Quantum affine Cartan matrices,Poincaré series of binary polyhedral groups,and reflection representations

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a, b, h; p, q, r) that one usually associates with such a group and hence with a simply-laced Coxeter–Dynkin diagram have a meaningful definition for the non-simply-laced diagrams, too, and as a byproduct we extend Saito’s formula for the determinant of the Cartan matrix to all cases. Returning to invariant theory we show that for each irreducible representation i of a binary tetrahedral, octahedral, or icosahedral group one can find a homomorphism into a finite complex reflection group whose defining reflection representation restricts to i.     

Existence of normal Hall subgroups by means of orders of products

Let G be a finite group, let π be a set of primes and let p be a prime. We characterize the existence of a normal Hall π‐subgroup in G in terms of the order of products of certain elements of G. This theorem generalizes a characterization of A. Moretó and the second author by using the orders of products of elements for those groups having a normal Sylow p‐subgroup 6 . As a consequence, we also give a π‐decomposability criterion for a finite group also by means of the orders of products.     

Uniquely factorizable elements and solvability of finite groups

In a finite group G every element can be factorized in such a way that there is one factor for each prime divisor p of | G |, and the order of this factor is p^α for some integer α ≧ 0. We define g ∈G to be uniquely factorizable if it has just one such factorization (whose factors must be pairwise commuting). We consider the existence of uniquely factorizable elements and its relation to the solvability of the group. We prove that G is solvable if and only if the set of all uniquely factorizable elements of G is the Fitting subgroup of G. We also prove various sufficient conditions for the non-existence of uniquely factorizable elements in non-solvable groups. Received: 9 June 2005     

Decomposition Numbers and Cartan Invariants of S p(4,q), q = 2 n,for p = 2*

Let G be the 4-dimensional sympletic group on a finite field of q elements, q a power of 2. We find all the decomposition numbers of G in characteristic 2, corresponding to the unipotent characters of G. We also find some of the Cartan invariants of G for p = 2.     

Cartan Subalgebras of Leibniz n-Algebras

The present article is devoted to the investigation of properties of Cartan subalgebras and regular elements in Leibniz n-algebras. The relationship between Cartan subalgebras and regular elements of given Leibniz n-algebra and Cartan subalgebras and regular elements of the corresponding factor n-Lie algebra is established.     

A note on the p-local rank

In this paper, we give an inductive definition of p-local rank of a p-block in a finite group G with p||G| and show a necessary and sufficient condition for a p-block B such that plr(B) = 2.Received: 10 March 2004     

Solutions and multiple solutions for problems with the <Emphasis Type="Italic">p</Emphasis>-Laplacian

In this paper we consider two nonlinear elliptic problems driven by the p-Laplacian and having a nonsmooth potential (hemivariational inequalities). The first is an eigenvalue problem and we prove that if the parameter λ < λ₂ = the second eigenvalue of the p-Laplacian, then there exists a nontrivial smooth solution. The second problem is resonant both near zero and near infinity for the principal eigenvalue of the p-Laplacian. For this problem we prove a multiplicity result. Our approach is variational based on the nonsmooth critical point theory. Second author is Corresponding author.     

On the amicability of orthogonal designs

Although it is known that the maximum number of variables in two amicable orthogonal designs of order 2ⁿp, where p is an odd integer, never exceeds 2n+2, not much is known about the existence of amicable orthogonal designs lacking zero entries that have 2n+2 variables in total. In this paper we develop two methods to construct amicable orthogonal designs of order 2ⁿp where p odd, with no zero entries and with the total number of variables equal or nearly equal to 2n+2. In doing so, we make a surprising connection between the two concepts of amicable sets of matrices and an amicable pair of matrices. With the recent discovery of a link between the theory of amicable orthogonal designs and space‐time codes, this paper may have applications in space‐time codes. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 240‐252, 2009     

A new characterization of periodic oscillations in periodic difference equations

In this paper, we characterize periodic solutions of p-periodic difference equations. We classify the periods into multiples of p and nonmultiples of p. We show that the elements of the set of multiples of p follow the well-known Sharkovsky’s ordering multiplied by p. On the other hand, we show that the elements of the set Γ_p of nonmultiples of p are independent in their existence. Moreover, we show the existence of a p-periodic difference equation with infinite Γ_p-set in which the maps are defined on a compact domain and agree exactly on a countable set. Based on the proposed classification, we give a refinement of Sharkovsky’s theorem for periodic difference equations.     

Realizable posets

Certain p-local orders in n-dimensional division algebras over the rational numbers occur as endomorphism rings of torsion-free abelian groups of rank n if and only if an associated finite poset P has a strict faithful representation of dimension less than |P| over the field with p elements. In this note we obtain a simple characterization of those finite posets which do not admit such a representation.Research supported in part by NSF grant DMS-8802833.     

On special types of nonholonomic contact elements

Our starting point has been a recent clarification of the role of semiholonomic contact elements in the theory of submanifolds of Cartan geometries, Kolá? and Vitolo (2010) [5]. We deduce some further properties of the iterated contact elements by using the general concept of contact (n,F)-element for a regular subcategory F of the category of nonholonomic r-jets. Special attention is paid to the incidence relation of contact F-elements of different dimensions.     

Homogeneous systems of parameters in cohomology algebras of finite groups

We shall show that mod p cohomology algebras of finite groups have systems of parameters with certain properties. Using this result, we can show that for a finite group G of p-rank at most three, the trivial kG-module k, where k is a field of characteristic p, has index zero. The index of kG-modules was introduced by J. F. Carlson in 1990 at the symposium Representations of algebras and related topics (Tsukuba).Received: 15 January 2001     

Riemannian manifolds carrying a pair of skew symmetric conformal vector fields

We deal with a Riemannian manifoldM carrying a pair of skew symmetric conformal vector fields (X, Y). The existence of such a pairing is determined by an exterior differential system in involution (in the sense of Cartan). In this case,M is foliated by 3-dimensional totally geodesic submanifolds. Additional geometric properties are proved. Supported by a JSPS postdoctoral fellowship.     

Best Simultaneous Local Approximation in the Lp Norms

We study the behavior of the best simultaneous approximation to two functions from a convex set in L^p spaces, 2<p<∞, on a finite union of intervals when its measure tends to zero. In particular, we give su?cient conditions over the differentiability of two functions to assure existence of the best simultaneous local approximation from the class of algebraic polynomials of a fixed degree. These conditions are weaker than the ordinary differentiability given in previous works. More precisely, we consider differentiable functions in the sense L^p.     

Improving two theorems of bose on difference families

In [2] R. C. Bose gives a sufficient condition for the existence of a (q, 5, 1) difference family in (GF(q), +)—where q ≡ 1 mod 20 is a prime power — with the property that every base block is a coset of the 5th roots of unity. Similarly he gives a sufficient condition for the existence of a (q, 4, 1) difference family in (GF(q, +)—where q ≡ 1 mod 12 is a prime power — with the property that every base block is the union of a coset of the 3rd roots of unity with zero. In this article we replace the mentioned sufficient conditions with necessary and sufficient ones. As a consequence, we obtain new infinite classes of simple difference families and hence new Steiner 2-designs with block sizes 4 and 5. In particular, we get a (p^4α, 5, 1)-DF for any odd prime p ≡ 2, 3 (mod 5), and a (p^2α, 4, 1)-DF for any odd prime p ≡ 2 (mod 3). © 1995 John Wiley & Sons, Inc.     

Vertices of simple modules and normal subgroups ofp-solvable groups

Let π be a set of prime numbers andG a finite π-separable group. Let θ be an irreducible π′-partial character of a normal subgroupN ofG and denote by I_π′ (G‖θ), the set of all irreducible π′-partial characters Φ ofG such that θ is a constituent of Φ_N. In this paper, we obtain some information about the vertices of the elements in I_π′ (G‖θ). As a consequence, we establish an analogue of Fong's theorem on defect groups of covering blocks, for the vertices of the simple modules (in characteristicsp) of a finitep-solvable group lying over a fixed simple module of a normal subgroup.     

A nonexistence result for abelian menon difference sets using perfect binary arrays

A Menon difference set has the parameters (4N ² ,2N ² -N, N ² -N). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian group contains a Menon difference set, wherep is an odd prime, |K|=p , andp ^j–1 (mod exp (H)) for somej. Using the viewpoint of perfect binary arrays we prove thatK must be cyclic. A corollary is that there exists a Menon difference set in the abelian group , where exp(H)=2 or 4 and |K|=3, if and only ifK is cyclic.This work is partially supported by NSA grant # MDA 904-92-H-3057 and by NSF grant # NCR-9200265. The author thanks the Mathematics Department, Royal Holloway College, University of London for its hospitality during the time of this researchThis work is partially supported by NSA grant # MDA 904-92-H-3067     

Existence and uniqueness for P-area minimizers in the Heisenberg group

In [3] we studied p-mean curvature and the associated p-minimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized p-area and associated ( p-) minimizers in general dimensions. We prove the existence and investigate the uniqueness of minimizers. Since this is reduced to solving a degenerate elliptic equation, we need to consider the effect of the singular set and this requires a careful study. We define the notion of weak solution and prove that in a certain Sobolev space, a weak solution is a minimizer and vice versa. We also give many interesting examples in dimension 2. An intriguing point is that, in dimension 2, a C ²-smooth solution from the PDE viewpoint may not be a minimizer. However, this statement is true for higher dimensions due to the relative smallness of the size of the singular set.     

New example of a variety of lie algebras with fractional exponent

It is shown that in the case of characteristic zero the variety generated by a simple infinitedimensional Lie algebra of Cartan type W ₂ has a fractional exponent.     

Distribution of matrices with restricted entries over finite fields

For a prime p, we consider some natural classes of matrices over a finite field F_p of p elements, such as matrices of given rank or with characteristic polynomial having irreducible divisors of prescribed degrees. We demonstrate two different techniques which allow us to show that the number of such matrices in each of these classes and also with components in a given subinterval [-H, H] [-(p - 1)/2, (p - 1)/2] is asymptotically close to the expected value.