709 research outputs found
Ampleness of normal bundles of base cycles in flag domains
Flag domains are open orbits of noncompact real forms of complex semisimple
Lie groups acting on flag manifolds. To each flag domain one can associate a
compact complex manifold called the base cycle. The ampleness of the normal
bundle of the base cycle in a flag domain measures the concavity near the base
cycle. In this paper we compute the ampleness of normal bundles of base cycles
in flag domains in various cases, including flag domains in the full flag
manifolds when is classical, and period domains parameterizing
polarized Hodge structures with fixed Hodge numbers.Comment: 27 page
Positivity of chromatic symmetric functions associated with Hessenberg functions of bounce number 3
We give a proof of Stanley-Stembridge conjecture on chromatic symmetric
functions for the class of all unit interval graphs with independence number 3.
That is, we show that the chromatic symmetric function of the incomparability
graph of a unit interval order in which the length of a chain is at most 3 is
positively expanded as a linear sum of elementary symmetric functions.Comment: 32 page
Prolongations, invariants, and fundamental identities of geometric structures
Working in the framework of nilpotent geometry, we give a unified scheme for
the equivalence problem of geometric structures which extends and integrates
the earlier works by Cartan, Singer-Sternberg, Tanaka, and Morimoto. By giving
a new formulation of the higher order geometric structures and the universal
frame bundles, we reconstruct the step prolongation of Singer-Sternberg and
Tanaka. We then investigate the structure function of the complete
step prolongation of a normal geometric structure by expanding it into
components and establish the fundamental
identities for , , . This then enables us to study the
equivalence problem of geometric structures in full generality and to extend
applications largely to the geometric structures which have not necessarily
Cartan connections.
Among all we give an algorithm to construct a complete system of invariants
for any higher order normal geometric structure of constant symbol by making
use of generalized Spencer cohomology group associated to the symbol of the
geometric structure. We then discuss thoroughly the equivalence problem for
geometric structure in both cases of infinite and finite type. We also give a
characterization of the Cartan connections by means of the structure function
and make clear where the Cartan connections are placed in the
perspective of the step prolongations
Bases of the equivariant cohomologies of regular semisimple Hessenberg varieties
We consider bases for the cohomology space of regular semisimple Hessenberg
varieties, consisting of the classes that naturally arise from the
Bialynicki-Birula decomposition of the Hessenberg varieties. We give an
explicit combinatorial description of the support of each class, which enables
us to compute the symmetric group actions on the classes in our bases. We then
successfully apply the results to the permutohedral varieties to explicitly
write down each class and to construct permutation submodules that constitute
summands of a decomposition of cohomology space of each degree. This resolves
the problem posed by Stembridge on the geometric construction of permutation
module decomposition and also the conjecture posed by Chow on the construction
of bases for the equivariant cohomology spaces of permutohedral varieties
- …