Exceptional Isomorphisms Research Articles

Correction: Stokes matrices and exceptional isomorphisms

Correction: Stokes matrices and exceptional isomorphisms

Stokes matrices and exceptional isomorphisms

AbstractWe prove that the integral Stokes matrices of size 4 with given Coxeter invariant of nonzero discriminant decompose into finitely many braid group orbits. We also establish exceptional isomorphisms between moduli spaces of points on algebraic unit 3-spheres and certain moduli of local systems on positive genus surfaces.

On the graph of non-degenerate linear [n,2]2 codes

Consider the Grassmann graph of k-dimensional subspaces of an n-dimensional vector space over the q-element field, 1<k<n−1. Every automorphism of this graph is induced by a semilinear automorphism of the corresponding vector space or a semilinear isomorphism to the dual vector space; the second possibility is realized only for n=2k. Let Γ(n,k)q be the subgraph of the Grassman graph formed by all non-degenerate linear [n,k]q codes. If q≥3 or k≥3, then every isomorphism of Γ(n,k)q to a subgraph of the Grassmann graph can be uniquely extended to an automorphism of the Grassmann graph. For q=k=2 there is an isomorphism of Γ(n,k)q to a subgraph of the Grassmann graph which does not have this property. In this paper, we show that such exceptional isomorphism is unique up to an automorphism of the Grassmann graph.

On the Bruhat-Tits stratification for GU(2,2) type Rapoport-Zink space: unramified case

In this note we study the supersingular locus of the GU(2,2) Shimura variety modulo a prime which is unramified in the imaginary quadratic extension. The supersingular locus of this Shimura variety can be related to the basic Rapoport-Zink space whose special fibre is described by the Bruhat-Tits stratification. The description for this supersingular locus in the case where the prime is inert in imaginary quadratic field is already known to Howard and Pappas by exploiting the exceptional isomorphism. Our method is more direct without using the exceptional isomorphism.

Exceptional isomorphisms between complements of affine plane curves

This article describes the geometry of isomorphisms between complements of geometrically irreducible closed curves in the affine plane $\mathbb{A}^2$, over an arbitrary field, which do not extend to an automorphism of $\mathbb{A}^2$. We show that such isomorphisms are quite exceptional. In particular, they occur only when both curves are isomorphic to open subsets of the affine line $\mathbb{A}^1$, with the same number of complement points, over any field extension of the ground field. Moreover, the isomorphism is uniquely determined by one of the curves, up to left composition with an automorphism of $\mathbb{A}^2$, except in the case where the curve is isomorphic to the affine line $\mathbb{A}^1$ or to the punctured line $\mathbb{A}^1 \setminus \{0\}$. If one curve is isomorphic to $\mathbb{A}^1$, then both curves are in fact equivalent to lines. In addition, for any positive integer $n$, we construct a sequence of $n$ pairwise non-equivalent closed embeddings of $\mathbb{A}^1 \setminus \{0\}$ with isomorphic complements. In characteristic~$0$ we even construct infinite sequences with this property. Finally, we give a geometric construction that produces a large family of examples of non-isomorphic geometrically irreducible closed curves in $\mathbb{A}^2$ that have isomorphic complements, answering negatively the Complement Problem posed by Hanspeter Kraft.. This also gives a negative answer to the holomorphic version of this problem in any dimension $n \geq 2$. The question had been raised by Pierre-Marie Poloni.

The group of K1-zero-cycles on the second generalized Severi–Brauer variety of an algebra of index 4

In this manuscript, it is shown that the group of K1-zero-cycles on the second generalized Severi–Brauer variety of an algebra A of index 4 is given by elements of the group K1(A) together with a square-root of their reduced norm. Utilizing results of Krashen concerning exceptional isomorphisms, we translate our problem to the computation of cycles on involution varieties. Work of Chernousov and Merkurjev then gives a means of describing such cycles in terms of Clifford and spin groups and corresponding R-equivalence classes. We complete our computation by giving an explicit description of these algebraic groups.

Equilateral Triangles and the Fano Plane

We formulate a definition of equilateral triangles in the complex line that makes sense over the field with seven elements. Adjacency of these abstract triangles gives rise to the Heawood graph, which is a way to encode the Fano plane. Through some reformulation, this gives a geometric construction of the Steiner systems S(2, 3, 7) and S(3, 4, 8). As a consequence, we embed the Heawood graph in a torus, and we derive the exceptional isomorphism PSL2(픽7) ≃ GL3(픽2). The study of equilateral triangles over other finite fields shows that seven is very specific.

Equilateral Triangles and the Fano Plane

AbstractWe formulate a definition of equilateral triangles in the complex line that makes sense over the field with seven elements. Adjacency of these abstract triangles gives rise to the Heawood graph, which is a way to encode the Fano plane. Through some reformulation, this gives a geometric construction of the Steiner systems S(2, 3, 7) and S(3, 4, 8). As a consequence, we embed the Heawood graph in a torus, and we derive the exceptional isomorphism PSL2(픽7) ≃ GL3(픽2). The study of equilateral triangles over other finite fields shows that seven is very specific.

Imprimitive irreducible modules for finite quasisimple groups

Acknowledgements Introduction Generalities Sporadic groups and the Tits group Alternating groups Exceptional Schur multipliers and exceptional isomorphisms Groups of Lie type: Induction from non-parabolic subgroups Groups of Lie type: Induction from parabolic subgroups Groups of Lie type: ${\rm char}(K) = 0$ Classical groups: ${\rm char}(K) = 0$ Exceptional groups Bibliography

An exceptional isomorphism between modular curves of level 13

In this note we use representation theory to compute an explicit equation over Q for the modular curve Xns(13) associated to the normalizer of a non-split Cartan subgroup of level 13. We also prove that the same equation defines the modular curve Xs(13) associated to the normalizer of a split Cartan subgroup of level 13. Hence, these two modular curves are isomorphic over Q. This isomorphism does not seem to have a “modular” explanation. For instance, Xs(13)(Q) contains a cusp but Xns(13)(Q) does not.

An exceptional isomorphism between level 13 modular curves via Torelli’s Theorem

The Jacobians of the modular curves Xns(13) and Xs(13) respectively associated to the normalizers of non-split and split Cartan subgroups of level 13 are isogenous over Q. In this note, we construct a Q-isomorphism between these Jacobians which respects their canonical principal polarizations. In particular we obtain a Q-isomorphism between Xns(13) and Xs(13); this has no known \modular explanation.

ON SUBGROUPS OF CLIFFORD GROUPS DEFINED BY JORDAN PAIRS OF RECTANGULAR MATRICES

AbstractSome embeddings of general linear groups into hyperbolic Clifford groups are constructed generically by using Jordan pairs of rectangular and alternating matrices over a ring. In low rank cases through exceptional isomorphisms, their direct description and relationships to some automorphisms of Clifford groups are given. Generic norms are calculated in detail, and equivariant embeddings of representation spaces are constructed.

An approach through big cells to Clifford groups of low rank

So-called exceptional isomorphisms of low-dimensional spinor groups are directly worked out for the terminal two cases of rank five and six (in the sense of quadratic modules) at the level of special Clifford groups with norm characters as group schemes over a ring. Explicit formulas of identifications are available on big cells, restrictions to which being justified by scheme theory.

Weierstrass points and Z 2 homology

In a recent paper (1993), Lustig established a beautiful connection between the six Weierstrass points on a Riemann surface M 2 of genus 2 and intersection points of closed geodesics for the associated hyperbolic metric. As a consequence, he was able to construct an action of the mapping class group Out( π 1 M 2) of M 2 on the set of Weierstrass points of M 2 and a virtual splitting of the natural homomorphism Aut( π 1 M 2) → Out( π 1 M 2). Our discussion in this paper begins with the observation that these two results of Lustig's are direct consequences of the work of Birman and Hilden (1973) on equivariant homotopies for surface homeomorphisms. It is well known that Γ 2 acts naturally on the Z 2 symplectic vector space of rank 4, H 1( M 2, Z 2). We identify this action with Lustig's action by constructing a natural correspondence between pairs of distinct Weierstrass points on M 2 and nonzero elements in H 1( M 2, Z 2). In this manner, the well-known exceptional isomorphism of finite group theory, S 6 ≅ Sp(4, Z 2), arises from a natural isomorphism of Γ 2 spaces.

Homological meaning of systems (ϕ, α, σ)

The present paper discusses, from a viewpoint of homology, the nature of systems (ϕ, α, σ) which induce exceptional isomorphisms Φασ of the three-dimensional linear groups over commutative rings.

The classification of the translation planes of order 16, II

The translation planes of order 16 are completely classified. The exceptional isomorphism A8≃GL(4, 2) gives a crucial computational approach to this problem