Emilie Dufresne Well-behaved separating algebras
Michael Ehrig MV-polytopes via affine buildings
Lucas Fresse Springer fibers admitting singular
components
Natalia Gherstega The solution of a Differential
System Given by Lie Algebras
Geordie Williamson Parity sheaves
Oded Yacobi An analysis of the multiplicity spaces
in classical symplectic branching
Vladimir Zhgoon On local structure theorem and
equivariant geometry of cotangent vector bundle
Abstracts
Emilie Dufresne Well-behaved separating algebras
The study of separating invariants has become quite popular
in the recent years. For a finite group acting on a finite dimensional vector
space, a separating algebra is a subalgebra of the ring of invariants which
separates the orbits. In this talk, we prove that there can exist polynomial
separating algebras only when the group is generated by reflections. We thus
generalize the classical result of Serre that only reflection groups may have a
polynomial ring of invariants. We also show that graded separating algebras can
be complete intersection only when the groups is generated by bireflections. We
end with results concerning the Cohen-Mcaulay property of separating algebras.
Michael
Ehrig MV-polytopes via affine buildings
We
give a construction of MV-polytopes of a complex semisimple algebraic group G
in terms of the geometry of the Bott-Samelson variety and the affine building.
This is done by using the construction of dense subsets of MV-cycles by
Gaussent and Littelmann. They used LS-gallery to define subsets in the
Bott-Samelson variety that map to subsets of the affine Grassmannian, whose
closure are MV-cycles. Since points in the Bott-Samelson variety correspond to
galleries in the affine building one can look at the image of a point in such a
special subset under all retractions at infinity. We prove that these images
can be used to construct the corresponding MV-polytope in an explicit way, by
using the GGMS strata. Furthermore we give a combinatorial construction for these
images by using the crystal structure of LS-galleries and the action of the
ordinary Weyl group on the coweight lattice.
Lucas
Fresse Springer fibers admitting singular components
Consider
a nilpotent endomorphism u of a finite dimensional vector space. The set of all
u-stable complete flags is an algebraic
projective
variety, called Springer fiber. This variety is reducible in general, and we
study the question of the singularity of its irreducible
components.
Specifically, we characterize the Jordan shapes for which every component in
the Springer fiber is nonsingular.
This
is joint work with Anna Melnikov.
Natalia Gherstega The solution of a Differential
System Given by Lie Algebras
Geordie
Williamson. Parity sheaves
Joint
work with Daniel Juteau and Carl Mautner
Modular
perverse sheaves are perverse sheaves on complex algebraic varieties with
coefficients in a field of positive
characteristic.
I will briefly discuss some relations (due to Soergel, Mirkovic-Vilonen, Fiebig
and Juteau) between categories of modular
perverse
sheaves and representation theory. I will then describe recent work in which we
introduce a new class of sheaves on certain stratified varieties which we call
parity sheaves. On the affine Grassmannian they correspond under geometric
Satake to tilting modules (except possibly in some small characteristics). I
will describe some of their properties, in particular how they allow one to
understand the failure of the decomposition theorem.
Oded
Yacobi An analysis of the multiplicity spaces in classical symplectic
branching
We
develop a new approach to Gelfand-Zeitlin theory for the symplectic group. Classical Gelfand-Zeitlin theory rests
on the fact that branching from GL(n,C) to GL(n-1,C) is multiplicity-free. Since branching from Sp(n,C) to
Sp(n-1,C) is not multiplicity-free the methods of Gelfand and Zeitlin cannot be
directly applied in this case.
Let L be the n-fold product of SL(2,C). Our main theorem asserts that each
multiplicity space that arises in the restriction of an irreducible
representation of Sp(n,C) to Sp(n-1,C) has a unique irreducible L-action
satisfying certain naturality conditions.
As an application we obtain a Gelfand-Zeitlin type basis for all
irreducible finite-dimensional representaions of Sp(n,C)
Vladimir Zhgoon On local structure theorem and
equivariant geometry of cotangent vector bundle
Let G be
a connected reductive group acting on an irreducible normal algebraic variety X. In this talk we discuss various
results describing an action of a certain parabolic subgroup of G on the open subset of X. Such results are usually called
”local structure theorems”. First results of that kind were discovered by
Grosshans, and independently by Brion, Luna and Vust. We should mention that this
theorem was improved by F.Knop. Later a refined version of this theorem was
obtained by D.A.Timashev. The aim of
this talk is to give a generalization of the result of
D.A.Timashev, mentioned above.
Let Q be
a parabolic subgroup that is a stabilizer of prime components for divisors of B-semiinvariant rational functions.
And let P be a
stabilizer of all prime B-invariant divisors. With the help of minimal Q-equivariant linear systems we
construct the mapping from the open subset of X onto the flag variety Q=P. This mapping allows us to derive
the stronger version of local structure theorem for non-quasiaffine varieties.
Last update: July 2009