Degenerate Principal Series Representations of U(p, q) and Spin0(p, q)

Let p>q and let G be the group U(p, q) or Spin₀(p, q). Let P=LN be the maximal parabolic subgroup of G with Levi subgroup where

Linear Widths of the Besov Classes of Periodic Functions of Many Variables. II

We obtain order estimates for linear widths of the Besov classes of periodic functions of many variables in the space L _q for certain values of parameters p and q different from those considered in the first part of the work.     

The closed ideal of (c o ,p,q)-summing operators

For 1/p+1/q1, we study the closed ideal formed by the (c _o ,p,q)-summing operators. It turns out thatT:XY does not belong to if and only if it factors the mapId:l _p ^*l _q . By localization, we get the ideal that consists of those operatorsT for which all ultrapowersT ^u are contained in . Operators in the complement of are characterized by the property that they factor the mapsId:l _p ^*n l _q ⁿ uniformly. Our main tools are ideal norms.Supported by DFG grant PI 322/1-2     

On flag-transitive affine planes of orderq 3

Let be a translation plane of orderq ³,q an odd prime power, whose kern GF(q). Letl be the line at infinity of . LetG be a solvable collineation group of in the linear translation complement, which acts transitively onl , and letH be a maximal normal cyclic subgroup ofG. Then the restriction ofH onl acts semiregularly onl and {1, 2, 3, 6}, where is the restriction ofG onl (ifq –1(mod 3), then {1, 2}). Ifq {3, 5} and {1, 2}, then is determined completely, using a computer.     

Unipotent Brauer Character Values of GL(n, \mathbb{F}_q ) and the Forgotten Basis of the Hall Algebra

We give a formula for the values of irreducible unipotent p-modular Brauer characters of at unipotent elements, where p is a prime not dividing q, in terms of (unknown!) weight multiplicities of quantum GL _n and certain generic polynomials S _,(q). These polynomials arise as entries of the transition matrix between the renormalized Hall-Littlewood symmetric functions and the forgotten symmetric functions. We also provide an alternative combinatorial algorithm working in the Hall algebra for computing S _,(q).     

On Estimates of the Kolmogorov Widths of the Classes $$B_{p,q}^r$$ in the Space Lq

We obtain an order estimate for the Kolmogorov width of the Besov classes of periodic functions of many variables in the space L _q for 2 < p < q < , which complements the result obtained earlier by the author.     

On quasi degree quadrature rules

Summary In this paper we examine quadrature rules for the integral which are exact for all with +d. We specify three distinct families of solutions which have properties not unlike the standard Gauss and Radau quadrature rules. For each integerd the abscissas of the quadrature rules lie within the closed integration interval and are expressed in terms of the zeros of a polynomialq _d(y). These polynomialsq _d(y), (d=0, 1, ...), which are not orthogonal, satisfy a three term recurrence relation of the type Q_d+1(y)=(y+_d+1)q_d(y)–_d+1yq_d–1(y) and have zeros with the standard interlacing property.This work was supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38     

Trigonometric Widths of the Classes B p,θ Ω of Periodic Functions of Many Variables

We obtain exact order estimates for the trigonometric widths of the classes of periodic functions of many variables in the space L _q, 1 < p 2 q < p/(p – 1).     

A remark on the irreducible characters and fake degrees of finite real reflection groups

Summary Beynon and Lusztig have shown that the fake degrees of almost all irreducible characters of finite real reflection groups are palindromes, and that the exceptions to this rule correspond to the non rational characters of the generic ringA defined overR=C[q]. Their proof consists of a case-by-case check. In this note we give an explanation for this phenomenon and some related facts about fake degrees. Moreover, in the situation where we allow for distinct parametersq in the definition ofA, we shall give a simple uniform proof of the fact that all the central idempotents of are elements of , where .Oblatum 10-XI-1994     

Note on the existence of large minimal blocking sets in galois planes

A subsetS of a finite projective plane of orderq is called a blocking set ifS meets every line but contains no line. For the size of an inclusion-minimal blocking setq+ +Sq +1 holds ([6]). Ifq is a square, then inPG(2,q) there are minimal blocking sets with cardinalityq +1. Ifq is not a square, then the various constructions known to the author yield minimal blocking sets with less than 3q points. In the present note we show that inPG(2,q),q1 (mod 4) there are minimal blocking sets having more thanqlog₂ q/2 points. The blocking sets constructed in this note contain the union ofk conics, whereklog₂ q/2. A slight modification of the construction works forq3 (mod 4) and gives the existence of minimal blocking sets of sizecqlog₂ q for some constantc.As a by-product we construct minimal blocking sets of cardinalityq +1, i.e. unitals, in Galois planes of square order. Since these unitals can be obtained as the union of parabolas, they are not classical.     

Commutator Subgroups of the Extended Hecke Groups \bar H(\lambda _q )

Hecke groups H(_q) are the discrete subgroups of generated by S(z) = –(z+ _q)^–1and T(z) = –1/z. The commutator subgroup of H(_q), denoted by H(_q), is studied in [2]. It was shown that H(_q) is a free group of rank q– 1.Here the extended Hecke groups obtained by adjoining to the generators of H(_q) are considered. The commutator subgroup of is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the H(_q) case, the index of H(_q) is changed by q, in the case of this number is either 4 for qodd or 8 for qeven.     

Distance-Regular Graphs Related to the Quantum Enveloping Algebra of sl(2)

We investigate a connection between distance-regular graphs and U _q(sl(2)), the quantum universal enveloping algebra of the Lie algebra sl(2). Let be a distance-regular graph with diameter d 3 and valency k 3, and assume is not isomorphic to the d-cube. Fix a vertex x of , and let (x) denote the Terwilliger algebra of with respect to x. Fix any complex number q {0, 1, –1}. Then is generated by certain matrices satisfying the defining relations of U _q(sl(2)) if and only if is bipartite and 2-homogeneous.     

Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

Let X _n and Y _n be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over , respectively. The orbits of GL _n( ) on X _n × X _n define an association scheme Qua(n, q). The orbits of GL _n( ) on Y _n × Y _n also define an association scheme Sym(n, q). Our main results are: Qua(n, q) and Sym(n, q) are formally dual. When q is odd, Qua(n, q) and Sym(n, q) are isomorphic; Qua(n, q) and Sym(n, q) are primitive and self-dual. Next we assume that q is even. Qua(n, q) is imprimitive; when (n, q) (2,2), all subschemes of Qua(n, q) are trivial, i.e., of class one, and the quotient scheme is isomorphic to Alt(n, q), the association scheme of alternating forms on V. The dual statements hold for Sym(n, q).     

On The Sum of Digits Function for Number Systems with Negative Bases

Let q 2 be an integer. Then –q gives rise to a number system in , i.e., each number n has a unique representation of the form n = c ₀ + c ₁ (–q) + ... + c _h (–q) ^h , with c _i {0,..., q – 1}(0 i h). The aim of this paper is to investigate the sum of digits function _–q (n) of these number systems. In particular, we derive an asymptotic expansion for

Triads, Flocks of Conics and Q -(5,q)

We show that if an ovoid of Q (4,q),q even, admits a flock of conics then that flock must be linear. It follows that an ovoid of PG (3,q),q even, which admits a flock of conics must be an elliptic quadric. This latter result is used to give a characterisation of the classical example Q ^-(5,q) among the generalized quadrangles T ₃( ), where is an ovoid of PG (3q) and q is even, in terms of the geometric configuration of the centres of certain triads.     

Blocking sets and partial spreads in finite projective spaces

A t-blocking set in the finite projective space PG(d, q) with dt+1 is a set of points such that any (d–t)-dimensional subspace is incident with a point of and no t-dimensional subspace is contained in . It is shown that | |q ^t +...+1+q ^t–1q and the examples of minimal cardinality are characterized. Using this result it is possible to prove upper and lower bounds for the cardinality of partial t-spreads in PG(d, q). Finally, examples of blocking sets and maximal partial spreads are given.     

Estimates for Approximation Characteristics of the Besov Classes $$B_{p,\theta }^r $$ of Periodic Functions of Many Variables in the Space Lq. II

We obtain order estimates for the approximation of the classes of periodic functions of many variables in the space L _q by using operators of orthogonal projection and linear operators satisfying certain conditions.     

On the number of nowhere zero points in linear mappings

LetA be a nonsingularn byn matrix over the finite fieldGF _q ,k=n/2,q=p ^a ,a1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF _q ) ⁿ such that bothx andAx have no zero component. We prove that forn2, and ,P(A,q)[(q–1)(q–3)] ^k (q–2) ^n–2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)1, is true for allqn+23 orqn+14.