras
and level two walled Brauer algebras. As an application, they give an explicit relationship
between the multiplicity of a Kac module in an indecomposable tilting modules and that
of a simple module in a cell module of level two walled Brauer algebras. This gives an
efficient way to compute decomposition matrices of such level two walled Brauer algebras.
Motivated by Brundan-Kleshchev’s work on higher Schur-Weyl duality, we establish the
mixed Schur-Weyl duality between general linear Lie algebras and cyclotomic walled Brauer
algebras in an arbitrary level. As an application, we determine decomposition matrices of
the cyclotomic walled Brauer algebras arising from mixed Schur-Weyl duality. We remark
that a special case of our result has been given by Sartori, previously. This is a joint work
with Rui.
Kraskiewicz-Pragacz modules and some positivity properties
of Schubert polynomials
Masaki Watanabe (ÞÜ ä)
Schubert polynomials, first introduced by Lascoux and Schutzenberger, are a family of
polynomials describing the Schubert classes in flag varieties. One of the important properties of Schubert polynomials is that they generalize the notion of Schur polynomials. As
well known, Schur polynomials appear as the characters of irreducible representations of
GLn . Generalizing Schur polynomials to Schubert polynomials, Kraskiewicz and Pragacz
introduced certain modules over the upper-triangular matrix group B whose characters are
Schubert polynomials. Motivated by some positivity properties and problems on Schubert
polynomials I studied the class of B-modules having filtrations by KP modules. In this talk
I will talk about my result about KP modules revealing some (old and new) properties of
Schubert polynomials. Time permitting I would also like to talk about my recent study on
Schubert polynomials and KP modules.
Relative singularity category vs singularity category
Dong Yang (
À)
The homological information of a singularity is encoded in the singularity category, a triangulated category introduced by Buchweitz and Orlov. Recently, a new relevant triangulated
category, the singularity category relative to a (non-commutative) resolution, has been introduced by Burban and Kalck. It can be viewed as a categorical version of the exceptional
locus. I will talk about the relationship between the singularity category and the relative
singularity category.
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Infinite ladders of triangulated categories
Pu Zhang (Ù â)
The concept of ladder of a triangulated category is introduced by [BGS] for studying filtrated triangulated category. For the recent studies for example see [AHKLY] and [HQ]. A
recollement in the sense of [BBD] is a ladder of height 1.
I will report that ladders of infinite height occur naturally in representation theory.
First, I will show that any recollement of a triangulated category with Serre functor induces
a ladder of infinite height and of period 1. Thus, the singularity category of a Gorenstein
triangular matrix algebra admits a ladder of infinite height and of period 1. Second, by using
Brown representability theorem and other things, the derived categories of a Gorenstein
triangular matrix algebra admits a ladder of infinite height. This is based on a joint work
Yuehui Zhang and Lin Zhu.
Bibliography
[AHKLY] L. Angeleri Hügel, S. König, Q. H. Liu, D. Yang, Ladders and simplicity of
derived module categories, arXiv: 1310.3479 v2.
[BBD] A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisgue 100,
Soc. Math. France, Paris, 1982.
[BGS] A. A. Beilinson, V. A. Ginsburg, V. V. Schechtman, Koszul duality, J. Geom.
Phys. 5(3)(1988), 317-350.
[HQ] Y. Han, Y. Y. Qin, Reducing homological conjectures by n-recollements, arXiv:
1410.3223v2.
[Z] P. Zhang, Gorenstein-projective modules and symmetric recollements, J. Algebra
388(2013), 65-80.
[ZZZ] P. Zhang, Y. H. Zhang, L. Zhu, Unbounded ladders induced by Gorenstein
algebras, arXiv:1507.07333
Functoriality of Drinfeld Centers
Hao Zheng (x Ó)
It is well known in classical algebras that the notion of center is not functorial unless we
consider only simple algebras. We will show that this rather trivial functoriality of center for
simple algebras becomes highly nontrivial for a categorical analogue of the simple algebra.
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