Simultaneous generating sets for flags

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 $m>3$.

16 pages, 6 figures, comments are welcome