Additive diameters and covering complexity of irreducible representations
Abstract
Let a group G act linearly on a finite-dimensional complex vector space V. The group-additive diameter of a subspace U ≤ V is the least number of translates of U whose sum is all of V. Counting dimensions, it is at least dim V / dim U. We show that when G is compact and V is irreducible, the diameter of every nonzero subspace is at most ⌈ (dim V / dim U) ln dim V ⌉, so the trivial bound is correct up to a logarithmic factor. We measure the discrepancy by the covering complexity C(V), the largest ratio between the diameter of any subspace and its trivial lower bound, so that 1 ≤ C(V) ≤ 2 + ln dim V, and we determine where in this range various representations lie. Every irreducible representation of SL2(C) has C(V) = 1. The logarithm can be genuinely present along families of symmetric and exterior powers of SLn(C) with n varying, and it is present for finite Heisenberg groups and $2$-transitive groups of small order such as PSL2(Fp). The complexity is bounded above by a constant on the conjugation representations of SLn(C) and on the representations Symk C3 of SL3(C). On the other hand, every fixed connected reductive group has a family of irreducible representations whose complexities tend to at least the dimension of the flag variety. Finally, for the Lie algebra sl3(C) acting on Symk C3, the monomial diameter with respect to Symk X for a plane X ≤ C3 is optimal, while the corresponding SL3(C) diameter is not.
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BibTeX
@misc{jezernik2026additive,
title = {{Additive diameters and covering complexity of irreducible representations}},
author = {Urban Jezernik and Špela Špenko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03882},
eprint = {2609.03882},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03882},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Additive diameters and covering complexity of irreducible representations
AU - Urban Jezernik
AU - Špela Špenko
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03882
UR - https://arxiv.org/abs/2609.03882
AB - Let a group G act linearly on a finite-dimensional complex vector space V. The group-additive diameter of a subspace U ≤ V is the least number of translates of U whose sum is all of V. Counting dimensions, it is at least dim V / dim U. We show that when G is compact and V is irreducible, the diameter of every nonzero subspace is at most ⌈ (dim V / dim U) ln dim V ⌉, so the trivial bound is correct up to a logarithmic factor. We measure the discrepancy by the covering complexity C(V), the largest ratio between the diameter of any subspace and its trivial lower bound, so that 1 ≤ C(V) ≤ 2 + ln dim V, and we determine where in this range various representations lie. Every irreducible representation of SL_2(C) has C(V) = 1. The logarithm can be genuinely present along families of symmetric and exterior powers of SL_n(C) with n varying, and it is present for finite Heisenberg groups and $2$-transitive groups of small order such as PSL_2(F_p). The complexity is bounded above by a constant on the conjugation representations of SL_n(C) and on the representations Sym^k C^3 of SL_3(C). On the other hand, every fixed connected reductive group has a family of irreducible representations whose complexities tend to at least the dimension of the flag variety. Finally, for the Lie algebra sl_3(C) acting on Sym^k C^3, the monomial diameter with respect to Sym^k X for a plane X ≤ C^3 is optimal, while the corresponding SL_3(C) diameter is not.
ER -