Plücker coordinates of finite-dimensional subspaces of ${\ell }^p$ and its direct sums: summability, reconstruction, stratification

Abstract

An $n$-dimensional subspace of ${\ell }^p$ has Plücker coordinates indexed by the $n$-element subsets of $\mathbb{N}$. We show these coordinates lie in $\ell^p\In{n}$ --- the exponent is preserved --- with multilinear norm exactly $1$ for $0<p\le 2$; for $p>2$ the sharp constant exceeds $1$ and its determination contains the Hadamard maximal determinant problem. A reconstruction lemma shows every nonzero solution of the quadratic Plücker relations in $\ell^p\In{n}$ is decomposable with frame in ${\ell }^p$; consequently $\Gr_n(\ell^p)$ is a closed Banach-analytic submanifold of $\mathbb{P}\big(\ell^p\In{n}\big)$ cut out by the Plücker relations alone, with no auxiliary summability condition and no polarization. For mixed sums $⨁{\ell }^{p_i}$ the exterior power is graded by compositions of $n$; the support of the grading is the lattice-point set of a generalized permutohedron determined by the intersection pattern of the subspace with partial sums, this stratification is canonical for the isometry group though not for $\GL$, and each stratum admits a tubular neighborhood whose normal coordinates are precisely the Plücker blocks vanishing on it. We record what is proved and what is conjectured; the finitary and single-space core of the theory, including full proofs of Cauchy--Binet and Hadamard's inequality, has been formally verified in Lean~4.

14 pages