International Conference on Representations of Algebras and
Related Topics
April 22 - 27, 2010
Woods Hole, Massachusetts, USA
Titles and Abstracts
Vladimir Bavula, The algebra of polynomial integro-differential operators and its group of automorphisms.
Abstract: Click here
Silvana Bazzoni, Flat Mittag-Leffler modules and approximations.
Abstract: Click here
Frauke Bleher, Deformation rings which are not local complete intersections.
Abstract: This is joint work with Ted Chinburg and Bart de Smit. In the
eighties, Mazur, using work of Schlessinger, introduced techniques of
deformation theory to the study of representations of Galois groups. In
this talk, we give examples of universal deformation rings that are not
local complete intersections, thus providing an answer to a question of
M. Flach in all characteristics.
Jon Carlson, Endotrivial modules over groups with quaternion and semidihedral Sylow 2-subgroups.
Abstract: This is joint work with Nadia Mazza and Jacques Thevenaz. I
will present a survey of recent efforts to classify the endotrivial
module over the group algebras of finite groups and discuss in some
detail two important cases. These are the two cases that are not of
finite representation type, but where the group of endotrivial modules
of the Sylow 2-subgroup has nontrivial torsion elements. One question
is whether these torsion elements lift from the Sylow subgroup to the
group. The constructions use some methods from the theory of support
varieties of modules as well as the technology of almost split sequence.
Andrew Carroll, An approach to semi-invariants of string algebras.
Abstract: We introduce the class of colored string algebras for which
the geometry of the representation spaces can be well-understood.
More precisely, the coordinate rings of the components can be
explicitly decomposed into Schur modules. We investigate the
rings of semi-invariants for particular components, and relate to the
coordinate rings of some toric varieties. This is thesis research
under the guidance of Jerzy Weyman.
Giovanni Cerulli Irelli, Quiver Grassmannians and cluster algebras of Kronecker type.
Abstract: This is joint work with Francesco Esposito. We study
some projective varieties which are closed subvarieties of partial flag
varieties and called quiver Grassmannians. In the last years many
authors have noticed the importance of quiver Grassmannians in the
theory of cluster algebras but many
geometric aspects, like the existence of a cellular decomposition, the
computation of Betti numbers and Euler characteristic are still open.
We make the first step in this direction by studying the "smallest"
interesting quiver Grassmannians, i.e. that ones associated with the
Kronecker quiver. Surprisingly this approach gives a geometric
realization of "canonical basis" of cluster algebras.
Calin Chindris, On the tameness of the representation type of a quiver.
Abstract: Click here
Lars Christensen, Algebras that satisfy Auslander's condition on vanishing of cohomology.
Abstract:
Late in his career, Auslander conjectured that every finitely generated
module M over an artin algebra would have a "latent projective
dimension". That is, a number a(M) such that if the cohomology
of M with coefficients in a finitely generated module N,
i.e. -- Ext(M,N) -- vanishes in high degrees, then it vanishes from
degree a(M).
The conjecture was disproved seven years ago. However, we know that
modules over several classes of algebras do have such a latent
projective dimension, and I will discuss recent work with H. Holm that
aims to improve our understanding of this ring theoretical property. In
the course of the talk, I will also explain why this question, rooted
in the representation theory of Artin algebras, has attracted much
attention in local algebra.
Grégoire Dupont, Geometric bases in affine cluster algebras.
Abstract: In this talk I will explain how to construct linear bases in
acyclic cluster algebras by means of geometric methods in
representation theory of quivers. We will introduce the notion of
generic variables in an arbitrary acyclic cluster algebra and show that
it provides an explicit basis in the affine case. We will also
provide a geometric realization of another kind of bases, putting into
context the results of Sherman-Zelevinsky and Cerulli on ``canonically
positive bases''.
Jennifer Froelich, Universal deformation rings for the symmetric group S5 and its double cover $\hat{S}_5$.
Abstract: This is joint work with Frauke Bleher. In this talk, we will determine the mod 2 representations V of S5 which belong to the principal 2-modular block of S5
and whose stable endomorphism ring is given by scalars when it is
inflated to $\hat{S}_5$. We will show their corresponding
universal deformation rings provide an affirmative answer to a question
raised by Bleher and Chinburg regarding the relationship between the
universal deformation ring of V and the Sylow 2-subgroups of S5 and $\hat{S}_5$, respectively.
Ryan Kinser, Tensor products of quiver representations via quivers over Q.
Abstract: The category of representations of a quiver Q has a natural
"pointwise" tensor product. To get around the lack of a
classification of representations of a general Q, we can study this
tensor product by forming the representation ring of Q (i.e., split
Grothendieck ring) and describing features of this ring (idempotents,
nilpotents, ideals, etc.). We will first survey here what is
known about this ring for various Q (Dynkin types, affine A type,
rooted trees), with some examples, and show how maps from other quivers
into Q can be used to study the tensor product. Then we'll see some new
results, which are joint with Ralf Schiffler, on combinatorial formulas
for tensor products of certain representations of acyclic quivers,
including an explicit formula for tensor product of projectives.
Patrick Le Meur, Crossed-products of Calabi-Yau algebras by finite groups.
Abstract: Click here
Shiping Liu, Representations of an infinite quiver.
Abstract: N/A
Dag Madsen, Filtrations determined by tilting objects of projective dimension two.
Abstract: Click here
Frank Marko, Algebra of supersymmetric polynomials in a positive characteristic.
Abstract: In the case of characteristic zero, generators of the algebra
of supersymmetric polynomials in the characteristic zero case were
described by Stembridge. That result is related to a work of Kantor and
Trishin who described generators of the algebra of invariants of
general linear supergroup. In a joint work with A.N. Grishkov and
A.N.Zubkov we described the generators of the algebra of supersymmetric
polynomials and the algebra of invariants of general linear supergroup
in the case of positive characteristic. In this talk I will explain how
to derive both results.
Hagen Meltzer, Nilpotent operators and stable vector bundle categories.
Abstract: Click here
Gregg Musiker, Positive formulas for loops in a cluster algebra from a surface.
Abstract: In 2009, Ralf Schiffler, Lauren Williams, and the speaker,
provided combinatorial formulas for cluster variables for those cluster
algebras arising from a surface, thus proving their positivity as
Laurent expansions with respect to any seed. This talk will
present recent work in progress with Schiffler and Williams which
generalizes such combinatorial interpretations to other important
elements of the cluster algebra. In particular, we provide graph
theoretic interpretations to the cluster algebra elements associated to
closed loops in the surface thus
establishing that they too have positive Laurent expansions with
respect to any seed. Following work of Fock-Goncharov and work in
progress of Fomin and Thurston, such loops are conjecturally the
elements needed to complete the cluster monomials to a positive basis.
Maria Ines Platzeck, Stratifying systems and proper systems.
Abstract: Click here
Steven Sam, Saturation for classical groups.
Abstract: Let G be a complex special orthogonal or symplectic group,
and let u, v, w be dominant weights for G. We prove that if the
tensor product of the irreducible representations with highest weights
Nu, Nv, and Nw contains a nonzero G-invariant for some N>0, then the
same is true when N=2. This extends some results of Knutson--Tao,
Belkale--Kumar, and Kapovich--Millson. Our techniques are based on and
extend the work of Derksen--Weyman on semi-invariants of quivers to the
case of symmetric quivers, and the work of Schofield on general
representations of quivers to the case of quivers with relations of
global dimension 2.
Ralf Schiffler, Cluster automorphisms.
Abstract: This talk is on a joint work with Assem and Shramchenko, in
which we introduce and study cluster automorphisms of cluster algebras.
These are automorphisms of the algebra which preserve the combinatorial
cluster algebra structure. We compute the group of cluster
automorphisms for Dynkin and Euclidean types using cluster categories
and Riemann surfaces with marked points.
Hugh Thomas, A combinatorial approach to B-matrices for cluster algebras.
Abstract: Let Q be a quiver without oriented cycles, and let T be a
cluster tilting object in the cluster category of Q. From a
representation-theoretic point of view, it is well-understood that the
B-matrix of the corresponding cluster is given by the Gabriel quiver of
T. In this talk I will present an alternative approach to finding
the B-matrix, which is more combinatorially explicit. This is
joint work with David Speyer.
Peter Tingley, Braidings, commutors, and quiver varieties.
Abstract: Consider the category of crystals associated to a
symmetrizable Kac-Moody algebra g. If g is of finite type, Henriques
and Kamnitzer constructed a system of isomorphisms between B
⊗ C and C ⊗ B for
all pairs of crystals B and C, endowing crystals with the structure of
a coboundary category. In joint work with Kamnitzer,
we explained how this system of isomorphisms is related to the braiding
on the corresponding category of representations. The situation outside
of finite type is less well understood. As shown by Alistair Savage,
crystals still form a coboundary category. However, the relationship
with the braiding has not yet been explained in
general. We will review Savage's proof of the coboundary
structure (a geometric argument using quiver varieties), and discuss
some attempts to complete the picture by including the braiding.
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