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PreprintCC BY 4.0arXiv

G2 and the Maximally Symmetric (3, 8) Distribution with 6-Dimensional Square

  • Nicklas Day
  • Boris Doubrov
  • Igor Zelenko
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Abstract

In 1910, Élie Cartan famously realized the split real form of the exceptional Lie group G2 as the symmetry group of the maximally symmetric rank 2 distribution on a 5-dimensional manifold with the small growth vector (2,3,5). In this paper, we discover a new appearance of G2 in the geometric theory of distributions, arising from a rank 3 distribution on an 8-dimensional manifold with the growth vector (3,6,8). The algebra of infinitesimal symmetries of this distribution at any point is 29-dimensional and isomorphic to (g2 ⊕ ℝ) ltimes W, where g2 is the Lie algebra of G2 and W is an adjoint module of g2. Our model possesses three remarkable properties. First, it is maximally symmetric among all bracket-generating rank 3 distributions with a 6-dimensional square (a family that includes both (3,6,8) and (3,6,7,8) distributions). To the best of our knowledge, this is the first example of a family of distributions defined by a set of prescribed small growth vectors in which maximal symmetry is achieved by a member whose growth vector is not the longest. Second, this model provides the first counterexample to the conjecture that all bracket-generating rank 3 distributions with a 6-dimensional square are of maximal class at a generic point (which is known to hold in dimensions 6 and 7). Third, further analysis yields the control-theoretic consequence that all abnormal extremal trajectories of this model originating at any point of the ambient manifold have a corank of at least 2. To our knowledge, this is the first example with this property among bracket-generating distributions with generic small growth vector for a given rank and ambient dimension. We also give an interpretation of our model in terms of split-octonions, more precisely, in terms of a natural algebraic structure on the tangent bundle to split octonions.

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Cite

BibTeXRIS
BibTeX
@misc{day2026maximally,
  title = {{$G_2$ and the Maximally Symmetric (3, 8) Distribution with 6-Dimensional Square}},
  author = {Nicklas Day and Boris Doubrov and Igor Zelenko},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.25910},
  eprint = {2605.25910},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.25910},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - G_2 and the Maximally Symmetric (3, 8) Distribution with 6-Dimensional Square
AU  - Nicklas Day
AU  - Boris Doubrov
AU  - Igor Zelenko
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2605.25910
UR  - https://arxiv.org/abs/2605.25910
AB  - In 1910, Élie Cartan famously realized the split real form of the exceptional Lie group G_2 as the symmetry group of the maximally symmetric rank 2 distribution on a 5-dimensional manifold with the small growth vector (2,3,5). In this paper, we discover a new appearance of G_2 in the geometric theory of distributions, arising from a rank 3 distribution on an 8-dimensional manifold with the growth vector (3,6,8). The algebra of infinitesimal symmetries of this distribution at any point is 29-dimensional and isomorphic to (g_2 ⊕ ℝ) ltimes W, where g_2 is the Lie algebra of G_2 and W is an adjoint module of g_2. Our model possesses three remarkable properties. First, it is maximally symmetric among all bracket-generating rank 3 distributions with a 6-dimensional square (a family that includes both (3,6,8) and (3,6,7,8) distributions). To the best of our knowledge, this is the first example of a family of distributions defined by a set of prescribed small growth vectors in which maximal symmetry is achieved by a member whose growth vector is not the longest. Second, this model provides the first counterexample to the conjecture that all bracket-generating rank 3 distributions with a 6-dimensional square are of maximal class at a generic point (which is known to hold in dimensions 6 and 7). Third, further analysis yields the control-theoretic consequence that all abnormal extremal trajectories of this model originating at any point of the ambient manifold have a corank of at least 2. To our knowledge, this is the first example with this property among bracket-generating distributions with generic small growth vector for a given rank and ambient dimension. We also give an interpretation of our model in terms of split-octonions, more precisely, in terms of a natural algebraic structure on the tangent bundle to split octonions.
ER  -