| Stanley's convolution and Koszul duality
for dual affine toric varieties
| Tom Braden*
(U Mass Amherst)
Valery Lunts
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Barthel-Brasselet-Fieseler-Kaup and Bressler-Lunts defined combinatorial
sheaves on fans which model intersection cohomology
on toric varieties (because of a result of Karu, this even makes sense for
non-rational fans, when there is no toric variety). We use complexes of these
sheaves to study more general perverse sheaves on toric varieties. Stanley's
convolution identity for g-polynomials has a natural interpretation in this
language: the multiplicity of simple objects in "standard perverse sheaves" on
a toric variety X are given by intersection cohomology betti numbers on a
dual toric variety Y.
This can be "lifted" from a numerical statement to a canonical, functorial
correspondence between sheaves on X and Y. This is exactly analogous to
the "Koszul duality" defined by Beilinson-Ginzburg-Soergel for
Schubert-constructible sheaves on flag varieties, which lifted a convolution
identity on Kazhdan-Lusztig polynomials analogous to Stanley's.
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Singularities of Schubert Varieties, Tangent Cones and Bruhat Graphs
| Jim Carrell*
(UBC)
Jochen Kuttler
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Let G be a semi-simple algebraic group over C,
and let X denote a Schubert variety in the flag variety G/B.
By a result of Dale Peterson, the Bruhat graph associated to X determines the
singular locus of X if G is
of type ADE. We will describe a generalization
of Peterson's result which gives a non-recursive algorithm
for finding the singular locus of X for any G in
terms of its Bruhat graph. The extra information required is a knowledge of the
length of an edge and the angle between edges.
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Quivers, long exact sequences and Horn type inequalities
| Calin Ioan Chindris
(Michigan, Ann Arbor)
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Horn's conjecture gives a recursive method for finding the eigenvalues of a sum
of two Hermitian matrices in terms of the eigenvalues of the summands. Other
problems that have the exact same solution as this eigenvalue problem concern:
the existence of short exact sequences of finite abelian p-groups and the
non-vanishing of the Littlewood-Richardson coefficients. We use methods from
quiver invariant theory to find necessary and sufficient inequalities for the
existence of long exact sequences of finite abelian p-groups. We also relate
this result with some generalized Littlewood-Richardson coefficients and
eigenvalues of Hermitian matrices satisfying some (in)equalities.
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| Quivers and Combinatorics
| Harm Derksen*
(Michigan, Ann Arbor)
Jerzy Weyman
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We will discuss various applications of the theory
of quiver representations to combinatorics.
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A combinatorial approach to describing the Mori Cone of the moduli
space of curves
| Angela Gibney*
(U Penn)
Diane Maclagan
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| We have reformulated a conjectural description of the cone of
curves on the moduli space of curves in an entirely combinatorial way.
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Almost-commuting variety, D-modules, and Cherednik Algebras
| Wee Liang Gan,
Victor Ginzburg*
(Chicago)
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We study a scheme M closely related to the
set of pairs of n by n matrices with rank 1 commutator.
We show that M is a reduced complete intersection with
n+1 irreducible components, which we describe.
There is a distinguished Lagrangian subvariety
N in M. We introduce a category, C,
of D-modules whose characteristic variety is contained in N.
Simple objects of that category are
analogous to Lusztig's character sheaves.
We construct a functor of Quantum Hamiltonian
reduction from category C to the
category O for type A rational Cherednik algebra.
It will be shown in a subsequent
paper that this functor is close to being an equivalence.
Thus, representation theory of rational Cherednik
algebras turns out to be `governed' by the theory of D-modules.
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Syzygies, multigraded regularity and toric varieties.
| Milena Hering*
(Michigan, Ann Arbor)
Hal Schenck
Greg Smith
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We study the equations defining a projective variety and the higher syzygies
between them using multigraded regularity as introduced by Maclagan and Smith.
As an application, we obtain a sufficient condition for the power of an ample
line bundle on a toric variety guaranteeing that the corresponding embedded
variety is projectively normal and generated by quadratic equations, and that
the first p syzygies are linear. This technique also yields new results for the
syzygies of Veronese-Segre embeddings. This is joint work with H. Schenck and
G. Smith.
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| Mirkovic-Vilonen cycles and polytopes
| Joel Kamnitzer
(UC Berkeley)
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Mirkovic-Vilonen showed that certain subvarieties of the affine Grassmannian
give bases for representations of complex semisimple groups. We will explain
recent work to give an explicit combinatorial description of these MV cycles
and their moment map images.
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Equivariant tableaux and a generalization of the
Robinson-Schensted-Knuth correspondence
| Victor Kreiman
(Virginia Tech)
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I will describe a set of tableaux which are in bijection with Knutson and Tao's
equivariant puzzles and which thus compute positive equivariant
Littlewood-Richardson coefficients. I will also describe a generalization of
the Robinson-Schensted-Knuth correspondence and show how it can be used to
prove a result of Kreiman-Lakshmibai and Kodiyalam-Raghavan which gives a
Grobner basis for the tangent cone at any point of a Schubert variety in the
Grassmannian.
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| Equivariant Schubert Classes
| V Lakshmibai*
(Northeastern),
K N Raghavan,
P Sankaran
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We compute a Schubert class as an element in the T-equivariant cohomology ring
of the Grassmannian variety. As a consequence, we obtain an equivariant
Giambelli formula expressing an equivariant Schubert class as a determinant
whose entries involve the equivariant classes of the special Schubert
varieties. The equivariant Giambelli formula specializes to the usual Giambelli
formula.
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Polyhedral geometry in the McKay correspondence
| Diane Maclagan*
(Rutgers University),
| Alistair Craw,
| Rekha Thomas
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When $G \subseteq SL(3,\mathbb C)$ the moduli space $M_{\theta}$ of
representations of the McKay quiver is a crepant resolution of the quotient
singularity $\mathbb C^n/G$. We give an explicit description of the component
of $M_{\theta}$ that is birational to $\mathbb C^n/G$ for abelian $G \subseteq
GL(n,\mathbb C)$ for arbitrary $n$ as a (not necessarily normal) toric variety.
This gives rise to an algorithm to determine for which $\theta$ the moduli
space $M_{\theta}$ is a crepant resolution of $\mathbb C^n/G$. This is joint
work with Alastair Craw (Stony Brook) and Rekha Thomas (Washington).
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| On Demazure modules of affine SL(n)
| Peter Magyar*
(Michigan State University),
| V. Kreiman,
| V. Lakshmibai,
| J. Weyman,
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Joint with V. Kreiman, V. Lakshmibai, J. Weyman.
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Horn systems and binomial ideals
| Laura Matusevich*
(University of Pennsylvania),
| | Alicia Dickenstein
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We show how the primary components of a binomial complete intersection
contribute to the formation of solutions of the associated Horn system of
differential equations.
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Cohomology of rational forms on toric varieties
| Anvar Mavlyutov*
(Oklahoma State University),
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We will show how to compute cohomology $H^k(P,\Omega_P^m(X))$ of differential
forms with simple poles along a semiample divisor $X$
on a complete simplicial toric variety $P$. The calculation is reduced to
finding the cohomology of an Ishida complex and we also get a dimension formula
for these groups in terms of the combinatorics of the fan, which generalizes
Bott's formula in the case of a projective space. The Bott vanishing theorem is
generalized in the toric case for semiample divisors. The cohomology of
rational forms is necessary for the description of the cohomology of semiample
quasismooth hypersurfaces in toric varieties.
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| Factorial Schur functions represent the
equivariant quantum
Schubert classes
| Leonardo Mihalcea
(University of Michigan)
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The (small) equivariant quantum cohomology (eq.q.coh.) of a
homogeneous variety X=G/P is an algebra which is a deformation of
both equivariant and quantum cohomology algebras of X. It was
introduced by A. Givental and B. Kim primarily to study the
quantum cohomology of X.
The eq.q.coh. has a distinguished basis determined by the Schubert
classes of X. The purpose of this talk is to show that with
respect to a certain presentation of the eq.q.coh. of the
Grassmannian, the Schubert classes are given by the factorial
Schur functions.
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A combinatorial approach to describing the Mori Cone of the moduli
space of curves
| Alexei Oblomkov*
(MIT),
Pavel Etingof,
Eric Rains
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To star-shaped simply laced affine Dynkin diagram $D$ one can use
a standard procedure to attach a crystallographic group $G$. We
define a flat deformation $H(t,q)$ of the group algebra
$\mathbb{C}[G]$. If $D=D_4$, then $H(t,q)$ is the double affine
Hecke algebra of rank $1$. We prove that $H(t,q)$ is the universal
deformation of the twisted group algebra of $G$, and that this
deformation is compatible with certain filtrations on $C[G]$. If
$q$ is a root of unity, then for generic $t$ the algebra $H(t,q)$
is an Azumaya algebra, and its center is the function algebra on
an affine del Pezzo surface. For generic $q$, the spherical
subalgebra $eH(t,q)e$ provides a quantization of such surfaces.
Talk is based on the joint paper with P. Etingof and E.Rains.
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| Ehrhart polynomials and stringy Betti numbers
| Sam Payne*
(University of Michigan),
| | Mircea Mustata
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We show that the coefficients of the numerator of a rational function
appearing as the generating function of the Ehrhart polynomial of a
reflexive polytope are the stringy Betti numbers of the projective toric
variety corresponding to the dual reflexive polytope. Using this fact,
along with a theorem of Yasuda relating stringy Betti numbers to orbifold
cohomology and results of Borisov, Chen, and Smith on the orbifold
cohomology of toric varieties, we give a formula for the $\delta$-vector
of a reflexive polytope (i.e.\ the coefficients of the numerator of the
Ehrhart generating function) as a positive linear combination of shifted
$h$-vectors of simplicial polytopes. We also give counterexamples to
Hibi's conjecture on the unimodality of $\delta$-vectors.
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| All the GIT quotients at once
| Nicholas J Proudfoot
(UT Austin)
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Let V be a smooth algebraic variety, and let T be an algebraic torus acting on
V. For any choice of ample equivariant line bundle L on V, one can define the
GIT quotient of (V,L) by T, and the topology of the quotient depends on L. I
will define the algebraic symplectic quotient of the cotangent bundle of V by
T, which may be thought of as a simultaneous complexification of each of the
GIT quotients. To study it is like studying all of the GIT quotients at once.
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| Geometry of the Horn recursion
| Kevin Purbhoo*
(University of British Columbia),
| | Frank Sottile
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I will discuss some of the geometry behind a generalisation of
the Horn recursion, a strange looking recursive method for determining which of
the Littlewood-Richardson numbers are non-zero. Our methods apply to all
minuscule flag varieties, however there is some hope of pushing the techniques
slightly beyond the minuscule case.
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| Tropical Compactifications
| Jenia Tevelev
(UT Austin)
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We study compactifications of very affine varieties defined by imposing a
polyhedral structure on the non-archimedean amoeba. These compactifications
have divisorial boundary with combinatorial normal crossings. We consider some
examples including
M_{0,n} \subset {\bar M}_{0,n} (and more generally log canonical models of
complements of hyperplane arrangements) and tropical recompactifications of
Chow quotients of Grassmannians.
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| How can flags meet?
| Ravi Vakil*
(Stanford),
Sara Billey
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How can m flags meet? In other words, given m flags in n-space, what are
the possible dimensions of intersections of pieces of the flags? When m=2,
we are led to Schubert varieties, which are beautiful in all ways. Hence for
general m, we are led to the natural generalization of Schubert varieties.
When m=3, the story is nice as well (and the generalized Schubert varieties
are too). However, for general m, they are terribly behaved --- they are not
irreducible, reduced, or equidimensional, and have nasty singularities. We
also give counterexamples to Eriksson and Linusson's Realizability Conjecture
identifying which of these generalized Schubert varieties is non-empty. Even
for four flags, we don't even know what intersection dimensions are
achievable!
This is part of a larger project with Sara Billey.
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Kazhdan-Lusztig polynomials for hypertoric varieties
| Ben Webster*
(UC Berkeley),
| Nick Proudfoot
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Hypertoric varieties are hyperkahler analogues of toric varieties. The
cohomology of smooth hypertoric varieties is well understood in terms of
invariants of hyperplane arrangments, but for singular hypertoric varieties,
the topological cohomology is less interesting. However, an analogue of
Kazhdan-Lusztig polynomials has a more subtle interpretation in the
combinatorics of matroids, via broken circuit complexes. Joint with N.
Proudfoot.
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| Gorenstein Schubert varieties
| Alexander K. Woo*
(UC Berkeley),
Alexander Yong
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We give a new combinatorial characterization for which
Schubert varieties (for SL(n)) are Gorenstein. Gorenstein-ness is a
technical algebro-geometric condition which is a weakening of
smoothness. In analogy with the now classic result of Lakshmibai and
Sandhya characterizing smooth Schubert varieties in terms of pattern
avoidance conditions, our results are in terms of a generalization of
pattern avoidance which we describe. When combined with other
geometric results, our results are further evidence that the
singularity structure of Schubert varieties is not too complicated.
If time permits we will also describe some open problems and ongoing work.
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Dual Schubert cells in full flag varieties and Poisson geometry
| Milen Yakimov*
(UCSB),
| | K. A. Brown,
| | K. R. Goodearl
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The affine space of $m \times n$ complex matrices possesses a complex
algebraic Poisson structure, invariant under the natural action of an
$m+n$ dimensional complex torus T, which recently appeared in works in
ring theory and cluster algebras. We will show that the T-orbits of
symplectic leaves of this Poisson structure are smooth, connected
locally closed subsets which are biregularly isomorphic to
intersections of dual Schubert cells. We will further identify the
poset of such orbits of leaves (under the inclusion of closures
relation) with a subset of $S_{m+n}$ of permutations under the Bruhat
order (e.g for $m=n$ this is the subset of permutations that move
indices by at most n position, with the inverse Bruhat order). We will
also describe explicit determinantal formulas for the ideals defining
the closures of orbits. At the end we will discuss a complex algebraic
Poisson structure on the full flag variety of any complex simple Lie
group whose torus orbits of symplectic leaves are exactly all
intersections of dual Schubert cells.
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