899 research outputs found
Compactified Jacobians and q,t-Catalan Numbers, I
J. Piontkowski described the homology of the Jacobi factor of a plane curve
singularity with one Puiseux pair. We discuss the combinatorial structure of
his answer, in particular, relate it to the bigraded deformation of Catalan
numbers introduced by A. Garsia and M. Haiman.Comment: Revised version, 24 page
Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics
We compute tautological integrals over Quot schemes on curves and surfaces.
After obtaining several explicit formulas over Quot schemes of dimension 0
quotients on curves (and finding a new symmetry), we apply the results to
tautological integrals against the virtual fundamental classes of Quot schemes
of dimension 0 and 1 quotients on surfaces (using also universality, torus
localization, and cosection localization). The virtual Euler characteristics of
Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler
characteristics of the moduli of bundles, is defined and studied. Complete
formulas for the virtual Euler characteristics are found in the case of
dimension 0 quotients on surfaces. Dimension 1 quotients are studied on K3
surfaces and surfaces of general type with connections to the Kawai-Yoshioka
formula and the Seiberg-Witten invariants respectively. The dimension 1 theory
is completely solved for minimal surfaces of general type admitting a
nonsingular canonical curve. Along the way, we find a new connection between
weighted tree counting and multivariate Fuss-Catalan numbers which is of
independent interest
Generalized -Catalan numbers
Recent work of the first author, Negut and Rasmussen, and of Oblomkov and
Rozansky in the context of Khovanov--Rozansky knot homology produces a family
of polynomials in and labeled by integer sequences. These polynomials
can be expressed as equivariant Euler characteristics of certain line bundles
on flag Hilbert schemes. The -Catalan numbers and their rational analogues
are special cases of this construction. In this paper, we give a purely
combinatorial treatment of these polynomials and show that in many cases they
have nonnegative integer coefficients.
For sequences of length at most 4, we prove that these coefficients enumerate
subdiagrams in a certain fixed Young diagram and give an explicit symmetric
chain decomposition of the set of such diagrams. This strengthens results of
Lee, Li and Loehr for rational -Catalan numbers.Comment: 33 pages; v2: fixed typos and included referee comment
Generating random density matrices
We study various methods to generate ensembles of random density matrices of
a fixed size N, obtained by partial trace of pure states on composite systems.
Structured ensembles of random pure states, invariant with respect to local
unitary transformations are introduced. To analyze statistical properties of
quantum entanglement in bi-partite systems we analyze the distribution of
Schmidt coefficients of random pure states. Such a distribution is derived in
the case of a superposition of k random maximally entangled states. For another
ensemble, obtained by performing selective measurements in a maximally
entangled basis on a multi--partite system, we show that this distribution is
given by the Fuss-Catalan law and find the average entanglement entropy. A more
general class of structured ensembles proposed, containing also the case of
Bures, forms an extension of the standard ensemble of structureless random pure
states, described asymptotically, as N \to \infty, by the Marchenko-Pastur
distribution.Comment: 13 pages in latex with 8 figures include
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