Simultaneous generating sets for flags

TL;DR

The paper addresses the problem of finding minimal simultaneous generating sets for m-tuples of complete flags in ℝ^d, extending the classical two-flag result tied to Bruhat decomposition. It introduces μ(m,d) and proves the exact formulas μ(m,d)=d for m=1 or d=1, μ(m,d)=md/2 for even m≥2, and μ(m,d)=md/2+⌊2d/3⌋−d/2 for odd m≥3, establishing μ(3,d)=⌊5d/3⌋ as a sharp bound. The core method for m=3 employs a graph-theoretic framework with two graphs G and ˜G, categorizing cycles into efficiency-based groups and applying a lattice-path cost monovariant to bound the number of necessary compatible sets, thereby achieving the 5d/3 bound. The work further analyzes generic versus non-generic flag configurations and discusses extensions to finite fields and matroids, including open questions on higher m and matroid analogues, with implications for representation theory and combinatorial geometry.

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 $m>3$.

Figures (6)

• Figure 1: The graphs $G$ (left) and $\widetilde{G}$ (right) for a triple of generic flags in $\mathbb{R}^3$. The graph $G$ consists of a triangle (in bold) and a $6$-cycle; these cycles cross. The resulting edges present in $\widetilde{G}$ but not in $G$ are drawn dashed.
• Figure 2: An illustration of \ref{['lem:tildeG']}. The left part of the diagram shows a pair of crossing edges in $G$, with the corresponding additional edge (drawn dotted) in $\widetilde{G}$. The right part of the diagram shows a pair of non-crossing edges in $G$, for which there is no additional edge in $\widetilde{G}$.
• Figure 3: The edge cover (shaded) used in the third part of \ref{['lem:graphmatching']}.
• Figure 4: A schematic of the types of cycles appearing in the groups $A,\space B,\space C$. In $A$, the isolated triangle represents a compatible triple, and the two odd cycles at the bottom are crossing (with the dotted edges representing edges present in $\widetilde{G}$ but not $G$).
• Figure 5: Schematic "snapshots" (from left to right) of four stages of the hopping procedure from the proof of \ref{['lem:bound_cost']}. The triangles in $C$ are drawn in bold. In each snapshot the already-hopped-on vertices are those in the shaded region. (i) We have just hopped on the vertices of the second triangle from the bottom. (ii) We have hopped on the vertices of $\IfNoValueTF{-NoValue-} { {{U}}_{}^{\bullet} } {{U}_{}^{-NoValue-}},\IfNoValueTF{-NoValue-} { {{V}}_{}^{\bullet} } {{V}_{}^{-NoValue-}}$ strictly between the second and third triangles. (iii) We have hopped on the vertices of $\IfNoValueTF{-NoValue-} { {{W}}_{}^{\bullet} } {{W}_{}^{-NoValue-}}$ strictly between the second and third triangles. (iv) We have hopped on the vertices of the third triangle.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 2.1
• proof : Proof of \ref{['thm:2flags']}
• Definition 3.1
• Proposition 3.2
• proof
• Proposition 4.1
• proof
• Corollary 4.2
• Proposition 4.3