Support Varieties of Non-Restricted Modules over Lie Algebras of Reductive Groups

Let G be a connected semisimple group over an algebraicallyclosed field K of characteristic p>0, and g=Lie (G). Fixa linear function g* and let Z_g() denote the stabilizer of in g. Set N_p(g)={xg|x^[p]=0}. Let C(g) denote the category offinite-dimensional g-modules with p-character . In [7], Friedlanderand Parshall attached to each MOb(C(g)) a Zariski closed, conicalsubset V_g(M)N_p(g) called the support variety of M. Suppose thatG is simply connected and p is not special for G, that is, p2if G has a component of type B_n, C_n or F₄, and p3 if G has acomponent of type G₂. It is proved in this paper that, for anynonzero MOb(C(g)), the support variety V_g(M) is contained inN_p(g)Z_g(). This allows one to simplify the proof of the Kac–Weisfeilerconjecture given in [18].     

Some Properties of Free Groups of Some Soluble Varieties of Groups

Let F be a free group, and let _n(F) be the nth term of the lowercentral series of F. It is proved that F/[_j(F), _i(F), _k(F)]and F/[_j(F), _i(F), _k(F), _l(F)] are torsion free and residuallynilpotent for certain values of i, j, k and i, j, k, l, respectively.In the process of proving this, it is proved that the analogousLie rings are torsion free.     

Faithful Representations of Free Products

In 1940 Nisnevi published the following theorem [3]. Let (G) be a family of groups indexed by some set and (F) a family of fields of the same characteristic p0. Iffor each the group G has a faithful representation of degreen over F then the free product* G has a faithful representationof degree n+1 over some field of characteristic p. In [6] Wehrfritzextended this idea. If (G) GL(n, F) is a family of subgroupsfor which there exists ZGL(n, F) such that for all the intersectionGF.1_n=Z, then the free product of the groups *_ZG with Z amalgamatedvia the identity map is isomorphic to a linear group of degreen over some purely transcendental extension of F. Initially, the purpose of this paper was to generalize theseresults from the linear to the skew-linear case, that is, togroups isomorphic to subgroups of GL(n, D) where the D are divisionrings. In fact, many of the results can be generalized to ringswhich, although not necessarily commutative, contain no zero-divisors.We have the following.     

Witt Groups of the Punctured Spectrum of a 3-Dimensional Regular Local Ring and a Purity Theorem

Let A be a regular local ring with quotient field K. Assumethat 2 is invertible in A. Let W(A)W(K) be the homomorphisminduced by the inclusion AK, where W( ) denotes the Witt groupof quadratic forms. If dim A4, it is known that this map isinjective [6, 7]. A natural question is to characterize theimage of W(A) in W(K). Let Spec¹(A) be the set of prime idealsof A of height 1. For PSpec¹(A), let _P be a parameter of thediscrete valuation ring A_P and k(P) = A_P/PA_P. For this choiceof a parameter _P, one has the second residue homomorphism _P:W(K)W(k(P))[9, p. 209]. Though the homomorphism _P depends on the choiceof the parameter _P, its kernel and cokernel do not. We havea homomorphism A part of the so-called Gersten conjecture is the followingquestion on ‘purity’. Is the sequence exact? This question has an affirmative answer for dim(A)2 [1;3, p. 277]. There have been speculations by Pardon and Barge-Sansuc-Vogelon the question of purity. However, in the literature, thereis no proof for purity even for dim(A) = 3. One of the consequencesof the main result of this paper is an affirmative answer tothe purity question for dim(A) = 3. We briefly outline our main result.     

Set Theory is Interpretable in the Automorphism Group of an Infinitely Generated Free Group

In [6] S. Shelah showed that in the endomorphism semi-groupof an infinitely generated algebra which is free in a varietyone can interpret some set theory. It follows from his resultsthat, for an algebra F which is free of infinite rank in avariety of algebras in a language L, if > |L|, then thefirst-order theory of the endomorphism semi-group of F, Th(End(F)),syntactically interprets Th(,L₂), the second-order theory ofthe cardinal . This means that for any second-order sentence of empty language there exists *, a first-order sentence ofsemi-group language, such that for any infinite cardinal >|L|, Th(,L₂)*Th(End(F)) In his paper Shelah notes that it is natural to study a similarproblem for automorphism groups instead of endomorphism semi-groups;a priori the expressive power of the first-order logic for automorphismgroups is less than the one for endomorphism semi-groups. Forinstance, according to Shelah's results on permutation groups[4, 5], one cannot interpret set theory by means of first-orderlogic in the permutation group of an infinite set, the automorphismgroup of an algebra in empty language. On the other hand, onecan do this in the endomorphism semi-group of such an algebra. In [7, 8] the author found a solution for the case of the varietyof vector spaces over a fixed field. If V is a vector spaceof an infinite dimension over a division ring D, then the theoryTh(, L₂) is interpretable in the first-order theory of GL(V),the automorphism group of V. When a field D is countable anddefinable up to isomorphism by a second-order sentence, thenthe theories Th(GL(V)) and Th(, L₂) are mutually syntacticallyinterpretable. In the general case, the formulation is a bitmore complicated. The main result of this paper states that a similar result holdsfor the variety of all groups.     

Residual Finiteness of Quasi-Positive One-Relator Groups

A criterion is given for showing that certain one-relator groupsare residually finite. This is applied to a one-relator groupwith torsion G = a₁,...,a_r|Wⁿ. It is shown that G is residuallyfinite provided that W is outside the commutator subgroup andn is sufficiently large. An important ingredient in the proofis a criterion which implies that a subgroup of a group is malnormal.A graded small-cancellation criterion is developed which detectswhether a map A B between graphs induces a ₁-injection, andwhether ₁A maps to a malnormal subgroup of ₁B.     

On Free Modules for Finite Subgroups of Algebraic Groups

We show that given an affine algebraic group G over a fieldK and a finite subgroup scheme H of G there exists a finitedimensional G-module V such that V|_H is free. The problem israised in the recent paper by Kuzucuglu and Zalesski [15] which containsa treatment of the special case in which K is the algebraicclosure of a finite field and H is reduced. Our treatment isdivided into two parts, according to whether K has zero or positivecharacteristic. The essence of the characteristic 0 case isa proof that, for given n, there exists a polynomial GL_n(Q)-moduleV of dimension , where the product is over all primes less than or equal to n+1, such thatV is free as a QH-module for every finite subgroup H of GL_n(Q).The module V is the tensor product of the exterior algebra *(E),on the natural GL_n(Q)-module E, and Steinberg modules St_p, onefor each prime not exceeding n+1. The Steinberg modules alsoplay the major role in the case in which K has characteristicp>0 and the key point in our treatment is to show that fora finite subgroup scheme H of a general linear group scheme(or universal Chevalley group scheme) G over K, the Steinbergmodule St_pⁿ for G is injective (and projective) on restrictionto H for n>>0. A curious consequence of this is that,despite the wild behaviour of the modular representation theoryof finite groups, one has the following. Let H be a finite groupand V a finite dimensional vector space. Then there exists a(well-understood) faithful rational representation GL(V)GL(W)such that, for each faithful representation : HGL(V), the compositeo: HGL(W) is free, in particular all representations o are equivalent.     

The Non-Finite Presentability of the Automorphism Group of the Free Z-Group of Rank Two

Let G be a group, and let F_n[G] be the free G-group of rankn. Then F_n[G] is just the natural non-abelian analogue of thefree ZG-module of rank n, and correspondingly the group _n(G)of equivariant automorphisms of F_n[G] is a natural analogueof the general linear group GL_n(ZG). The groups _n(G) have beenstudied recently in [4, 8, 5]. In particular, in [5] it wasshown that if G is not finitely presentable (f.p.) then neitheris _n(G), and conversely, that _n(G) is f.p. if G is f.p. andn2. It is a common phenomenon that for small ranks the automorphismgroups of free objects may behave unstably (see the survey article[2]), and the main aim of the present paper is to show thatthis turns out to be the case for the groups ₂(G).     

Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rankn defined over the field k, where n3. Let be the root systemassociated with G(k) and let ={₁, ₂, ..., _n} be a set of fundamentalroots of , with ⁺ being the set of positive roots of with respectto . For and ⁺, let n() be the coefficient of in the expressionof as a sum of fundamental roots; so =n(). Also we recall thatht(), the height of , is given by ht()=n(). The highest rootin ⁺ will be denoted by . We additionally assume that the Dynkindiagram of G(k) is connected.     

Metrical Theorems on Prime Values of the Integer Parts of Real Sequences II

Let [ ] denote the integer part. Among other results in [3]we gave a complete solution to the following problem. PROBLEM. Given an increasing sequence a_n R⁺, n = 1, 2, ...,where a_n as n , are there infinitely many primes in the sequence[a_n] for almost all ?     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.     

Homology of GLn over algebraically closed fields

In this paper we define higher pre-Bloch groups _n(F) of a fieldF. When the base field is algebraically closed, we study itsconnection to the homology of the general linear groups withcoefficients in /l , where l is a positive integer. As a resultof our investigation we give a necessary and sufficient conditionfor the natural map H_n(GL_n–1(F), /l ) H_n(GL_n(F), /l )to be bijective. We prove that this map is bijective for n4.We also demonstrate that a certain property of _n() is equivalentto the validity of the Friedlander–Milnor isomorphismconjecture for (n+1)th homology of GL_n().     

The Bestvina-Brady Construction Revisited: Geometric Computation of {sum}-Invariants for Right-Angled Artin Groups

The starting point of our investigation is the remarkable paper[2] in which Bestvina and Brady gave an example of an infinitelyrelated group of type FP₂. The result about right-angled Artingroups behind their example is best interpreted by means ofthe Bieri–Strebel–Neumann–Renz -invariants. For a group G the invariants ⁿ(G) and ⁿ(G, Z) are sets of non-trivialhomomorphisms :GR. They contain full information about finitenessproperties of subgroups of G with abelian factor groups. Themain result of [2] determines for the canonical homomorphism, taking each generator of the right-angled Artin group G to1, the maximal n with ⁿ(G), respectively ⁿ(G, Z). In [6] Meier, Meinert and VanWyk completed the picture by computingthe full -invariants of right-angled Artin groups using as wellthe result of Bestvina and Brady as algebraic techniques from-theory. Here we offer a new account of their result which istotally geometric. In fact, we return to the Bestvina–Bradyconstruction and simplify their argument considerably by bringinga more general notion of links into play. At the end of thefirst section we re-prove their main result. By re-computingthe full -invariants, we show in the second section that thesimplification even adds some power to the method. The criterionwe give provides new insight on the geometric nature of the‘n-domination’ condition employed in [6].     

The Consistency of Holt's Conjectures on Cohomological Dimension of Locally Finite Groups

Let G be a locally finite group of cardinality _n where n isa natural number. Let (G) be the set of primes p for which Ghas an element of order p. In [5], Holt conjectures that ifk is a finite field with char k (G) then (1) G has cohomological dimension n+1 over k; (2) Hⁿ⁺¹(G, kG) has cardinality 2^_n; (3) Hⁱ(G, kG) = 0 for 0 i n.     

An extension of the recursively enumerable Turing degrees

Consider the countable semilattice _T consisting of the recursivelyenumerable Turing degrees. Although _T is known to be structurallyrich, a major source of frustration is that no specific, naturaldegrees in _T have been discovered, except the bottom and topdegrees, 0 and 0'. In order to overcome this difficulty, weembed _T into a larger degree structure which is better behaved.Namely, consider the countable distributive lattice _w consistingof the weak degrees (also known as Muchnik degrees) of massproblems associated with non-empty ₀¹ subsets of 2. It is knownthat _w contains a bottom degree 0 and a top degree 1 and isstructurally rich. Moreover, _w contains many specific, naturaldegrees other than 0 and 1. In particular, we show that in _wone has 0 < d < r₁ < f(r₂, 1) < 1. Here, d is theweak degree of the diagonally non-recursive functions, and r_nis the weak degree of the n-random reals. It is known that r₁can be characterized as the maximum weak degree of a ₀¹ subsetof 2 of positive measure. We now show thatf(r₂, 1) can be characterizedas the maximum weak degree of a ₀¹ subset of 2, the Turing upwardclosure of which is of positive measure. We exhibit a naturalembedding of _T into _w which is one-to-one, preserves the semilatticestructure of _T, carries 0 to 0, and carries 0' to 1. Identifying_T with its image in _w, we show that all of the degrees in _Texcept 0 and 1 are incomparable with the specific degrees d,r₁, andf(r₂, 1) in _w.     

On the Discreteness and Convergence in n-Dimensional Mobius Groups

Throughout this paper, we adopt the same notations as in [1,6, 8] such as the Möbius group M(Rⁿ), the Clifford algebraC_n–1, the Clifford matrix group SL(2, _n), the Cliffordnorm of ||A||=(|a|²+|b|²+|c|²+|d|²) (1) and the Clifford metric of SL(2, _n) or of the Möbius groupM(Rⁿ) d(A₁,A₂)=||A₁–A₂||(|a₁–a₂|²+|b₁–b₂|²+|c₁–c₂|²+|d₁–d₂|²)(2) where |·| is the norm of a Clifford number and represents f_i M(), i = 1,2, and so on. In addition, we adopt some notions in [6, 12]:the elementary group, the uniformly bounded torsion, and soon. For example, the definition of the uniformly bounded torsionis as follows.     

On the Left Cell Representations of Iwahori-Hecke Algebras of Finite Coxeter Groups

In this paper we investigate the left cell representations ofthe Iwahori-Hecke algebras associated to a finite Coxeter groupW. Our main result shows that , where w₀ is the element of longest length in W, acts essentiallyas an involution upon the canonical bases of a cell representation.We describe some properties of this involution, use it to furtherdescribe the left cells, and finally show how to realize eachcell representation as a submodule of . Our results rely uponcertain positivity properties of the structure constants ofthe Kazhdan-Lusztig bases of the Hecke algebra and so have notyet been shown to apply to all finite Coxeter groups.     

Base Sizes and Regular Orbits for Coprime Affine Permutation Groups

Let G be a permutation group on a finite set . A sequence B=(₁,..., _b) of points in is called a base if its pointwise stabilizerin G is the identity. Bases are of fundamental importance incomputational algorithms for permutation groups. For both practicaland theoretical reasons, one is interested in the minimal basesize for (G, ), For a nonredundant base B, the elementary inequality2^|B||G|||^|B| holds; in particular, |B|log|G|/log||. In the casewhen G is primitive on , Pyber [8, p. 207] has conjectured thatthe minimal base size is less than Clog|G|/log|| for some (large)universal constant C. It appears that the hardest case of Pyber's conjecture is thatof primitive affine groups. Let H=GV be a primitive affine group;here the point stabilizer G acts faithfully and irreduciblyon the elementary abelian regular normal subgroup V of H, andwe may assume that =V. For positive integers m, let mV denotethe direct sum of m copies of V. If (v₁, ..., v_m)mV belongsto a regular G-orbit, then (0, v₁, ..., v_m) is a base for theprimitive affine group H. Conversely, a base (₁, ..., _b) forH which contains 0V= gives rise to a regular G-orbit on (b–1)V. Thus Pyber's conjecture for affine groups can be viewed asa regular orbit problem for G-modules, and it is therefore aspecial case of an important problem in group representationtheory. For a related result on regular orbits for quasisimplegroups, see [4, Theorem 6].     

Immanant Inequalities, Induced Characters, and Rank Two Partitions

If = {₁, ₂, ..., _s}, where _{1 2} ... _s > 0, is a partitionof n then denotes the associated irreducible character of S_n,the symmetric group on {1, 2, ..., n}, and, if cCS_n, the groupalgebra generated by C and S_n, then d_c(·) denotes thegeneralized matrix function associated with c. If c₁, c₂ CS_nthen we write c₁ c₂ in case (A) (A) for each n x n positivesemi-definite Hermitian matrix A. If cCS_n and c(e) 0, wheree denotes the identity in S_n, then or denotes (c(e))^–1 c. The main result, an estimate for the norms of tensors of a certainanti-symmetry type, implies that if = {₁, ₂, ..., _s, 1^t} isa partition of n such that s > 1 and _s = 2, and ' denotes{₁, ₂, ..., _s-1, 1^t} then (, _{2}) where denotes characterinduction from S_n–2 x S₂ to S_n. This in turn implies thatif = {₁, ₂, ..., _s, 1^t} with s > 1, _s = 2, and ßdenotes {₁ + 2, ₂, ..., _s-1, 1^t} then _ß which,in conjunction with other known results, provides many new inequalitiesamong immanants. In particular it implies that the permanentfunction dominates all normalized immanants whose associatedpartitions are of rank 2, a result which has proved elusivefor some years. We also consider the non-relationship problem for immanants– that is the problem of identifying pairs, (,ß)such that _ß and _ß are both false.