Normalizers of Chevalley Groups of Type G 2 Over Local Rings Without 1/2

In this paper, we prove that every element of the linear group GL₁₄(R) normalizing the Chevalley group of type G ₂ over a commutative local ring R without 1/2 belongs to this group up to some multiplier. This allows us to improve our classification of automorphisms of these Chevalley groups showing that an automorphism-conjugation can be replaced by an inner automorphism. Therefore, it is proved that every automorphism of a Chevalley group of type G ₂ over a local ring without 1/2 is a composition of a ring and an inner automorphisms.     

Standardness and Standard Automorphisms of Chevalley Groups, I: the Case of Rank at Least Two

The author studies the linkage between the standardness and the standard automorphisms of Chevalley groups over rings.It is proved that if H is any standard subgroup of G(R),then each of its automorphisms can be extended to an automorphism of G(R,I),restricted to an automorphism of E(R,I),and an automorphism of E(R,I) can be extended to one of G(R,I).The case of Chevalley groups of rank at least two is treated in this paper.Further results about the case of Chevalley groups of rank one,the case of non-commutative ground ring and some others exceptions will appear elsewhere.     

Strong quadratic modules

For a Chevalley groupG over a field of characteristic 2 we determine all irreducible modulesV overGF(2) such that [V, R, Q]=0, whereR is a long root group andQ=Z ₂(O ₂(N _G(R))). As a corollary we obtain a classification of those irreducible modules admitting a quadratic fours groupE which intersect a long root group nontrivially but is not contained in such a group.     

Representations of the Klein Group Determined by Quadruples of Polynomials Associated with the Double Confluent Heun Equation

The canonical representation of the Klein group K₄ = ?₂⊕?₂ on the space ?* = ? {0} induces a representation of this group on the ring L = C[z, z^?1], z ∈ ?*, of Laurent polynomials and, as a consequence, a representation of the group K₄ on the automorphism group of the group G = GL(4,L) by means of the elementwise action. The semidirect product ?G = GK₄ is considered together with a realization of the group ? as a group of semilinear automorphisms of the free 4-dimensional L-module M⁴. A three-parameter family of representations R of K₄ in the group ? and a three-parameter family of elements X ∈ M⁴ with polynomial coordinates of degrees 2(? ? 1), 2?, 2(? ? 1), and 2?, where ? is an arbitrary positive integer (one of the three parameters), are constructed. It is shown that, for any given family of parameters, the vector X is a fixed point of the corresponding representation R. An algorithm for calculating the polynomials that are the components of X was obtained in a previous paper of the authors, in which it was proved that these polynomials give explicit formulas for automorphisms of the solution space of the doubly confluent Heun equation.     

K-theory for finite covariant systems

LetA be a real or complex Banach algebra and assume that is an action of a finite groupG onA by means of continuous automorphisms. To such a finite covariant system (A, G, ), we associate an Abelian groupK(A, G, ). We obtain some classical exact sequences for an algebraA and a closed invariant idealI. We also compute the group in a few important special cases. Doing so, we relate our new invariant to the classicalK ₀ andK ₁ of a Banach algebra and to theK-theory of ₂-graded Banach algebras. Finally, we obtain a result that gives a close relationship of our groupK(A, G, ) with theK-theory of the crossed productA G. In particular, we prove a six-term exact sequence involving our groupK(A, G, ) and theK-groups ofA G. In this way, we hope to contribute to the well-known problem of finding theK-theory of the crossed productA G in the case of an action of a finite group.     

On geodesic structures of weakly median graphs—II: Compactness, the role of isometric rays

We prove that the vertex set of a K_ℵ0-free weakly median graph G endowed with the weak topology associated with the geodesic convexity on V(G) is compact if and only if G has one of the following equivalent properties: (1) G contains no isometric rays; (2) any chain of interval of G ordered by inclusion is finite; (3) every self-contraction of G fixes a non-empty finite regular weakly median subgraph of G. We study the self-contractions of K_ℵ0-free weakly median graphs which fix no finite set of vertices. We also follow a suggestion of Imrich and Klavzar [Product Graphs, Wiley, New York, 2000] by defining different centers of such a graph G, each of them giving rise to a non-empty finite regular weakly median subgraph of G which is fixed by all automorphisms of G.     

Lifting representations of ℤ-groups

LetK be the kernel of an epimorphismG→ℤ, whereG is a finitely presented group. IfK has infinitely many subgroups of index 2,3 or 4, then it has uncountably many. Moreover, ifK is the commutator subgroup of a classical knot groupG, then any homomorphism fromK onto the symmetric groupS ₂ (resp. ℤ₃) lifts to a homomorphism ontoS ₃ (resp. alternating groupA ₄). Both authors partially supported by NSF grants DMS-0071004 and DMS-0304971.     

Products of conjugacy classes in Chevalley groups I. Extended covering numbers

This paper is concerned with products of conjugacy classes in Chevalley groups. We prove that in any quasisimple Chevalley groupG proper or twisted, over any field, the extended covering number is bounded above linearly in terms of the rank ofG, that is, for some constante, for any Chevalley groupG, the product of anye · rank(G) non-central classes covers all ofG. We give estimates for the constante in different cases. The authors gratefully acknowledge the support of EPSRC through a Visiting Fellowship number GR/M58542 and a Research grant number GR/L92174.     

Automorphisms and holomorphic differentials in characteristic p

Let G be a group of automorphisms of a function field F of genus g_F over an algebraically closed field K. The space $\text{Ω}{}_{\mathrm{F}}$ of holomorphic differentials of F is a g_F^? dimensional K-space. In response to a query of Hecke, Chevalley and Weil (Abh. Math. Sem. Univ. Hamburg, 10 (1934), 358–361) completely determined the structure of $\text{Ω}{}_{\mathrm{F}}$ as representation space for G in the classical case. They carried out the proof for the special case in which F is unramified over the fixed field of G. The case of cyclic ramified extensions had been previously considered by Hurwitz (Math. Ann., 41 (1893), 37–45). Weil (Abh. Math. Sem. Univ. Hamburg, 11 (1935), 110–115) gave a proof in the general case. The treatment in the last two papers is analytical. In characteristic p, the problem is open. If G is cyclic and F is unramified over the fixed field E of G, Tamagawa (Proc. Japan Acad., 27 (1951), 548–551) proved that the representation is the direct sum of one identity representation of degree 1 and g_E ? 1 regular representations. The principal object of this paper is an extension of Tamagawa's result to arbitrary cyclic extensions of p-power degree. The number of times an indecomposable representation of given degree occurs in the representation of G on $\text{Ω}{}_{\mathrm{F}}$ is explicitly determined in terms of g_E and the Witt vector characterizing the extension $\text{F}\text{E}$. The paper also contains a purely algebraic proof of the result of Chevalley and Weil for arbitrary cyclic extensions of degree relatively prime to p. Using character theory, it can be extended to arbitrary groups of order relatively prime to the characteristic.     

Embeddings in finitely presented groups which preserve the center

V.N. Remeslennikov proposed in 1976 the following problem: is any countable abelian group a subgroup of the center of some finitely presented group? We prove that every finitely generated recursively presented group G is embeddable in a finitely presented group K such that the center of G coincide with that of K. We prove also that there exists a finitely presented group H with soluble word problem such that every countable abelian group is embeddable in the center of H. This gives a strong positive answer to the question raised by V.N. Remeslennikov.     

Automorphism groups of finite posets

For any finite group G, we construct a finite poset (or equivalently, a finite T₀-space) X, whose group of automorphisms is isomorphic to G. If the order of the group is n and it has r generators, X has n(r+2) points. This construction improves previous results by G. Birkhoff and M.C. Thornton. The relationship between automorphisms and homotopy types is also analyzed.     

Shephard–Todd–Chevalley Theorem for Skew Polynomial Rings

We prove the following generalization of the classical Shephard–Todd–Chevalley Theorem. Let G be a finite group of graded algebra automorphisms of a skew polynomial ring \(A:=k_{p_{ij}}[x_1,\cdots,x_n]\). Then the fixed subring A ^G has finite global dimension if and only if G is generated by quasi-reflections. In this case the fixed subring A ^G is isomorphic to a skew polynomial ring with possibly different p _ij ’s. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings.     

A complete parametrization of cyclic field extensions of 2-power degree

Letq be a power of 2 at least equal to 8 and ζ be a primitiveq-th root of unity, and letK be any field of characteristic zero. We define the group of special projective conormsS _K as a quotient of the group of elements ofK(ζ) of norm 1:S _K is obviously trival if the groul Gal (K(ζ)/K) is cyclic. We prove that for some fieldsK, the groupS _K is finite, and it is even trivial for certain fields such as ? or ?(X ₁,...,X _m). We then prove that the groupS _K completely paramatrizes the cycle extensions ofK of degreeq. We exhibit an explicit polynomial defined over ?(T ₀,...,T _q/2) which parametrizes all cyclic extensions ofK of degreeq associated to the trivial element ofS _K. In particular, this polynomial parametrizes all cyclic extensions ofK of degreeq whenever the groupS _K is trivial.     

Generalized hexagons of order (t,t)

A generalized hexagon of order (t,t) in which certain subsets are maximal may be characterized as the generalized hexagon associated with Dickson’s groupG ₂(t). From this geometric result, it follows that ifG is a group of automorphisms of a generalized hexagon of order (p,p) for a primep and ifG has rank 4 on points, thenG ⊵G ₂(p).     

Doubly transitive automorphism groups of block designs

If G is a doubly transitive group of automorphisms of a block design with λ = 1, then for any block Δ of the design and any point α in Δ, the set Δ?{α} is a block of imprimitivity for G_α. What are sufficient conditions for a doubly transitive but not doubly primitive permutation group G to be a group of automorphisms of a non-trivial block design with λ = 1 ? Can the design or the group G be identified if there is a nonidentity automorphism in G fixing every point of some block of the design? Both of these questions are investigated and some answers are given.     

Groups with infinitely many types of fixed subgroups

It is a theorem of Shor that ifG is a word-hyperbolic group, then up to isomrphism, only finitely many groups appear as fixed subgroups of automorphisms ofG. We give an example of a groupG acting freely and cocompactly on a CAT(0) square complex such that infinitely many non-isomorphic groups appear as fixed subgroups of automorphisms ofG. Consequently, Shor’s finiteness result does not hold if the negative curvature condition is relaxed to either biautomaticity or nonpositive curvature. D. T. Wise was supported by grants from FCAR and NSERC.     

Banach space representations and Iwasawa theory

We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringo _K[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗o _K[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL₂(ℤ _p ).     

Automorphisms of Chevalley groups of types A l , D l , or E l over local rings with 1/2

In this paper, we prove that every automorphism of an (elementary) Chevalley group of type A _l , D _l , or E _l , l ≥ 2, over a commutative local ring with 1/2 is standard, i.e., is the composition of inner, ring, graph, and central automorphisms.     

Intermediate subgroups of chevalley groups over a field of fractions of a ring of principal ideals

Let K be a field of fractions of a principal ideal ring R and G_K be a Chevalley group (of normal type) over K. For each subring P ⊂ K, denote by G_P a subgroup of all elements of G_K with coefficients in P. Let M be intermediate between G_R and G_K, i.e., G_R ⊆ M ⊆ G_K. We prove that M=G_P for some intermediate subring P (R ⊆ P ⊆ K). Supported by RFFR grant No. 96-01-00409. Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 347–358, May–June, 2000.