Narrow Results

Joint spectral measures associated to the rank two Lie group $G_2$, including the representation graphs for the irreducible representations of $G_2$ and its maximal torus, nimrep graphs associated to the $G_2$ modular invariants have been studied. In this paper, we study the joint spectral measures for the McKay graphs (or representation graphs) of finite subgroups of $G_2$. Using character theoretic methods we classify all non-conjugate embeddings of each subgroup into the fundamental representation of $G_2$ and present their McKay graphs, some of which are new.

This paper focuses on a geometric structure defined by a system of closed exterior differential forms and develops a new approach to deformation problems of geometric structures. We obtain a criterion for unobstructed deformations from a cohomological point of view (Theorem 1.7). Further we show that under a cohomological condition, the moduli space of the geometric structures becomes a smooth manifold of finite dimension (Theorem 1.8). We apply our approach to the geometric structures such as Calabi–Yau, HyperKähler, G₂ and Spin(7) structures and then obtain a unified construction of smooth moduli spaces of these four geometric structures. We generalize the Moser's stability theorem to provide a direct proof of the local Torelli type theorem in these four geometric structures (Theorem 1.10).

There are two well-known parabolic split ${\mathrm{\text{G}}}_2$ geometries in dimension 5, $(2,3,5)$ distributions and ${\mathrm{\text{G}}}_2$ contact structures. Here we link these two geometries with yet another ${\mathrm{\text{G}}}_2$ related contact structure, which lives on a $7$-manifold. More precisely, we present a natural geometric construction that associates to a $(2,3,5)$ distribution a $7$-dimensional bundle endowed with a canonical Lie contact structure. We further study the relation between the canonical normal Cartan connections associated with the two structures and we show that the Cartan holonomy of the induced Lie contact structure reduces to ${\mathrm{\text{G}}}_2$. This motivates the study of the curved orbit decomposition associated with a ${\mathrm{\text{G}}}_2$ reduced Lie contact structure on a $7$-manifold. It is shown that, provided an additional curvature condition is satisfied, in a neighborhood of each point in the open curved orbit the structure descends to a $(2,3,5)$ distribution on a local leaf space. The closed orbit carries an induced ${\mathrm{\text{G}}}_2$ contact structure.

We use the companion matrix construction for ${\mathrm{\text{GL}}}_n$ to build canonical sections of the Chevalley map $[𝔤/G]\to 𝔤//G$ for classical groups $G$ as well as the group $G_2$. To do so, we construct canonical tensors on the associated spectral covers. As an application, we make explicit lattice descriptions of affine Springer fibers and Hitchin fibers for classical groups and $G_2$.

We derive a number of results related to the Gaudin model associated to the simple Lie algebra of type G₂.

We compute explicit formulas for solutions of the Bethe ansatz equations associated to the tensor product of an arbitrary finite-dimensional irreducible module and the vector representation. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic.

We show that the points of the spectrum of the Gaudin model in type G₂ are in a bijective correspondence with self-self-dual spaces of polynomials. We study the set of all self-self-dual spaces — the self-self-dual Grassmannian. We establish a stratification of the self-self-dual Grassmannian with the strata labeled by unordered sets of dominant integral weights and unordered sets of nonnegative integers, satisfying certain explicit conditions. We describe closures of the strata in terms of representation theory.

The purpose of this paper is to provide an octonionic description of the Lie group SL(2, 𝕆). The main result states that it can be obtained as a free group generated by invertible and determinant preserving transformations from 𝔥₂(𝕆) onto itself. An interesting characterization is given for the generators of G₂. Also, explicit isomorphisms are constructed between the Lie algebras 𝔰𝔩(2, 𝕂), for 𝕂 = ℝ, ℂ, ℍ, 𝕆, and their corresponding Lorentz algebras.

We show how to compute the SO(7)-invariants from the G₂-invariant functions of a curve in the purely imaginary octonions.