Dual instability measures of a subspace of $P^{n}(K)$ under a subgroup of $\operatorname{Aut}(K)$

TL;DR

This paper introduces dual instability measures for subspaces of projective space under a subgroup of automorphisms, quantifying how the dimension of joins and meets varies under the group action using Plücker coordinates.

Contribution

It develops new dual irrationality measures for subspaces, linking their instability under group actions to Plücker coordinates and invariant fields.

Findings

Defines dual instability measures based on join and meet dimensions.

Expresses measures in terms of Plücker coordinates and invariant fields.

Provides a framework for quantifying subspace instability under automorphism groups.

Abstract

Let $K$ be a commutative field and let $V$ be a subspace of $P^{n}(K)$. Let $\Gamma$ be a subgroup of $\operatorname{Aut}(K)$ and let $\Gamma$ act on $P^{n}(K)$ by $\sigma((x_{i})_{0\leq i\leq n})=(\sigma(x_{i}))_{0\leq i\leq n}$ for $\sigma\in\Gamma$ and $(x_{i})_{0\leq i\leq n}\in P^{n}(K)$. In this paper, we ask `how much' unstable $V$ is under $\Gamma$ by asking how much higher (or lower) dimension the join (or the meet) of $\sigma(V)$ ($\sigma \in\Gamma$) has than $V$, and answer it in terms of the Pl\"{u}cker coordinates of $V$ and the invariant field $k$ of $\Gamma$, through presenting dual `irrationality' measures of $V$ over $k$.