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The height of an infinite parallelotope is infinite

Alexandre Kosyak

TL;DR

The paper proves a general divergence phenomenon for Gram determinants of $m+1$ infinite vectors $(f_r)_{r=0}^m$ under the non-$l_2$-combination condition, establishing that the ratio of full to reduced Gram determinants tends to infinity as projections grow. The approach combines quadratic-form analysis, Cramer's rule in Gram-structured systems, and the distance-to-hyperplane viewpoint to derive explicit growth and boundedness properties of minimizers, culminating in a general lemma valid for all $m$. It also develops explicit formulas for $C^{-1}(\lambda)$ and the bilinear form $(C^{-1}(\lambda)a,a)$ in the Gram-matrix setting, via the generalized characteristic polynomial and Gram determinants, with closed forms in low-dimensional cases and a general construction for arbitrary $m$. These results underpin the irreducibility of certain unitary representations of infinite-dimensional groups by controlling Gram- and determinant-based obstructions. $\,$

Abstract

We show that $\frac{Γ(f_0,f_1,\dots,f_m)} {Γ(f_1,\dots,f_m)}=\infty$ for $m+1$ vectors having the properties that no non-trivial linear combination of them belongs to $l_2(\mathbb N)$. This property is essential in the proof of the irreducibility of unitary representations of some infinite-dimensional groups.

The height of an infinite parallelotope is infinite

TL;DR

The paper proves a general divergence phenomenon for Gram determinants of infinite vectors under the non--combination condition, establishing that the ratio of full to reduced Gram determinants tends to infinity as projections grow. The approach combines quadratic-form analysis, Cramer's rule in Gram-structured systems, and the distance-to-hyperplane viewpoint to derive explicit growth and boundedness properties of minimizers, culminating in a general lemma valid for all . It also develops explicit formulas for and the bilinear form in the Gram-matrix setting, via the generalized characteristic polynomial and Gram determinants, with closed forms in low-dimensional cases and a general construction for arbitrary . These results underpin the irreducibility of certain unitary representations of infinite-dimensional groups by controlling Gram- and determinant-based obstructions.

Abstract

We show that for vectors having the properties that no non-trivial linear combination of them belongs to . This property is essential in the proof of the irreducibility of unitary representations of some infinite-dimensional groups.
Paper Structure (16 sections, 12 theorems, 96 equations)

This paper contains 16 sections, 12 theorems, 96 equations.

Table of Contents

  1. One lemma about $m$ infinite vectors
  2. The particular cases
  3. Case $m=1$
  4. Case $m=2$
  5. The proof of Lemma \ref{['l.min=proj.m']}
  6. How far is a vector from a hyperplane?
  7. The distance of a vector from a hyperplane
  8. Cramer's rule reformulated
  9. Some estimates
  10. Gram determinants and Gram matrices
  11. The generalized characteristic polynomial and its properties
  12. The explicit expression for $C^{-1}(\lambda)$ and $(C^{-1}(\lambda)a,a)$
  13. The case where $C$ is the Gram matrix
  14. Case $m=2$
  15. Case $m=3$
  16. ...and 1 more sections

Key Result

Lemma 1.1

Let $m\in \mathbb N$ and $(f_r)_{r=0}^{m}$ be $m+1$ infinite real vectors $f_r=(f_{rk})_{k\in \mathbb N},$$0\leq r\leq m$ such that for all $(C_0,\dots,C_{m})\in {\mathbb R}^{m+1}\setminus\{0\}$ holds Denote by $f_r^{(n)}=(f_{rk})_{k=1}^n\in \mathbb R^n$ the projections of the vectors $f_r$ on the subspace $\mathbb R^n$. Then for all $s$ with $0\leq s\leq m$ where $\hat{f_s}$ means that the vecto

Theorems & Definitions (12)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 2.1: KosJFA17Kos_B_09
  • Lemma 2.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Theorem 5.1
  • Lemma 5.2: Kos-hpl-arx23, Lemma 2.2
  • ...and 2 more