Related papers: A Giambelli formula for isotropic Grassmannians
We study the torus-equivariant homology $H_*^T(\mathrm{Gr}_G)$ of the affine Grassmannian $\mathrm{Gr}_G$, where $G=\mathrm{Sp}_{2n}(\mathbb{C})$ is the symplectic group. This homology admits a natural ring structure and a Schubert basis,…
We prove that if $\sigma \in S_m$ is a pattern of $w \in S_n$, then we can express the Schubert polynomial $\mathfrak{S}_w$ as a monomial times $\mathfrak{S}_\sigma$ (in reindexed variables) plus a polynomial with nonnegative coefficients.…
In the framework of the study of the $Sp(n)$-orbits in the real Grassmannian $G^\R(k,4n)$ of $k$-dimensional non oriented subspaces of a real $4n$-dimensional vector space $V$, here we consider the case of the isoclinic subspaces whose set…
We show that sigma models with orthogonal and symplectic Grassmannian target spaces admit chiral Gross-Neveu model formulations, thus extending earlier results on unitary Grassmannians. As a first application, we calculate the one-loop…
We show that general isotropic flags for odd-orthogonal and symplectic groups are general for Schubert calculus on the classical Grassmannian in that Schubert cells defined by such flags meet transversally. This strengthens a result of…
We classify all normal Schubert varieties in the affine Grassmannian of a semisimple group over an arbitrary field with special attention to small positive characteristic. The proof is elementary and relies on tangent space calculations for…
We give an algebra-combinatorial constructions of (noncommutative) generating functions of double Schubert and double $\beta$-Grothendieck polynomials corresponding to the full flag varieties associated to the Lie groups of classical types…
We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of orthogonal flag varieties. We use these polynomials to describe the arithmetic…
We develop a theory of Jacobi polynomials for parabolic subgroups of finite reflection groups that specializes to the cases studied by Heckman and Opdam in which the whole group and the trivial group are considered. For the intermediate…
We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special Schubert…
Jacobi-Trudy formula for a generalisation of Schur polynomials related to any sequence of orthogonal polynomials in one variable is given. As a corollary we have Giambelli formula for generalised Schur polynomials.
We introduce two classes of right quaternionic Hilbert spaces in the context of slice polyregular functions, generalizing the so-called slice and full hyperholomorphic Bargmann spaces. Their basic properties are discussed, the explicit…
We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…
We prove a collection of formulas for products of Schubert classes in the quantum $K$-theory ring $QK(X)$ of a cominuscule flag variety $X$. This includes a $K$-theory version of the Seidel representation, stating that the quantum product…
An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special…
We prove the abelian-nonabelian correspondence for quasimap $I$-functions. That is, if $Z$ is an affine l.c.i. variety with an action by a complex reductive group $G$, we prove an explicit formula relating the quasimap $I$-functions of the…
We prove a Chevalley formula for the equivariant quantum multiplication of two Schubert classes in the homogeneous variety X=G/P. As in the case when X is a Grassmannian, studied by the author in a previous paper, this formula implies an…
The Euler characteristic of a very affine variety encodes the algebraic complexity of solving likelihood (or scattering) equations on this variety. We study this quantity for the Grassmannian with $d$ hyperplane sections removed. We provide…
Present notes can be viewed as an attempt to extend the notion of Schubert/Grothendieck polynomial to the context of an arbitrary algebraic oriented cohomology theory and, hence, of a commutative one-dimensional formal group law.
In previous work, we employed a geometric method of Kazarian to prove Pfaffian formulas for a certain class of degeneracy loci in types B, C, and D. Here we refine that approach to obtain formulas for more general loci, including those…