Rigidity of symmetric holomorphic functions on infinite-dimensional sequence spaces
Abstract
We study holomorphic functions on permutation-invariant subsets of an infinite-dimensional sequence space that are invariant under all finite permutations of coordinates. Working on domains modeled on the space czero of null sequences, we show that permutation symmetry forces the first differential to vanish along every constant tail: the coordinate derivatives agree there and an averaging argument annihilates their common value. This yields rigidity---constancy---on domains of constant sequences, and more generally on any cc-chain-connected domain on which the differential vanishes on the finitely supported directions. The method has a sharp boundary: a precise obstruction prevents the elementary argument from reaching the finitely many exceptional coordinates of a non-constant point, so unconditional rigidity on general eventually-constant domains remains open. In the other direction, an explicit example---any Banach limit on ℓ∞---shows that the chain-connectedness and boundedness hypotheses cannot be dropped: mere topological connectedness admits non-constant symmetric entire functions. We record the vector-valued corollary and discuss the motivating application to holomorphic cross-sections of the zero-set map, isolating the lifting problem that remains open.
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BibTeX
@misc{feldman2026rigidity,
title = {{Rigidity of symmetric holomorphic functions on infinite-dimensional sequence spaces}},
author = {David Feldman and Jon Bannon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20196},
eprint = {2609.20196},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20196},
}
RIS
TY - UNPB
TI - Rigidity of symmetric holomorphic functions on infinite-dimensional sequence spaces
AU - David Feldman
AU - Jon Bannon
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20196
UR - https://arxiv.org/abs/2609.20196
AB - We study holomorphic functions on permutation-invariant subsets of an infinite-dimensional sequence space that are invariant under all finite permutations of coordinates. Working on domains modeled on the space czero of null sequences, we show that permutation symmetry forces the first differential to vanish along every constant tail: the coordinate derivatives agree there and an averaging argument annihilates their common value. This yields rigidity---constancy---on domains of constant sequences, and more generally on any cc-chain-connected domain on which the differential vanishes on the finitely supported directions. The method has a sharp boundary: a precise obstruction prevents the elementary argument from reaching the finitely many exceptional coordinates of a non-constant point, so unconditional rigidity on general eventually-constant domains remains open. In the other direction, an explicit example---any Banach limit on ℓ^∞---shows that the chain-connectedness and boundedness hypotheses cannot be dropped: mere topological connectedness admits non-constant symmetric entire functions. We record the vector-valued corollary and discuss the motivating application to holomorphic cross-sections of the zero-set map, isolating the lifting problem that remains open.
ER -