Frame eversion and contextual geometric rigidity
Alexandru Chirvasitu
TL;DR
The paper addresses rigidity of contextually constrained maps between Grassmannians and spaces of line tuples, aiming to reconstruct linear or semilinear structures from partial compatibility. It develops a unifying, functorial framework for $\mathbb{G}(V)$, $\mathbb{F}^{\perp}(V)$ and $\mathbb{F}(V)$, introducing partition-linkage and the eversion symmetry to capture contextual constraints. The authors prove three main rigidity results: Commuting-Lattice Rigidity (CLR) for dimension-preserving, lattice-commuting maps between Grassmannians induced by semilinear injections; Partition-Frame Rigidity, Perpendicular (PFR$_{\perp}$) for $\mathbb{F}^{\perp}(V)\to \mathbb{F}(W)$ yielding semilinear injections, linear or conjugate-linear under measurability; and Partition-Frame Rigidity, General (PFR) for $\mathbb{F}(V)\to \mathbb{F}(W)$ allowing semilinear bijections or these composed with a novel global contextsymmetry called eversion. These results are connected to Wigner-type rigidity and preserver theory, and extend prior work by relaxing continuity assumptions, with a reduction to the $n=3$ case via projective-geometry arguments. The framework yields a comprehensive classification of contextually constrained maps and highlights the role of eversion as a purely-contextual global symmetry.
Abstract
We prove rigidity results describing contextually-constrained maps defined on Grassmannians and manifolds of ordered independent line tuples in finite-dimensional vector or Hilbert spaces. One statement in the spirit of the Fundamental Theorem of Projective Geometry classifies maps between full Grassmannians of two $n$-dimensional Hilbert spaces, $n\ge 3$, preserving dimension and lattice operations for pairs with commuting orthogonal projections, as precisely those induced by semilinear injections unique up to scaling. In a different but related direction, denote the manifolds of ordered orthogonal (linearly-independent) $n$-tuples of lines in an $n$-dimensional Hilbert space $V$ by $\mathbb{F}^{\perp}(V)$ (respectively $\mathbb{F}(V)$) and, for partitions $π$ of the set $\{1..n\}$, call two tuples $π$-linked if the spans along $π$-blocks agree. A Wigner-style rigidity theorem proves that the symmetric maps $\mathbb{F}^{\perp}(\mathbb{C}^n)\to \mathbb{F}(\mathbb{C}^n)$, $n\ge 3$ respecting $π$-linkage are precisely those induced by semilinear injections, hence by linear or conjugate-linear maps if also assumed measurable. On the other hand, in the $\mathbb{F}(\mathbb{C}^n)$-defined analogue the only other possibility is a qualitatively new type of purely-contextual-global symmetry transforming a tuple $(\ell_i)_i$ of lines into $\left(\left(\bigoplus_{j\ne i}\ell_j\right)^{\perp}\right)_i$.