Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA K-theoretic Fulton class
115
0
0.0
(
0
)
Authors
Richard P. Thomas
Ask ChatGPT about the research
No Arabic abstract
Fulton defined classes in the Chow group of a quasi-projective scheme $M$ which reduce to its Chern classes when $M$ is smooth. When $M$ has a perfect obstruction theory, Siebert gave a formula for its virtual cycle in terms of its total Fulton class. We describe K-theory classes on $M$ which reduce to the exterior algebra of differential forms when $M$ is smooth. When $M$ has a perfect obstruction theory, we give a formula for its K-theoretic virtual structure sheaf in terms of these classes.
rate research
Read More
Using raising operators and geometric arguments, we establish formulas for the K-theory classes of degeneracy loci in classical types. We also find new determinantal and Pfaffian expressions for classical cases considered by Giambelli: the loci where a generic matrix drops rank, and where a generic symmetric or skew-symmetric matrix drops rank. In an appendix, we construct a K-theoretic Euler class for even-rank vector bundles with quadratic form, refining the Chow-theoretic class introduced by Edidin and Graham. We also establish a relation between top Chern classes of maximal isotropic subbundles, which is used in proving the type D degeneracy locus formulas.
We show that normalized quantum K-theoretic vertex functions for cotangent bundles of partial flag varieties are the eigenfunctions of quantum trigonometric Ruijsenaars-Schneider (tRS) Hamiltonians. Using recently observed relations between quantum Knizhnik-Zamolodchikov (qKZ) equations and tRS integrable system we derive a nontrivial identity for vertex functions with relative insertions.
Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator $X$ and a nondecreasing function $h$. The family of Hessenberg varieties for regular $X$ is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and $K$-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on $h$. Our formula and our methods are different from a recent result of Abe, Fujita, and Zeng that gives the class of a regular Hessenberg variety with more restrictions on $h$ than here.
This paper aims to focus on Richardson varieties on symplectic groups, especially their combinatorial characterization and defining equations. Schubert varieties and opposite Schubert varieties have profound significance in the study of generalized flag varieties which are not only research objects in algebraic geometry but also ones in representation theory. A more general research object is Richardson variety, which is obtained by the intersection of a Schubert variety and an opposite Schubert variety. The structure of Richardson variety on Grassmannian and its combinatorial characterization are well known, and there are also similar method on quotients of symplectic groups. In the first part of this paper, we calculate the orbit of the symplectic group action, and then rigorously give a method to describe the corresponding quotient by using the nesting subspace sequence of the linear space, i.e. flags. At the same time, the flag is used to describe the Schubert variety and Richardson variety on quotient of symplectic group. The flag varieties of Sp_{2n}(k)/P_d can be viewed as closed subvarieties of Grassmannian. Using the standard monomial theory, we obtain the generators of its ideal, i.e. its defining equations, in homogeneous coordinate ring of Grassmannian. Furthermore, we prove several properties of the type C standard monomial on the symplectic group flag variety. Defining equations of Richardson varieties on Sp_{2n}(k)/P_d are given as well.
Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of convolution operators on the Hilbert space L^2(G/H) associated to test functions. In this paper we present a cohomological interpretation of the trace formula involving the K-theory of the maximal group C*-algebras of G and H. As an application, we exploit the role of group C*-algebras as recipients of higher indices of elliptic differential operators and we obtain the index theoretic version of the Selberg trace formula developed by Barbasch and Moscovici from ours.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
