644 research outputs found
Perverse sheaves, Koszul IC-modules, and the quiver for the category O
For a stratified topological space we introduce the category of IC-modules,
which are linear algebra devices with the relations described by the equation
d^2=0. We prove that the category of (mixed) IC-modules is equivalent to the
category of (mixed) perverse sheaves for flag varieties. As an application, we
describe an algorithm calculating the quiver underlying the BGG category O for
arbitrary simple Lie algebra, thus answering a question which goes back to I.
M. Gelfand.Comment: NEW: final version to appear in Inventiones Mathematicae. Some
exposition revisions. Quiver computed explicitly for A_2 in the appendix. 29
page
Hurwitz numbers and intersections on moduli spaces of curves
This article is an extended version of preprint math.AG/9902104. We find an
explicit formula for the number of topologically different ramified coverings
of a sphere by a genus g surface with only one complicated branching point in
terms of Hodge integrals over the moduli space of genus g curves with marked
points.Comment: 30 pages (AMSTeX). Minor typos are correcte
Signatures of partition functions and their complexity reduction through the KP II equation
A statistical amoeba arises from a real-valued partition function when the
positivity condition for pre-exponential terms is relaxed, and families of
signatures are taken into account. This notion lets us explore special types of
constraints when we focus on those signatures that preserve particular
properties. Specifically, we look at sums of determinantal type, and main
attention is paid to a distinguished class of soliton solutions of the
Kadomtsev-Petviashvili (KP) II equation. A characterization of the signatures
preserving the determinantal form, as well as the signatures compatible with
the KP II equation, is provided: both of them are reduced to choices of signs
for columns and rows of a coefficient matrix, and they satisfy the whole KP
hierarchy. Interpretations in term of information-theoretic properties,
geometric characteristics, and the relation with tropical limits are discussed.Comment: 42 pages, 11 figures. Section 7.1 has been added, the organization of
the paper has been change
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