Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

As the main result, we show that if G is a finite group such that Γ(G) = Γ(² F ₄(q)), where q = 2^2m+1 for some m ≧ 1, then G has a unique nonabelian composition factor isomorphic to ² F ₄(q). We also show that if G is a finite group satisfying |G| =|² F ₄(q)| and Γ(G) = Γ(² F ₄(q)), then G ≅ ² F ₄(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for ² F ₄(q). The third author was supported in part by a grant from IPM (No. 87200022).     

A characterization of Grassmann graphs

Let q be a prime power and suppose that e and n are integers satisfying 1 e n − 1. Then the Grassmann graph Γ(e, q, n) has as vertices the e-dimensional subspaces of a vector space of dimension n over the field F_q, where two vertices are adjacent iff they meet in a subspace of dimension e − 1. In this paper, a characterization of Γ(e, q, n) in terms of parameters is obtained provided that and ( if q ε {2, 3}) and if q = 3). As a consequence we can show that these Grassmann graphs are uniquely determined as distance-regular graphs by their intersection arrays.     

Isomorphism criterion for monomial graphs

Let q be a prime power, ??_q be the field of q elements, and k, m be positive integers. A bipartite graph G = G_q(k, m) is defined as follows. The vertex set of G is a union of two copies P and L of two‐dimensional vector spaces over ??_q, with two vertices (p₁, p₂) ∈ P and [ l₁, l₂] ∈ L being adjacent if and only if p₂ + l₂ = pl. We prove that graphs G_q(k, m) and G_q′(k′, m′) are isomorphic if and only if q = q′ and {gcd (k, q ? 1), gcd (m, q ? 1)} = {gcd (k′, q ? 1),gcd (m′, q ? 1)} as multisets. The proof is based on counting the number of complete bipartite INFgraphs in the graphs. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 322–328, 2005     

Limq → ∞ \frac{c(G)}{r(G)} for Simple Groups

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. For a given finite group G, let c(G) denote the minimal degree of a faithful representation of G by complex quasi-permutation matrices and let r(G) denote the minimal degree of a faithful rational valued character of G. Also let G denote one of the symbols A_l, B_l, C_l, D_l, E₆, E₇, E₈, G₂, F₄, ²B₂, ²E₄, ²G₂, and ³D₄. Let G(q) denote simple group of type G over GF(q). Let c(q) = c(G(q)) and r(q) = r(G(q)). Then we will show that lim Lim_q = 1.     

Some results on the achromatic number

Let G be a simple graph. The achromatic number ψ(G) is the largest number of colors possible in a proper vertex coloring of G in which each pair of colors is adjacent somewhere in G. For any positive integer m, let q(m) be the largest integer k such that ≤ m. We show that the problem of determining the achromatic number of a tree is NP-hard. We further prove that almost all trees T satisfy ψ (T) = q(m), where m is the number of edges in T. Lastly, for fixed d and ϵ > 0, we show that there is an integer N₀ = N₀(d, ϵ) such that if G is a graph with maximum degree at most d, and m ≥ N₀ edges, then (1 - ϵ)q(m) ≤ ψ (G) ≤ q(m). © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 129–136, 1997     

The existence of even cycles with specific lengths in Wenger’s graph

Wenger＇s graph Hm（q） is a q-regular bipartite graph of order 2qm constructed by using the mdimensional vector space Fq^m over the finite field Fq. The existence of the cycles of certain even length plays an important role in the study of the accurate order of the Turan number ex（n; C2m） in extremal graph theory. In this paper, we use the algebraic methods of linear system of equations over the finite field and the “critical zero-sum sequences” to show that： if m ≥ 3, then for any integer l with l ≠ 5, 4 ≤ l ≤ 2ch（Fq） （where ch（Fq） is the character of the finite field Fq） and any vertex v in the Wenger＇s graph Hm（q）, there is a cycle of length 21 in Hm（q） passing through the vertex v.     

On recognizability of some finite simple orthogonal groups by spectrum

It is proved that, if G is a finite group that has the same set of element orders as the simple group D _p (q), where p is prime, p ≥ 5 and q ∈ {2, 3, 5}, then the commutator group of G/F(G) is isomorphic to D _p (q), the subgroup F(G) is equal to 1 for q = 5 and to O _q (G) for q ∈ {2, 3}, F(G) ≤ G′, and |G/G′| ≤ 2.     

Growth for the Central Polynomials

We study the growth of the central polynomials for the infinite dimensional Grassmann algebra G, and for the algebra M_k(F) of the k × k matrices, both over a field F of characteristic zero.     

The central polynomials for the Grassmann algebra

In this paper we describe the central polynomials for the infinite-dimensional unitary Grassmann algebra G over an infinite field F of characteristic ≠ 2. We exhibit a set of polynomials that generates the vector space C(G) of the central polynomials of G as a T-space. Using a deep result of Shchigolev we prove that if charF = p > 2 then the T-space C(G) is not finitely generated. Moreover, over such a field F, C(G) is a limit T-space, that is, C(G) is not a finitely generated T-space but every larger T-space W ≩ C(G) is. We obtain similar results for the infinite-dimensional nonunitary Grassmann algebra H as well.     

Spreads, arcs, and multiple wavelength codes

We present several new families of multiple wavelength (2-dimensional) optical orthogonal codes (2D-OOCs) with ideal auto-correlation λ_a=0 (codes with at most one pulse per wavelength). We also provide a construction which yields multiple weight codes. All of our constructions produce codes that are either optimal with respect to the Johnson bound (J-optimal), or are asymptotically optimal and maximal. The constructions are based on certain pointsets in finite projective spaces of dimension k over GF(q) denoted PG(k,q).     

Quasirecognition by prime graph of F4(q) where q?=?2n?>?2

The prime graph of a finite group G is denoted by Γ(G). In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(F ₄(q)), where q = 2 ⁿ  > 2, then G has a unique nonabelian composition factor isomorphic to F ₄(q). We also show that if G is a finite group satisfying |G| = |F ₄(q)| and Γ(G) = Γ(F ₄(q)), where q = 2 ⁿ  > 2, then G @ F₄(q){G \cong F_4(q)}. As a consequence of our result we give a new proof for a conjecture of Shi and Bi for F ₄(q) where q = 2 ⁿ  > 2.     

Totally geodesic submanifolds in Lie groups

Summary We study minimal and totally geodesic submanifolds in Lie groups and related problems. We show that: (1) The imbedding of the Grassmann manifold G_F(n,N) in the Lie group G_F(N) defined naturally makes G_F(n,N) a totally geodesic submanifold; (2) The imbedding S⁷→SO(8) defined by octonians makes S⁷a totally geodesic submanifold inSO(8); (3) The natural inclusion of the Lie group G_F(N) in the sphere S^{cN^2-1}(√N) of gl(N,F)is minimal. Therefore the natural imbedding G_F(N)<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>→gl(N,F)is formed by the eigenfunctions of the Laplacian on G_F(N).     

Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points

ABSTRACT

Let F _m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F _m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F _m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) ^G is not finitely generated.     

Constructing quasi-cyclic codes from linear algebra theory

Let F _q be a finite field of cardinality q, l and m be positive integers and M _l (F _q ) the F _q -algebra of all l × l matrices over F _q . We investigate the relationship between monic factors of X ^m ? 1 in the polynomial ring M _l (F _q )[X] and quasi-cyclic (QC) codes of length lm and index l over F _q . Then we consider the idea of constructing QC codes from monic factors of X ^m ? 1 in polynomial rings over F _q -subalgebras of M _l (F _q ). This idea includes ideas of constructing QC codes of length lm and index l over F _q from cyclic codes of length m over a finite field F _q l, the finite chain ring F _q  + uF _q  + · · · + u ^{l ? 1} F _q (u ^l  = 0) and other type of finite chain rings.     

The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants

We give conditions on a curve class that guarantee the vanishing of the structure constants of the small quantum cohomology of partial flag varieties F(k ₁, ..., k _r ; n) for that class. We show that many of the structure constants of the quantum cohomology of flag varieties can be computed from the image of the evaluation morphism. In fact, we show that a certain class of these structure constants are equal to the ordinary intersection of Schubert cycles in a related flag variety. We obtain a positive, geometric rule for computing these invariants (see Coskun in A Littlewood–Richardson rule for partial flag varieties, preprint). Our study also reveals a remarkable periodicity property of the ordinary Schubert structure constants of partial flag varieties.     

Folding Wheels and Fans

 If two non-adjacent vertices of a connected graph that have a common neighbor are identified and the resulting multiple edges are reduced to simple edges, then we obtain another graph of order one less than that of the original graph. This process can be repeated until the resulting graph is complete. We say that we have folded the graph onto complete graph. This process of folding a connected graph G onto a complete graph induces in a very natural way a partition of the vertex-set of G. We denote by F(G) the set of all complete graphs onto which G can be folded. We show here that if p and q are the largest and smallest orders, respectively, of the complete graph in F(W _n ) or F(F _n ), then K _s is in F(W _n ) or F(F _n ) for each s, q≤s≤p. Lastly, we shall also determine the exact values of p and q. Received: October, 2001 Final version received: June 26, 2002     

On the multiplicity of the eigenvalues of a graph

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q ₁(t)q ₂(t)² ... q _m (t)^m, where each q _i (t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of q _j (t). We give an algorithm to construct the polynomials q _i (t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = |λ_i| of G, where λ₁, λ₂, ..., λ_n are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. This work was done during a visit of the second named author to UNAM.     

Root games on Grassmannians

We recall the root game, introduced in [8], which gives a fairly powerful sufficient condition for non-vanishing of Schubert calculus on a generalised flag manifold G/B. We show that it gives a necessary and sufficient rule for non-vanishing of Schubert calculus on Grassmannians. In particular, a Littlewood-Richardson number is non-zero if and only if it is possible to win the corresponding root game. More generally, the rule can be used to determine whether or not a product of several Schubert classes on Gr _l (ℂ ⁿ ) is non-zero in a manifestly symmetric way. Finally, we give a geometric interpretation of root games for Grassmannian Schubert problems. Research partially supported by an NSERC scholarship.     

On <Emphasis Type="Italic">T</Emphasis> -spaces and relations in relatively free,lie nilpotent,associative algebras

The purpose of this work is to obtain the commutator relations and Frobenius relations in a relatively free algebra F ^(l) specified by the identity [x ₁ , . . . , x _l ] = 0 over a field of characteristic p > 0. These relations for l > 3 are analogous to the relations in the algebra F ⁽³⁾ and are applied to the T-spaces in the algebra F ^(l). In order to study the relations in F ^(l) in more detail, we construct a model algebra analogous to the Grassmann algebra.