Morita Equivalent Principal 3-Blocks of the Chevalley Groups G2(q)

The principal 3-block of a Chevalley group G₂(q) with q a powerof 2 satisfying q 2 or 5 mod 9 and the principal 3-block ofG₂(2) are Morita equivalent. 2000 Mathematical Subject Classification:20C05, 20C20, 20C33.     

The determination of ovoids of PG(3,q) containing a pointed conic

It is shown that if a plane of PG(3,q),q even, meets an ovoid in a pointed conic, then eitherq=4 and the ovoid is an elliptic quadric, orq=8 and the ovoid is a Tits ovoid.     

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.     

SCALED ASYMPTOTICS FOR SOME q-SERIES

This work, investigates the asymptotics for Euler’s q-exponentialE_q(z), Ramanujan’s function A_q(z), Jackson’s q-Besselfunction J_v⁽²⁾ (z; q), the Stieltjes–Wigert orthogonalpolynomials S_n(x; q) and q-Laguerre polynomials L_n⁽⁾ (x; q)as q approaches 1.     

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.     

Maximal Subgroups of Large Rank in Exceptional Groups of Lie Type

Let G = G(q) be a finite almost simple exceptional group ofLie type over the field of q elements, where q = p^a and p isprime. The main result of the paper determines all maximal subgroupsM of G(q) such that M is an almost simple group which is alsoof Lie type in characteristic p, under the condition that rank(M)> rank(G). The conclusion is that either M is a subgroupof maximal rank, or it is of the same type as G over a subfieldof F_q, or (G, M) is one of (, F₄(q)), (, C₄(q)), (E₇(q),³D₄(q)). This completes work of the first author with Saxl andTesterman, in which the same conclusion was obtained under someextra assumptions.     

Well matched systems

The product (3.10) on page 33 is incorrectly called a cartesianproduct on pages 33 and 35. This misnomer in effect amountsto a wrong definition. The product (3.10) should be definedso that the right-hand member of (3.10) is the set of all sums_f=1 f_j (not the set of all ordered q-tuples) such that f₁ F(m₁,d₁), ..., f_q F(m_q, d_q).     

De Rham Cohomology and Hodge Decomposition For Quantum Groups

Let be one of the N²-dimensionalbicovariant first order differential calculi for the quantumgroups GL_q(N), SL_q(N), SO_q(N), or Sp_q(N), where q is a transcendentalcomplex number and z is a regular parameter. It is shown thatthe de Rham cohomology of Woronowicz' external algebra coincides with the de Rham cohomologiesof its left-coinvariant, its right-coinvariant and its (two-sided)coinvariant subcomplexes. In the cases GL_q(N) and SL_q(N) thecohomology ring is isomorphic to the coinvariant external algebra and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in thesecases. The main technical tool is the spectral decompositionof the quantum Laplace-Beltrami operator. 2000 MathematicalSubject Classification: 46L87, 58A12, 81R50.     

Specht Filtrations for Hecke Algebras of Type A

Let H_q(d) be the Iwahori–Hecke algebra of the symmetricgroup, where q is a primitive 1th root of unity. Using resultsfrom the cohomology of quantum groups and recent results aboutthe Schur functor and adjoint Schur functor, it is proved that,contrary to expectations, for l 4 the multiplicities in a Spechtor dual Specht module filtration of an H_q(d)-module are welldefined. A cohomological criterion is given for when an H_q(d)-modulehas such a filtration. Finally, these results are used to givea new construction of Young modules that is analogous to theDonkin–Ringel construction of tilting modules. As a corollary,certain decomposition numbers can be equated with extensionsbetween Specht modules. Setting q = 1, results are obtainedfor the symmetric group in characteristic p 5. These resultsare false in general for p = 2 or 3.     

A Hilbert Basis Theorem for Quantum Groups

A noncommutative version of the Hilbert basis theorem is usedto show that certain R-symmetric algebras S_R(V) are Noetherian.This result applies in particular to the coordinate ring ofquantum matrices A_R(V) associated with an R-matrix R operatingon the tensor square of a vector space V, to show that, undera natural set of hypotheses on R, the algebra A_R(V) is Noetherianand its augmentation ideal has a polynormal set of generators.As a corollary we deduce that these properties hold for thegeneric quantized function algebras R_q[G] over any field ofcharacteristic zero, for G an arbitrary connected, simply connected,semisimple group over C. That R_q[G] is Noetherian recovers aresult due to Joseph [10], with a different proof.1991 MathematicsSubject Classification 17B37, 16P40.     

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.     

Blocking Sets Of External Lines To A Conic In PG(2,q), q ODD

We determine all point-sets of minimum size in PG(2,q), q odd that meet every external line to a conic in PG(2,q). The proof uses a result on the linear system of polynomials vanishing at every internal point to the conic and a corollary to the classification theorem of all subgroups of PGL(2,q). * Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni and by the Hungarian-Italian Intergovernemental project “Algebraic and Geometric Structures”.     

The determination of a coefficient in a parabolic equation from input sources

The authors consider the question of recovering the coefficientq from the equation u_t – u_xx + q(x)u = f_j(x) with homogeneousinitial and boundary conditions. The nonhomogeneous source terms form a basis for L²(0,1).It is proved that a unique determination is possible from datameasurements consisting of either the flux at one end of thebar or the net flux leaving the bar, taken at a single fixedtime for each input source. An algorithm that allows efficientnumerical reconstruction of q(x) from finite data is given.     

Mahler Measure of Alexander Polynomials

Let l be an oriented link of d components in a homology 3-sphere.For any nonnegative integer q, let l(q) be the link of d–1components obtained from l by performing 1/q surgery on itsdth component l_d. The Mahler measure of the multivariable Alexanderpolynomial _l(q) converges to the Mahler measure of _l as q goesto infinity, provided that l_d has nonzero linking number withsome other component. If l_d has zero linking number with eachof the other components, then the Mahler measure of _l(q) hasa well defined but different limiting behavior. Examples aregiven of links l such that the Mahler measure of _l is small.Possible connections with hyperbolic volume are discussed.     

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.     

Geometrical Spines of Lens Manifolds

We introduce the concept of ‘geometrical spine’for 3-manifolds with natural metrics, in particular, for lensmanifolds. We show that any spine of L_p,q that is close enoughto its geometrical spine contains at least E(p,q) – 3vertices, which is exactly the conjectured value for the complexityc(L_p,q). As a byproduct, we find the minimal rotation distance(in the Sleator–Tarjan–Thurston sense) between atriangulation of a regular p-gon and its image under rotation.     

Crystal Bases for Quantum Generalized Kac-Moody Algebras

In this paper, we develop the crystal basis theory for quantumgeneralized Kac–Moody algebras. For a quantum generalizedKac–Moody algebra U_q(g), we first introduce the categoryO_int of U_q(g)-modules and prove its semisimplicity. Next, wedefine the notion of crystal bases for U_q(g)-modules in thecategory O_int and for the subalgebra . We then prove the tensor product rule and the existence theoremfor crystal bases. Finally, we construct the global bases forU_q(g)-modules in the category O_int and for the subalgebra . 2000 Mathematics Subject Classification17B37, 17B67.     

Conversion of Polynomials between Different Polynomial Bases

The purpose of this paper is to derive a recursive scheme forthe evaluation of the coefficients in the expansion , in terms of the coefficients in the expansion , where both q_k(x) and Q_k(x) are polynomials in xof degree k, and where both q_k(x) and Q_k{x} satisfy recursionformulae of the type satisfied by orthogonal polynomials. Thesets {Q_k(x)} and {q_k(x)} need not be orthogonal polynomials,though they usually are in the applications. An applicationis made to the evaluation of integrals with oscillatory andsingular integrands.     

Classification of Quantum Groups SUq(n)

It is shown that the compact matrix quantum groups SU_q(2) arenon-isomorphic to each other for q[–1, 1]\{0}, and thatthe compact matrix quantum groups SU_q(n) are non-isomorphicto each other for q(0, 1]. Some invariants for compact quantumgroups are also discussed.     

Cyclic Arcs in PG(2, q)

B.C. Kestenband [9], J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas [3], E. Boros, and T. Szönyi [1] constructed complete (q ² ? q + l)-arcs in PG(2, q ²), q ≥ 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q ² ? q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.