739 research outputs found
Closure relations for totally nonnegative cells in G/P
The totally nonnegative part of a partial flag variety G/P is known to have a
decomposition into semi-algebraic cells. We show that the closure of a cell is
again a union of cells and give a combinatorial description of the closure
relations. The totally nonnegative cells are defined by intersecting the
totally nonnegative part with a certain stratification of G/P defined by
Lusztig. We also verify the same closure relations for these strata.Comment: 12 pages, 1 figure, improved reference
A mirror construction for the totally nonnegative part of the Peterson variety
We explain how A. Givental's mirror symmetric family to the type A flag
variety and its proposed generalization to partial flag varieties by Batyrev,
Ciocan-Fontanine, Kim and van Straten relate to the Peterson variety Y in
SL_n/B. We then use this theory to describe the totally nonnegative part of Y.Comment: Now published, corrected minor typo in the introduction. An early
version appeared as ESI preprint 1674; 27 pages, 2 figure
A mirror symmetric solution to the quantum Toda lattice
We use representation theory to construct integral formulas for solutions to
the quantum Toda lattice in general type. This result generalizes work of
Givental for SL(n)/B in a uniform way to arbitrary type and can be interpreted
as a kind of mirror theorem for the full flag variety G/B. We also prove the
existence of a totally positive critical point of the 'superpotential' in every
mirror fiber.Comment: 25 pages, various improvements including to references, to appear in
CM
An introduction to perverse sheaves
These notes aim to give a first introduction to intersection cohomology and
perverse sheaves with applications to representation theory or quantum groups
in mind.Comment: 39 pages, to appear in Proceedings of the ICRA
Discrete Morse theory for totally non-negative flag varieties
In a seminal 1994 paper, Lusztig extended the theory of total positivity by
introducing the totally non-negative part (G/P)_{\geq 0} of an arbitrary
(generalized, partial) flag variety G/P. He referred to this space as a
"remarkable polyhedral subspace", and conjectured a decomposition into cells,
which was subsequently proven by the first author. Subsequently the second
author made the concrete conjecture that this cell decomposed space is the next
best thing to a polyhedron, by conjecturing it to be a regular CW complex that
is homeomorphic to a closed ball. In this article we use discrete Morse theory
to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that
the boundaries of the cells are homotopic to spheres, and the closures of cells
are contractible. The latter part generalizes a result of Lusztig's that
(G/P)_{\geq 0} -- the closure of the top-dimensional cell -- is contractible.
Concerning our result on the boundaries of cells, even the special case that
the boundary of the top-dimensional cell (G/P)_{> 0} is homotopic to a sphere,
is new for all G/P other than projective space.Comment: 30 page
A comparison of Landau-Ginzburg models for odd-dimensional Quadrics
In [Rie08], the second author defined a Landau-Ginzburg model for homogeneous spaces G/P, as a regular function on an affine subvariety of the Langlands dual group. In this paper, we reformulate this LG model (X^, W_t) in the case of the odd-dimensional quadric, as a rational function on a Langlands dual projective space, in the spirit of work by R. Marsh and the second author for type A Grassmannians and by both authors for Lagrangian Grassmannians. We also compare this LG model with the one obtained independently by Gorbounov and Smirnov, and we use this comparison to deduce part of a conjecture of the second author for odd-dimensional quadrics
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