5,215 research outputs found
On the cohomology of stable map spaces
We describe an approach to calculating the cohomology rings of stable map
spaces. The method we use is due to Akildiz-Carrell and employs a C^*-action
and a vector field which is equivariant with respect to this C^*-action. We
give an explicit description of the big Bialynicky-Birula cell of the
C^*-action on Mbar_00(P^n,d) as a vector bundle on Mbar_0d. This is used to
calculate explicitly the cohomology ring of Mbar_00(P^n,d) in the cases d=2 and
d=3. Of particular interest is the case as n approaches infinity.Comment: 63 page
Osculating Paths and Oscillating Tableaux
The combinatorics of certain osculating lattice paths is studied, and a
relationship with oscillating tableaux is obtained. More specifically, the
paths being considered have fixed start and end points on respectively the
lower and right boundaries of a rectangle in the square lattice, each path can
take only unit steps rightwards or upwards, and two different paths are
permitted to share lattice points, but not to cross or share lattice edges.
Such paths correspond to configurations of the six-vertex model of statistical
mechanics with appropriate boundary conditions, and they include cases which
correspond to alternating sign matrices and various subclasses thereof.
Referring to points of the rectangle through which no or two paths pass as
vacancies or osculations respectively, the case of primary interest is tuples
of paths with a fixed number of vacancies and osculations. It is then shown
that there exist natural bijections which map each such path tuple to a
pair , where is an oscillating tableau of length (i.e., a
sequence of partitions, starting with the empty partition, in which the
Young diagrams of successive partitions differ by a single square), and is
a certain, compatible sequence of weakly increasing positive integers.
Furthermore, each vacancy or osculation of corresponds to a partition in
whose Young diagram is obtained from that of its predecessor by
respectively the addition or deletion of a square. These bijections lead to
enumeration formulae for osculating paths involving sums over oscillating
tableaux.Comment: 65 pages; expanded versio
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