Related papers: Standard monomials and invariant theory for arc sp…

This is the third in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field $K$, we prove the arc space analogue of the first and second fundamental theorems of invariant theory…

This is the second in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field $K$, we construct a standard monomial basis for the arc space of the Pfaffian variety over $K$. As…

Consider the diagonal action of the special orthogonal group on the direct sum of a finite number of copies of the standard representation--the underlying field is assumed to be algebraically closed and of characteristic not equal to two.…

Consider the diagonal action of $SL_n(K)$ on the affine space $X=V^{\oplus m}\oplus (V^*)^{\oplus q}$ where $V=K^n, K$ an algebraically closed field of arbitrary characteristic and $m,q>n$. We construct a "standard monomial" basis for the…

We study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals. We start with the standard generic monomial ideals to find invariants for rational maps of spheres and…

Let $\sigma$ be a simple involution of an algebraic semisimple group $G$ and let $H$ be the subgroup of $G$ of points fixed by $\sigma$. If the restricted root system is of type $A$, $C$ or $BC$ and $G$ is simply connected or if the…

Regular algebraic $K$-theory for groups is a homology theory for discrete groups closely connected (but different from) group homology. It also gives a version of algebraic $K$-theory for rings by the simple functorial mapping assigning to…

We introduce a class of $G$-invariant connections on a homogeneous principal bundle $Q$ over a hermitian symmetric space $M=G/K$. The parameter space carries the structure of normal variety and has a canonical anti-holomorphic involution.…

Let G be a complex semisimple Lie group. The aim of this article is to compare two basis for G-modules, namely the standard monomial basis and the dual canonical basis. In particular, we give a sufficient condition for a standard monomial…

For a closed locally symmetric space M=\Gamma\G/K and a representation of G we consider the push-forward of the fundamental class in the homology of the linear group and a related invariant in algebraic K-theory. We discuss the…

Let $k$ be a field, let $G$ be a reductive algebraic group over $k$, and let $V$ be a linear representation of $G$. Geometric invariant theory involves the study of the $k$-algebra of $G$-invariant polynomials on $V$, and the relation…

In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while…

This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of a linear space. In this article we still work…

A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally…

We use representation theory to construct spaces of matrices of constant rank. These spaces are parametrized by the natural representation of the general linear group or the symplectic group. We present variants of this idea, with more…

This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical…

The First Fundamental Theorem of Invariant Theory describes a minimal generating set of the invariant polynomial ring under the action of some group $G$. In this note we give an elementary and direct proof for the $\operatorname{GL}_2(K)$…

We develop the non-commutative polynomial version of the invariant theory for the quantum general linear supergroup ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$. A non-commutative ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-module superalgebra…

The goal of invariant theory is to find all the generators for the algebra of representations of a group that leave the group invariant. Such generators will be called \emph{basic invariants}. In particular, we set out to find the set of…

A general setting for a standard monomial theory on a multiset is introduced and applied to the Cox ring of a wonderful variety. This gives a degeneration result of the Cox ring to a multicone over a partial flag variety. Further, we deduce…