Peterson varieties are a special class of Hessenberg varieties that have been extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley-Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety Y in ...

We study exponentially asymptotic behaviors for the trajectories of additive functionals of Harris Markov chains. In the main result, we establish a moderate deviation principle for a pair of additive functionals of different growth rates. Moreover, we give explicit formulas for the rate functions which exhibit a non-quadratic behavior. In particular, we achieve the functional moderate deviatio...

The classical Thom–Porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of Schur polynomials. A proof of this formula was given by Pragacz by expressing this alternating sum as the Euler characteristic of a Schur complex, which gives an explanation for the signs. Fulton later generalized this formula to the situa...

LetX be a symplectic or odd orthogonal Grassmannian parametrizing isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in H∗(X,Z) as a polynomial in certain special Schubert classes. We introduce and study theta polynomials, a family of polynomials which are positive linear combination...

This paper develops a new method for studying the cohomology of orthogonal flag varieties. Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to (not necessarily isotropic) flags. They interpolate between Schubert varieties in orthogonal flag varieties and the restrictions of general Schubert varieties in ordinary flag varieties. We give ...

For each infinite series of the classical Lie groups of type B, C or D, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the corresponding flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynom...

Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torus-equivariant cohomology classes of certain determinantal loci in the vector space of n×n complex matrices. Our central result is that the minors defining these “matrix Schubert varieties” are Gröbner bases for any antidiagonal term order. The Schubert polynomials are...

Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torus-equivariant cohomology classes of certain determinantal loci in the vector space of n×n complex matrices. Our central result is that the minors defining these “matrix Schubert varieties” are Gröbner bases for any antidiagonal term order. The Schubert polynomials are...

In this article, we describe explicitely the Gorenstein locus of all minuscule Schubert varieties. This proves a special case of a conjecture of A. Woo and A. Yong [WY06b] on the Gorenstein locus of Schubert varieties.