ASL structures of some quadrics

TL;DR

This paper investigates the algebraic structure of ideals generated by 1x1 minors of matrix products, demonstrating that certain quotient rings possess an algebra with a straightening law (ASL) structure.

Contribution

It establishes that quotient rings formed from ideals of 1x1 minors of matrix products have an ASL structure under specific conditions.

Findings

Proves quotient rings have an ASL structure.

Identifies conditions on matrices for ASL structure.

Enhances understanding of algebraic properties of matrix minors.

Abstract

Let $K$ be a field and $X$, $Y$ denote matrices such that, the entries of $X$ are either indeterminates over $K$ or $0$ and the entries of $Y$ are indeterminates over $K$ which are different from those appearing in $X$. We consider ideals of the form $I_{1}(XY)$, which is the ideal generated by the $1\times 1$ minors of the matrix $XY$. We prove that the quotient ring $K[X, Y]/I_{1}(XY)$ admits an ASL structure for certain $X$ and $Y$.