Seminormal Representations of Weyl Groups and Iwahori-Hecke Algebras

The purpose of this paper is to describe a general procedurefor computing analogues of Young's seminormal representationsof the symmetric groups. The method is to generalize the Jucys-Murphyelements in the group algebras of the symmetric groups to arbitraryWeyl groups and Iwahori-Hecke algebras. The combinatorics ofthese elements allows one to compute irreducible representationsexplicitly and often very easily. In this paper we do thesecomputations for Weyl groups and Iwahori-Hecke algebras of typesA_n, B_n, D_n, G₂. Although these computations are in reach fortypes F₄, E₆ and E₇, we shall postpone this to another work.1991 Mathematics Subject Classification: primary 20F55, 20C15;secondary 20C30, 20G05.     

Factor Algebras of Free Algebras: On a Problem of G. Bergman

Let A_n = K x₁,...,x_n be a free associative algebra over a fieldK. In this paper, examples are given of elements u A_n, n 3,such that the factor algebra of A_n over the ideal generatedby u is isomorphic to A_n–1, and yet u is not a primitiveelement of A_n (that is, it cannot be taken to x₁ by an automorphismof A_n). If the characteristic of the ground field K is 0, thisyields a negative answer to a question of G. Bergman. On theother hand, by a result of Drensky and Yu, there is no suchexample for n = 2. It should be noted that a similar questionfor polynomial algebras, known as the embedding conjecture ofAbhyankar and Sathaye, is a major open problem in affine algebraicgeometry. 2000 Mathematics Subject Classification 16S10, 16W20(primary); 14A05, 13B25 (secondary).     

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.     

Locally Finite Groups All of Whose Subgroups are Boundedly Finite over Their Cores

For n a positive integer, a group G is called core-n if H/H_Ghas order at most n for every subgroup H of G (where H_G is thenormal core of H, the largest normal subgroup of G containedin H). It is proved that a locally finite core-n group G hasan abelian subgroup whose index in G is bounded in terms ofn. 1991 Mathematics Subject Classification 20D15, 20D60, 20F30.     

Minimal Immersions of Kahler Manifolds into Euclidean Spaces

It is proved here that a minimal isometric immersion of a Kähler-Einsteinor homogeneous Kähler-manifold into an Euclidean spacemust be totally geodesic. As an application, it is shown thatan open subset of the real hyperbolic plane RH² cannot be minimallyimmersed into the Euclidean space. As another application, aproof is given that if an irreducible Kähler manifold isminimally immersed in a Euclidean space, then its restrictedholonomy group must be U(n), where n = dim_CM. 2000 MathematicsSubject Classification 53B25 (primary); 53C42 (secondary).     

Nonoscillation of Elliptic Integrals Related to Cubic Polynomials with Symmetry of Order Three

We study zeros of elliptic integrals I(h)=_HhR(x,y)dxdy, whereH(x,y) is a real cubic polynomial with a symmetry of order three,and R(x,y) is a real polynomial of degree at most n. It turnsout that the vector space A_n formed by such integrals is a Chebishevsystem: the number of zeros of each elliptic integral I(h)A_nis less than the dimension of the vector space A_n. 1991 MathematicsSubject Classification 34C10.     

The Solubility of Diagonal Cubic Diophantine Equations

This paper proves conditional existence results for non-trivialsolutions of the equation where the coefficients a_i and the unknowns X_i are taken to berational integers. No such results were previously known for n6. The proofs useelementary facts about the 3-descent procedure for ellipticcurves of the form E_A: X³ + Y³ = AZ³. Thus, when n=4, and the a_i are each prime, and are all congruentto 2 modulo 3, it is shown that (*) will have non-trivial solutions,providing that the Selmer conjecture holds for the curves E_A.One may replace the Selmer conjecture by an appropriate formof the Generalized Riemann Hypothesis, when n=5 and the a_i areagain taken to be primes, all congruent to 8 modulo 9. Finally,when n=5, one may require only that the a_i be square-free andcoprime to 3, providing one assumes both the Selmer conjectureand a special case of Schinzel's conjecture (on the representationof primes by cubic polynomials). 1991 Mathematics Subject Classification:11D25, 11G05, 14G05.     

An error bound for the modified successive overrelaxation method

In this paper we consider the modified successive overrelaxation(MSOR)methodto appropriate the solution of the linear system D^-1/2 Ax =D^-1/2b, where A is a symmetric, positive definite and consistentlyordered matrix and D is a diagonal matrix with the diagonalidentical to that of A. The main purpose of this paper is to obtain some theoreticalresults, namely a bound for the norm of _n = v –v_n in termsof the norms _n –v_n-1, _n+1 –v_n and their inner product,where v =D^-1/2 x and v_n is the nth iteration vector, obtainedusing the (MSOR)method.     

On Decomposition Numbers and Branching Coefficients for Symmetric and Special Linear Groups

In this paper we find the multiplicities dim L()_– where is an arbitrary root and L() is an irreducible SL_n-module withhighest weight . We provide different bases of the correspondingweight spaces and outline some applications to the symmetricgroups. In particular we describe certain composition multiplicitiesin the modular branching rule. 1991 Mathematics Subject Classification:20C05, 20G05.     

Rado Partition Theorem for Random Subsets of Integers

For an l x k matrix A = (a_ij) of integers, denote by L(A) thesystem of homogenous linear equations a_i1x₁ + ... + a_ikx_k =0, 1 i l. We say that A is density regular if every subsetof N with positive density, contains a solution to L(A). Fora density regular l x k matrix A, an integer r and a set ofintegers F, we write if for any partition F = F₁ ... F_r there exists i {1, 2,..., r} and a column vector x such that Ax = 0 and all entriesof x belong to F_i. Let [n]_N be a random N-element subset of{1, 2, ..., n} chosen uniformly from among all such subsets.In this paper we determine for every density regular matrixA a parameter = (A) such that lim_n P([n]_N (A)_r)=0 if N =O(n) and 1 if N = (n). 1991 Mathematics Subject Classification:05D10, 11B25, 60C05     

On Tensor Products of Modular Representations of Symmetric Groups

Let F be a field, and let _n be the symmetric group on n letters.In this paper we address the following question: given two irreducibleF_n-modules D₁ and D₂ of dimensions greater than 1, can it happenthat D₁ D₂ is irreducible? The answer is known to be ‘no’if char F = 0 [12] (see also [2] for some generalizations).So we assume from now on that F has positive characteristicp. The following conjecture was made in [4]. CONJECTURE. Let D₁ and D₂ be two irreducible F_n-modules of dimensionsgreater than 1. Then D₁ D₂ is irreducible if and only if p= 2, n = 2 + 4l for some positive integer l; one of the modulescorresponds to the partition (2l + 2, 2l) and the other correspondsto a partition of the form (n – 2j – 1, 2j 1), 0 j < l. Moreover, in the exceptional cases, one has The main result of this paper is the following theorem, whichestablishes a big part of the conjecture. 1991 Mathematics SubjectClassification 20C20, 20C30.     

Cm Elliptic Curves and Bicolorings of Steiner Triple Systems

It is shown that a necessary condition for the existence ofa bicolored Steiner triple system of order n is that n can bewritten in the form A²+3B² for integers A and B. In the casewhen n=q is either a prime congruent to 1 mod 3, or the squareof a prime congruent to 2 mod 3, it is shown that the numbersof colored vertices in the triple system would be unique, andare given by the number of points on specific twists of theCM elliptic curve y²=x³–1 over the finite field F_q. 2000Mathematics Subject Classification 05B07, 11G20, 14G15 (primary);11G15, 14K22 (secondary).     

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.     

Modular Subgroup Arithmetic and a Theorem of Philip Hall

A surprising relationship is established in this paper, betweenthe behaviour modulo a prime p of the number S_n G of index nsubgroups in a group G, and that of the corresponding subgroupnumbers for a normal subgroup in G normal subgroup in p-powerorder. The proof relies, among other things, on a twisted versiondue to Philip Hall of Frobenius' theorem concerning the equationx^m=1 in finite groups. One of the applications of this result,presented here, concerns the explicit determination modulo pof S_n G in the case when G is the fundamental group of a treeof groups all of whose vertex groups are cyclic of p-power order.Furthermore, a criterion is established (by a different technique)for the function S_n G to be periodic modulo p. 2000 MathematicsSubject Classification 20E06, 20F99 (primary); 05A15, 05E99(secondary).     

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.     

A WKB analysis of the buckling condition for a cylindrical shell of arbitrary thickness subjected to an external pressure

We consider the plane-strain buckling of a cylindrical shellof arbitrary thickness which is made of a Varga material andis subjected to an external hydrostatic pressure on its outersurface. The WKB method is used to solve the eigenvalue problemthat results from the linear bifurcation analysis. We show thatthe circular cross-section buckles into a non-circular shapeat a value of µ₁ which depends on A₁/A₂ and a mode number,where A₁ and A₂ are the undeformed inner and outer radii, andµ₁ is the ratio of the deformed inner radius to A₁. Inthe large mode number limit, we find that the dependence ofµ₁ on A₁/A₂ has a boundary layer structure: it is constantover almost the entire region of 0 < A₁/A₂ < 1 and decreasessharply from this constant value to unity as A₁/A₂ tends tounity. Our asymptotic results for A₁ – 1 = O(1) and A₁– 1 = O(1/n) are shown to agree with the numerical resultsobtained by using the compound matrix method.     

Enumerating Transitive Finite Permutation Groups

Denote by f(n) the number of subgroups of the symmetric groupSym(n) of degree n, and by f_trans(n) the number of its transitivesubgroups. It was conjectured by Pyber [9] that almost all subgroupsof Sym(n) are not transitive, that is, f_trans(n)/f(n) tendsto 0 when n tends to infinity. It is still an open questionwhether or not this conjecture is true. The difficulty comesfrom the fact that, from many points of view, transitivity isnot a really strong restriction on permutation groups, and thereare too many transitive groups [9, Sections 3 and 4]. In thispaper we solve the problem in the particular case of permutationgroups of prime power degree, proving the following result.1991 Mathematics Subject Classification 20B05, 20D60.     

Hochschild (Co)Homology Dimension

In 1989 Happel asked the question whether, for a finite-dimensionalalgebra A over an algebraically closed field k, gl.dim A < if and only if hch.dim A < . Here, the Hochschild cohomologydimension of A is given by hch.dim A := inf{n N₀ | dim HHⁱ(A) = 0 for i > n}. Recently Buchweitz, Green, Madsen andSolberg gave a negative answer to Happel's question. They founda family of pathological algebras A_q for which gl.dim A_q = but hch.dim A_q = 2. These algebras are pathological in manyaspects. However, their Hochschild homology behaviors are notpathological any more; indeed one has hh.dim A_q = = gl.dimA_q. Here, the Hochschild homology dimension of A is given byhh.dim A := inf{n N₀ | dim HH_i(A) = 0 for i > n}. This suggestsposing a seemingly more reasonable conjecture by replacing theHochschild cohomology dimension in Happel's question with theHochschild homology dimension: gl.dim A < if and only ifhh.dim A < if and only if hh.dim A = 0. The conjecture holdsfor commutative algebras and monomial algebras. In the casewhere A is a truncated quiver algebra, these conditions areequivalent to the condition that the quiver of A has no orientedcycles. Moreover, an algorithm for computing the Hochschildhomology of any monomial algebra is provided. Thus the cyclichomology of any monomial algebra can be read off when the underlyingfield is characteristic 0.     

A Sharp Lp Inequality for Dyadic A1 Weights in Rn

The exact best possible range of p is determined such that anydyadic A₁ weight w on Rⁿ satisfies a reverse Hölder inequalityfor p, which depends on the dimension n and the correspondingA₁ constant of w. The proof is based on an effective linearizationof the dyadic maximal operator applied to dyadic step functions.2000 Mathematics Subject Classification 42B25.     

Common Multiples of Complete Graphs

A graph H is said to divide a graph G if there exists a setS of subgraphs of G, all isomorphic to H, such that the edgeset of G is partitioned by the edge sets of the subgraphs inS. Thus, a graph G is a common multiple of two graphs if eachof the two graphs divides G. This paper considers common multiples of a complete graph oforder m and a complete graph of order n. The complete graphof order n is denoted K_n. In particular, for all positive integersn, the set of integers q for which there exists a common multipleof K₃ and K_n having precisely q edges is determined. It is shown that there exists a common multiple of K₃ and K_nhaving q edges if and only if q 0 (mod 3), q 0 (mod n2) and (1) q 3 n2 when n 5 (mod 6); (2) q (n + 1) n2 when n is even; (3) q {36, 42, 48} when n = 4. The proof of this result uses a variety of techniques includingthe use of Johnson graphs, Skolem and Langford sequences, andequitable partial Steiner triple systems. 2000 MathematicalSubject Classification: 05C70, 05B30, 05B07.