Symmetric power of the Grassmann variety

TL;DR

This paper provides an explicit basis and simplifies the dimension formula for a subspace related to the symmetric power of the Grassmann variety, building on prior algebraic geometry work.

Contribution

It introduces an explicit basis for the subspace and simplifies the existing formula for its dimension, advancing understanding of Grassmann variety symmetric powers.

Findings

Explicit basis of the subspace is constructed.

Dimension formula is significantly simplified.

Connections to previous work by Sharafutdinov and Hodge are clarified.

Abstract

In this work, we find base and dimension of a subspace appeared in works by V. A. Sharafutdinov. The same problem, but expressed in terms of polynomials from matrix minors, was initially solved by W. Hodge. The new result of this paper is the explicit base of the space and considerable simplification of the formula for its dimension.