Hong You

Lattices Generated by Orbits of Subspaces Under Finite Singular Classical Groups and Its Characteristic Polynomials

Abstract Let be the (2ν + l)-dimensional vector space over the finite field 𝔽 q , and Sp 2ν+l,ν(𝔽 q ) the singular symplectic groups of degree 2ν + l over 𝔽 q . Let ℳ be any orbit of subspaces under Sp 2ν+l,ν(𝔽 q ). Denote by ℒ the set of subspaces which are intersections of subspaces in ℳ and the intersection of the empty set of subspaces of is assumed to be . By ordering ℒ by ordinary or reverse inclusion, two lattices are obtained. This paper studies the inclusion relations between different lattices, a characterization of subspaces contained in a given lattice ℒ, when the lattices form geometric lattice, and the characteristic polynomial of ℒ.

Vertex Operator Construction for the Toroidal Lie Algebra of Type F 4

Abstract The purpose of this work is to construct completely reducible vertex representations for the toroidal Lie algebra of type F 4.

OVERGROUPS OF SYMPLECTIC GROUP IN LINEAR GROUP OVER LOCAL RINGS

For a commutative local ring R, we give a complete description of all overgroups of symplectic group S p 2n R in general linear group GL 2n R.

Products of commutators of transvections over local rings

Abstract Let R be a commutative local ring, and M the maximal ideal of R . We prove that if |R/M|>3 , then every matrix in SL n R (n⩾2) is a product of at most [n/2]+2 commutators of transvections and the length [ res A/2]+2 cannot be reduced for n=2 .

SOME ANZAHL THEOREMS IN VECTOR SPACE OVER Z/pkZ

Abstract Let R = Z/pkZ be the ring of integers modulo pk where p is a prime and k > 1. Denote by n(r, m × n) the number of m by n matrices with real rank r over R. In the present paper, we compute n(r, m × n) and the number of the orbits of Mm,n(R) under GLm(R) × GLn(R), where Mm,n(R) is the set of all m by n matrices over R.

The Augmentation Quotients of the Groups of Order 25, II

We have described the structure of Q n (G) = Δ n (G)/Δ n+1(G) for 35 particular classes of groups G with order 25 in the previous article. In this article, the structure of Q n (G) for all the remaining classes of groups G with order 25 are presented.

Two-dimensional representations of the free group with two generators over a finite field

Over a finite field, this paper characterizes absolute irreducibility of two-dimensional representations of the free group with two generators. Then, representations and their similarity classes are enumerated.

Generalized inverses of a sum of morphisms

Abstract Let C be an additive category. Suppose that ϕ:X→X is a morphism of C with group inverse ϕ # , and η:X→X is a morphism of C such that 1 X +ϕ # η is invertible. Let e=(1 X −ϕϕ # )ηα(1 X −ϕ # ϕ) , γ=α(1 X −ϕ # ϕ)ηϕ # β,δ=αϕ # η(1 X −ϕϕ # )β , where α=(1 X +ϕ # η) −1 ,β=(1 X +ηϕ # ) −1 . Then f=ϕ+η−e has a group inverse if and only if 1 X −γ,1 X −δ are invertible. Similar result for MP-inverse is also given, which completed Huylebroucks result.

The conjugacy classes of fixed point free elements in GLn(F) and SLn(F)

Abstract If there is a least positive integer m such that A m =I for an element A of GL n (F), then we call A a fixed point free element of order m. In the present paper, we determine the formula of the conjugacy classes of fixed point free elements in GL n (F) and SL n (F) completely.

A distance regular graph of typeE 1° E d

In this note, the distance regular graph of typeE1°Ed is considered and some characterization of the type graph is given. The results generalize the characterization of tight distance regular graphs.