Brauer Diagrams, Updown Tableaux and Nilpotent Matrices

We interpret geometrically a variant of the Robinson-Schensted correspondence which links Brauer diagrams with updown tableaux, in the spirit of Steinberg's result [32] on the original Robinson-Schensted correspondence. Our result uses the variety of all where is a complete flag in is a nondegenerate alternating bilinear form on and N is a nilpotent element of the Lie algebra of the simultaneous stabilizer of both and instead of Steinberg's variety of where are two complete flags in and N is a nilpotent element of the Lie algebra of the simultaneous stabilizer of both .     

Hyperplane Sections of Kantor's Unitary Ovoids

The projective plane is embedded as a variety of projective points in , where M is a nine dimensional -module for the groupG=GL(3,q ²). The hyperplane sections of thisvariety and their stabilizers in the group G aredetermined. When q 2 (mod 3) one such hyperplanesection is a member of the family of Kantor's unitary ovoids.We furtherdetermine all sections whereD has codimension two in M and demonstratethat these are never empty. Consequences are drawn for Kantor'sovoids.     

Derangements and Tensor Powers of Adjoint Modules for \mathfrak{s}\mathfrak{l}_n

We obtain the decomposition of the tensor space as a module for , find an explicit formula for the multiplicities of its irreducible summands, and (when n 2k) describe the centralizer algebra = ( ) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of is given by the number of derangements of a set of 2k elements.     

A Geometric Characterization of Fischer's Baby Monster

The sporadic simple group F ₂ known as Fischer's Baby Monster acts flag-transitively on a rank 5 P-geometry . P-geometries are geometries with string diagrams, all of whose nonempty edges except one are projective planes of order 2 and one terminal edge is the geometry of the Petersen graph. Let be a flag-transitive P-geometry of rank 5. Suppose that each proper residue of is isomorphic to the corresponding residue in . We show that in this case is isomorphic to . This result realizes a step in classification of the flag-transitive P-geometries and also plays an important role in the characterization of the Fischer–Griess Monster in terms of its 2-local parabolic geometry.     

Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p

The aim of this paper is to give a geometric interpretation of the continued fraction expansion in the field of formal Laurent series in X ^–1 over , in terms of the action of the modular group on the Bruhat–Tits tree of , and to deduce from it some corollaries for the diophantine approximation of formal Laurent series in X ^–1 by rational fractions in X.     

Proof of Two Conjectures of Brenti and Simion on Kazhdan-Lusztig Polynomials

We find an explicit formula for the Kazhdan-Lusztig polynomials P _ui,a,v _i of the symmetric group (n) where, for a, i, n such that 1 a i n, we denote by u _i,a = s _a s _a+1 ··· s _i–1 and by v _i the element of (n) obtained by inserting n in position i in any permutation of (n – 1) allowed to rise only in the first and in the last place. Our result implies, in particular, the validity of two conjectures of Brenti and Simion [4, Conjectures 4.2 and 4.3], and includes as a special case a result of Shapiro, Shapiro and Vainshtein [13, Theorem 1]. All the proofs are purely combinatorial and make no use of the geometry of the corresponding Schubert varieties.     

On the Asymptotic Behaviour of General Partition Functions,II

Let = {a ₁, a ₂,...} be a set of positive integers and let p (n) and q (n) denote the number of partitions of n into a's, resp. distinct a's. In an earlier paper the authors studied large values of log(max (2,p (n)))/log(max(2,q (n))). In this paper the small values of the same quotient are studied.     

Localizations of Transfors

Let be n-dimensional teisi, i.e., higher-dimensional Gray-categorical structures. The following questions can be asked. Does a left q-transfor , i.e., a functor 2 _q , induce a right q-transfor , i.e., a functor More generally, does a functor induce a functor For k-arrows c and whose (k – 1)-sources and targets agree, does a q-transfor induce a q-transfor , for appropriate k-arrows For k-arrows c and whose (k – 1)-sources and targets agree, does a q-transfor induce a (q + k + 1)-transfor , for appropriate k-arrows I give answers to these questions in the cases where n-dimensional teisi and their tensor product have been defined, i.e., for n 3, and for n up to 5 in some cases that do not need all data and axioms of n-dimensional teisi.I apply the above to compositions in teisi, in particular to braidings and syllepses. One of the results is that a braiding on a monoidal 2-category induces a pseudo-natural transformation , where is the reverse of ? –, which is almost, but not quite, equal to – ?. However, in higher dimensions need not be reversible, so a braiding on a higher-dimensional tas can not be seen as a transfor A B B A.     

Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let denote a Bose-Mesner algebra on a finite nonempty set X. Fix p X, and let and denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of with respect to p. By a hyper-duality of , we mean an automorphism of such that for all ; and is a duality of . is said to be hyper-self-dual whenever there exists a hyper-duality of . We say that is strongly hyper-self-dual whenever there exists a hyper-duality of which can be expressed as conjugation by an invertible element of . We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.     

Nonuniqueness Sequences for Weighted Algebras of Holomorphic Functions in the Unit Circle

Suppose that is a system of continuous subharmonic functions in the unit disk and is the class of holomorphic functions f in such that log|f(z)| ≤ B _f p _f (z) + C _f , z ∈ , where B _f and C _f are constants and p _f ∈ . We obtain sufficient conditions for a given number sequence Λ = { λ_n} ⊂ to be a subsequence of zeros of some nonzero holomorphic function from , i.e., Λ is a nonuniqueness sequence for .__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 775–787.Original Russian Text Copyright ©2005 by L. Yu. Cherednikova.     

Generating Random Elements in SL n (F q ) by Random Transvections

This paper studies a random walk based on random transvections in SL _n(F _q ) and shows that, given > 0, there is a constant c such that after n + c steps the walk is within a distance from uniform and that after n – c steps the walk is a distance at least 1 – from uniform. This paper uses results of Diaconis and Shahshahani to get the upper bound, uses results of Rudvalis to get the lower bound, and briefly considers some other random walks on SL _n(F _q ) to compare them with random transvections.     

Zeros of H ∞-Functions on Hyperplanes in \mathbb{B} n

Let ⁿ be the unit ball in ℂⁿ, n ≥ 2. Let T_α = {z ∈ ⁿ : (z, a) = |a|²} for a ∈ ⁿ and denote for a discrete set A in ⁿ. We find a sharp necessary condition for a set A to be a part of the zero-set for a function in H^∞( ⁿ). Bibliography 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 272–278.     

Stickelberger Codes

Let p be an odd prime and be a primitive p th root of unity over . The Galois group G of over is a cyclic group of order p-1. The integral group ring [G] contains the Stickelberger ideal S _p which annihilates the ideal class group of K. In this paper we investigate the parameters of cyclic codes S _p (q) obtained as reductions of S _p modulo primes q which we call Stickelberger codes. In particular, we show that the dimension of S _p (p) is related to the index of irregularity of p, i.e., the number of Bernoulli numbers B _2k , , which are divisible by p. We then develop methods to compute the generator polynomial of S _p (p). This gives rise to anew algorithm for the computation of the index of irregularity of a prime. As an application we show that 20,001,301 is regular. This significantly improves a previous record of 8,388,019 on the largest explicitly known regular prime.     

Type II Self-Dual Codes over Finite Rings and Even Unimodular Lattices

In this paper, we investigate self-dual codes over finite rings, specifically the ring of integers modulo 2m. Type II codes over are introduced as self-dual codes with Euclidean weights which are a multiple of 2m +1. We describe a relationship between Type II codes and even unimodular lattices. This relationship provides much information on Type II codes. Double circulant Type II codes over are also studied.     

On K3 of Witt Vectors of Length Two Over Finite Fields

We prove that for W₂ the Witt vectors of length two over the finite field , we have in characteristic at least 5 and for (3,f) = 1. The result is proved by using the identity and calculating the right term with a group homology spectral sequence. Some information on the spectral sequence is achieved by using the action of the outer automorphism of SL on the homology groups and recent results on K-groups of local rings and the ring of dual numbers over finite fields.     

Combinatorial Statistics on Alternating Permutations

We consider two combinatorial statistics on permutations. One is the genus. The other, , is defined for alternating permutations, as the sum of the number of descents in the subwords formed by the peaks and the valleys. We investigate the distribution of on genus zero permutations and Baxter permutations. Our q-enumerative results relate the statistic to lattice path enumeration, the rank generating function and characteristic polynomial of noncrossing partition lattices, and polytopes obtained as face-figures of the associahedron.     

The Weight Distribution of C 5(1, n)

In [2] the codes C _q (r,n) over were introduced. These linear codes have parameters , can be viewed as analogues of the binary Reed-Muller codes and share several features in common with them. In [2], the weight distribution of C ₃(1,n) is completely determined.In this paper we compute the weight distribution of C ₅(1,n). To this end we transform a sum of a product of two binomial coefficients into an expression involving only exponentials an Lucas numbers. We prove an effective result on the set of Lucas numbers which are powers of two to arrive to the complete determination of the weight distribution of C ₅(1,n). The final result is stated as Theorem 2.     

Unimodular Triangulations and Coverings of Configurations Arising from Root Systems

Existence of a regular unimodular triangulation of the configuration , where ⁺ is the collection of the positive roots of a root system and where (0, 0,...,0 ) is the origin of , will be shown for = B _n , C _n , D _n and BC _n . Moreover, existence of a unimodular covering of a certain subconfiguration of the configuration A _n+1 ⁺ will be studied.     

Some Combinatorial Properties of Schubert Polynomials

Schubert polynomials were introduced by Bernstein et al. and Demazure, and were extensively developed by Lascoux, Schützenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polynomial in terms of the reduced decompositions of the permutation w. Using this result, a variation of Schensted's correspondence due to Edelman and Greene allows one to associate in a natural way a certain set of tableaux with w, each tableau contributing a single term to . This correspondence leads to many problems and conjectures, whose interrelation is investigated. In Section 2 we consider permutations with no decreasing subsequence of length three (or 321-avoiding permutations). We show for such permutations that is a flag skew Schur function. In Section 3 we use this result to obtain some interesting properties of the rational function , where denotes a skew Schur function.Sara C. Billey: Supported by the National Physical Science Consortium. William Jockusch: Supported by an NSF Graduate Fellowship. Richard P. Stanley: Partially supported by NSF grants DMS-8901834 and DMS-9206374