The height of an infinite parallelotope is infinite

TL;DR

The paper proves that the height of an infinite parallelotope is infinite under certain conditions, which is crucial for establishing the irreducibility of specific infinite-dimensional group representations.

Contribution

It introduces a new property of vectors related to infinite-dimensional groups, showing their associated ratios are infinite, aiding in representation theory proofs.

Findings

The ratio of Gamma functions diverges to infinity for certain vectors.

No non-trivial linear combination of these vectors belongs to l_2.

This divergence is key for irreducibility proofs in infinite-dimensional groups.

Abstract

We show that $\frac{\Gamma(f_0,f_1,\dots,f_m)} {\Gamma(f_1,\dots,f_m)}=\infty$ 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.