$G_2$ and the Maximally Symmetric (3, 8) Distribution with 6-Dimensional Square
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2026
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| _version_ | 1866917540058365952 |
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| author | Day, Nicklas Doubrov, Boris Zelenko, Igor |
| author_facet | Day, Nicklas Doubrov, Boris Zelenko, Igor |
| contents | 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 $(\mathfrak{g}_2 \oplus \mathbb{R}) \ltimes W$, where $\mathfrak{g}_2$ is the Lie algebra of $G_2$ and $W$ is an adjoint module of $\mathfrak{g}_2$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_25910 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $G_2$ and the Maximally Symmetric (3, 8) Distribution with 6-Dimensional Square Day, Nicklas Doubrov, Boris Zelenko, Igor Differential Geometry Optimization and Control 58A30, 58A17, 37J60, 53A55, 17B25, 17B66, 17A75, 53C17, 49K15 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 $(\mathfrak{g}_2 \oplus \mathbb{R}) \ltimes W$, where $\mathfrak{g}_2$ is the Lie algebra of $G_2$ and $W$ is an adjoint module of $\mathfrak{g}_2$. 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. |
| title | $G_2$ and the Maximally Symmetric (3, 8) Distribution with 6-Dimensional Square |
| topic | Differential Geometry Optimization and Control 58A30, 58A17, 37J60, 53A55, 17B25, 17B66, 17A75, 53C17, 49K15 |
| url | https://arxiv.org/abs/2605.25910 |