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).     

Nilpotent-by-Finite Cofinitary Groups

Throughout this paper, D denotes a division ring (possibly commutative)and V a left vector space over D, usually, but not exclusively,infinite-dimensional. We consider irreducible subgroups G ofGL(V) and are particularly interested in such G that containan element g the fixed-point set C_V(g) of which is non-zerobut finite-dimensional (over D). We then use this to deriveconclusions about cofinitary groups, an element g of GL(V) beingcofinitary if dim_DC_V(g) is finite, and a subgroup of GL(V) beingcofinitary if all its non-identity elements are cofinitary. Suppose that G is a cofinitary subgroup of GL(V). There aretwo extreme cases. If dim_DV is finite the cofinitary conditionis vacuous. At the other extreme, if G acts fixed-point freelyon V then the fixed-point sets C_V(g) for gG\1 are as small aspossible, namely {0}. Work of Blichfeldt and his successorsshows that certain irreducible linear groups G of dimensionat least 2 over, for example, the complexes are always imprimitive.This is the case if G is nilpotent, or supersoluble, or metabelian.Apart from the two extreme cases, the same is frequently truefor irreducible cofinitary subgroups G of GL(V). For example,this is the case if G is finitely generated nilpotent [9, 1.2]or more generally if G is supersoluble [10, 1.1], but not ingeneral if G is metabelian [10, 7.1] or parasoluble (a groupG is parasoluble if it has a normal series of finite lengthsuch that every subgroup of each of its factors is Abelian andnormalised by G) (see [10, 7.2]). Further, it is also the caseif G is Abelian-by-finite [10, 3.4], and every supersolublegroup is finitely generated and nilpotent-by-finite. Collectively,these results suggest that one should consider nilpotent-by-finitegroups.     

Oscillation Properties of a Semi-discrete Approximation to the Beam Equation with a Second Order Term

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx–[p(x)u_x]_x=0, 0< x < L, t > 0, where it is assumed that w, p,and q are positive on the interval [0, L], is approximated bythe method of straight lines. The resulting approximation isa linear system of differential equations with coefficient matrixS. The matrix S is studied under a variety of boundary conditionswhich result in a conservative system. In all cases the matrixS is shown to be similar to an oscillation matrix.     

A Matrix Setting for the q-Schur Algebra

The Schur algebra S(n, r) has a basis (described in [6, 2.3])consisting of certain elements _i,j, where i, jI(n, r), the setof all ordered r-tuples of elements from the set n={1, 2, ...,n}. The multiplication of two such basis elements is given bya formula known as Schur's product rule. In recent years, aq-analogue S_q(n, r) of the Schur algebra has been investigatedby a number of authors, particularly Dipper and James [3, 4].The main result of the present paper, Theorem 3.6, shows howto embed the q-Schur algebra in the rth tensor power T^r(M_n)of the nxn matrix ring. This embedding allows products in theq-Schur algebra to be computed in a straightforward manner,and gives a method for generalising results on S(n, r) to S_q(n,r). In particular we shall make use of this embedding in subsequentwork to prove a straightening formula in S_q(n, r) which generalisesthe straightening formula for codeterminants due to Woodcock[12]. We shall be working mainly with three types of algebra: thequantized enveloping algebra U(gl_n) corresponding to the Liealgebra gl_n, the q-Schur algebra S_q(n, r), and the Hecke algebra,H(A_r–1). It is often convenient, in the case of the q-Schuralgebra and the Hecke algebra, to introduce a square root ofthe usual parameter q which will be denoted by v, as in [5].This corresponds to the parameter v in U(gl_n). We shall denotethis ‘extended’ version of the q-Schur algebra byS_v(n, r), and we shall usually refer to it as the v-Schur algebra.All three algebras are associative and have multiplicative identities,and the base field will be the field of rational functions,Q(v), unless otherwise stated. The symbols n and r shall bereserved for the integers given in the definitions of thesethree algebras.     

The Steinberg Lattice of a Finite Chevalley Group and its Modular Reduction

Let p be a prime and let q = p^a, where a is a positive integer.Let G 7equals; G(F_q) be a Chevalley group over F_q, with associatedsystem of roots and Weyl group W. Steinberg showed in 1957that G has an irreducible complex representation whose degreeequals the p-part of |G| [11]. This representation, now knownas the Steinberg representation, has remarkable properties,which reflect the structure of G, and there have been many researchpapers devoted to its study. The module constructed in [11]is in fact a right ideal in the integral group ring ZG of G,and is thus a ZG-lattice, which we propose to call the Steinberglattice of G. It should be noted that lattices not integrallyisomorphic to the Steinberg lattice may also afford the Steinbergrepresentation, and such lattices may differ considerably intheir properties compared with the Steinberg lattice.     

On the division of polynomial matrices

The main purpose of this paper is to determine two new algorithmsfor the division of the polynomial matrix B(s) R[s]^pxq by A(s) R[s]^pxp (a) based on the Laurent matrix expansion at s = =of the inverse of A(s), i.e. A(s)^–1, and (b) in a waysimilar to the one presented by Gantmacher (1959).     

A Bound on the Presentation Rank of a Finite Group

Let G be a finite group, and let I_G be the augmentation idealof ZG. We denote by d(G) the minimum number of generators forthe group G, and by d(I_G) the minimum number of elements ofI_G needed to generate I_G as a G-module. The connection betweend(G) and d(I_G) is given by the following result due to Roggenkamp]14]: where pr(G) is a non-negative integer, called the presentationrank of G, whose definition comes from the study of relationmodules (see [4] for more details). 1991 Mathematics SubjectClassification 20D20.     

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].     

Kazhdan-Lusztig Cells, q-Schur Algebras and James' Conjecture

We consider the Dipper–James q-Schur algebra S_q(n, r)_k,defined over a field k and with parameter q 0. An understandingof the representation theory of this algebra is of considerableinterest in the representation theory of finite groups of Lietype and quantum groups; see, for example, [6] and [11]. Itis known that S_q(n, r)_k is a semisimple algebra if q is nota root of unity. Much more interesting is the case when S_q(n,r)_k is not semisimple. Then we have a corresponding decompositionmatrix which records the multiplicities of the simple modulesin certain ‘standard modules’ (or ‘Weyl modules’).A major unsolved problem is the explicit determination of thesedecomposition matrices.     

The Complete Reducibility of Locally Completely Reducible Finitary Linear Groups

Let V be a vector space over some division ring D, and G a finitarysubgroup of GL(V). If G is locally completely reducible, thenthe D-G modules V, [V, G] and V/C_V(G) need not be completelyreducible, even if dim_DV is finite. Moreover, if F is a field,then V and V/C_V(G) need not be completely reducible. We provehere that if D is a finite-dimensional division algebra andG is locally completely reducible, then [V, G] is always a completelyreducible D-G module. 1991 Mathematics Subject Classification20H25.     

Multistep approximation of linear sectorial evolution equations

This paper concerns the linear multistep approximation of alinear sectorial evolution equation u_t = Au on a complex Banachspace X. Given a strictly A()-stable q-step method of orderp whose stability region includes a sectorial region containingthe spectrum of the operator A, the corresponding evolutionsemigroup for the method is Cⁿ(hA), n 0, defined on X^q, whereC(z) L (C^q) denotes the one-step map associated with the method.It is shown that for appropriately chosen V, Y: C C^q, basedon the principal right and left eigenvectors of C(z), Cⁿ(hA)approximates the semigroup V(hA)e^nhAY^H(hA) with optimal orderp.     

Invariants of Finite Reflection Groups and the Mean Value Problem for Polytopes

Let P be an n-dimensional polytope admitting a finite reflectiongroup G as its symmetry group. Consider the set H_P(k) of allcontinuous functions on Rⁿ satisfying the mean value propertywith respect to the k-skeleton P(k) of P, as well as the setH_G of all G-harmonic functions. Then a necessary and sufficientcondition for the equality H_P(k) = H_G is given in terms of adistinguished invariant basis, called the canonical invariantbasis, of G. 1991 Mathematics Subject Classification 20F55,52B15.     

On Waring's Problem in Number Fields

Let K be an algebraic number field of degree n over the rationals,and denote by J_k the subring of K generated by the kth powersof the integers of K. Then G_K(k) is defined to be the smallests1 such that, for all totally positive integers vJ_k of sufficientlylarge norm, the Diophantine equation (1.1) is soluble in totally non-negative integers _i of K satisfying N(_i)<<N(v)^1/k (1is). (1.2) In (1.2) and throughout this paper, all implicit constants areassumed to depend only on K, k, and s. The notation G_K(k) generalizesthe familiar symbol G(k) used in Waring's problem, since wehave G_Q(k) = G(k). By extending the Hardy–Littlewood circle method to numberfields, Siegel [8, 9] initiated a line of research (see [1–4,11]) which generalized existing methods for treating G(k). Thistypically led to upper bounds for G_K(k) of approximate strengthnB(k), where B(k) was the best contemporary upper bound forG(k). For example, Eda [2] gave an extension of Vinogradov'sproof (see [13] or [15]) that G(k)(2+o(1))k log k. The presentpaper will eliminate the need for lengthy generalizations assuch, by introducing a new and considerably shorter approachto the problem. Our main result is the following theorem.     

Finite Modules of Finite Injective Dimension Over a Noetherian Algebra

Let R be a commutative Noetherian ring. Let P(R) (respectively,I(R)) be the category of all finite R-modules of finite projective(respectively, injective) dimension. Sharp [9] constructed acategory equivalence between I(R) and P(R) for certain Cohen–Macaulaylocal rings R. Thus many properties about finite modules offinite projective dimension can be connected with those of finiteinjective dimension through this equivalence.     

Algebraic Cycles and Even Unimodular Lattices

The number (up to isomorphism) of positive-definite, even, unimodularlattices of rank 8r grows rapidly with r. However, Bannai [1]has shown that, when counted according to weight, those withnon-trivial automorphisms make up a fraction of the whole, whichgoes rapidly to zero as r. Therefore it is of some interestto produce families of positive-definite, even, unimodular latticeswith large automorphism groups and unbounded ranks. Suppose that G is a finite group and V is an irreducible Q[G]-modulesuch that VR is still irreducible. Then, as observed by Gross[8], the space of G-invariant symmetric bilinear forms on Vis one-dimensional and is necessarily generated by a positive-definiteform, unique up to scaling by non-zero positive rationals. Thompson[23] showed that, if V is also irreducible modp for all primesp, then it contains an invariant lattice (unique up to scaling)which is even and unimodular with appropriate scaling of thequadratic form. Examples arising in this manner are the E₈-latticeof rank 8, the Leech lattice of rank 24 and the Thompson–Smithlattice of rank 248. Gow [6] has also constructed some examplesassociated with the basic spin representations of 2A_n and 2S_n.     

Spaces of Harmonic Functions

It is important and interesting to study harmonic functionson a Riemannian manifold. In an earlier work of Li and Tam [21]it was demonstrated that the dimensions of various spaces ofbounded and positive harmonic functions are closely relatedto the number of ends of a manifold. For the linear space consistingof all harmonic functions of polynomial growth of degree atmost d on a complete Riemannian manifold Mⁿ of dimension n,denoted by H_d(Mⁿ), it was proved by Li and Tam [20] that thedimension of the space H₁(M) always satisfies dimH₁(M) dimH₁(Rⁿ)when M has non-negative Ricci curvature. They went on to askas a refinement of a conjecture of Yau [32] whether in generaldim H_d(Mⁿ) dimH_d(Rⁿ)for all d. Colding and Minicozzi made animportant contribution to this question in a sequence of papers[5–11] by showing among other things that dimH_d(M) isfinite when M has non-negative Ricci curvature. On the otherhand, in a very remarkable paper [16], Li produced an elegantand powerful argument to prove the following. Recall that Msatisfies a weak volume growth condition if, for some constantA and , (1.1) for all x M and r R, where V_x(r) is the volume of the geodesicball B_x(r) in M; M has mean value property if there exists aconstant B such that, for any non-negative subharmonic functionf on M, (1.2) for all p M and r > 0.     

Classification Des Ideaux a Droite De A1(k)

Les études récentes sur les idéaux àdroite de A₁(k), la première algèbre de Weyl surun corps algébriquement clos et de caractéristiquenulle k, nous montrent que : pour tout idéal I 0 àdroite de A₁(k), il existe x Q = frac(A₁(k)), et V V telsque : I = xD(R, V) o V est l'ensemble des sous-espaces primairementdécomposables de k[t] = R, et D(R, V), l'idéalà droite {d A₁(k/d(R V}. Dans cet article nous montreronsprincipalement que: pour tout 0 I idéal à droitede A₁(k, !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),où X_n est la courbe d'algèbre des fonctions régulières: O(X_n = k+tⁿ⁺¹k[t]. La forme des idéaux décriteci-dessus permet de voir dans une hypothèse de Letzteret Makar-Limanov, pour deux courbes algébriques affinesX et X' on a : D(XD(X') co dim D(X = co dim D(X'). Recent studies on right ideals of the first Weyl algebra A₁(k)over an algebraic closed field k with characteristic zero showthat: for each right ideal I 0 of A₁(k), there exist x Q =fracA₁(k)) and a primary decomposable sub-space V of k[t] suchthat I=xD(R,V), where D(R,V) : = {d A₁(k)/d(R) V} is a rightideal of A₁(k). In this paper, we show that for all right idealsI 0 of A₁(k), !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),where X_n denotes the affine algebraic curve with ring of regularfunctions O(X_n=k+tⁿ⁺¹k[t]. With ideals as described above, onecan easily see, under a hypothesis given by Letzter and Makar-Limanov,that for two affine algebraic curves X and X', D(X)D(X') codim D(X) = co dim D(X'). 2000 Mathematics Subject Classification16S32.     

Simultaneous prime specializations of polynomials over finite fields

Recently the author proposed a uniform analogue of the Bateman–Hornconjectures for polynomials with coefficients from a finitefield (that is, for polynomials in F_q[T] rather than Z[T]).Here we use an explicit form of the Chebotarev density theoremover function fields to prove this conjecture in particularranges of the parameters. We give some applications includingthe solution of a problem posed by Hall.     

On the Number of Solutions of an Equation Over a Finite Field

The Stöhr–Voloch approach is used to obtain a newbound for the number of solutions in (F_q)² of an equation f(X,Y) = 0, where f(X, Y) is an absolutely irreducible polynomialwith coefficients in a finite field F_q.     

Reduction of CalderON Commutators

Let A₁,..., A_n be Lipschitz functions on R such that A'₁,...,A'_nVMO. We show that on any bounded interval, the Calderóncommutator associated with the kernel (A₁(x)–A₁(y)) ...(A_n(x) – A_n(y))/(x–y) ⁿ¹ is a compact perturbationof , where H is the Hilberttransform. 1991 Mathematics Subject Classification 47B38, 47B47,47G10, 45E99.