Simultaneous generating sets for flags

TL;DR

This paper establishes the minimal size of common generating sets for triples of complete flags in real vector spaces, extending classical results for pairs and providing sharp bounds for larger tuples.

Contribution

It proves that any three complete flags in  admit a common generating set of size  floor 5d/3, extending the classical pairwise result and establishing sharp bounds.

Findings

Bound of  floor 5d/3 for triples of flags

Extension of classical pairwise results to triples and larger m-tuples

Sharpness of the bounds proved

Abstract

We prove that any triple of complete flags in $\mathbb R^d$ admits a common generating set of size $\lfloor 5d/3\rfloor$ and that this bound is sharp. This result extends the classical linear-algebraic fact -- a consequence of the Bruhat decomposition of $\text{GL}_d(\mathbb R)$ -- that any pair of complete flags in $\mathbb R^d$ admits a common generating set of size $d$. We also deduce an analogue for $m$-tuples of flags with $m>3$.