Rational invariants of certain orthogonal groups over finite fields of characteristic two

In this article we establish the conjecture of Huah Chu [8] on modular rational invariants of finite orthogonal groups over finite fields of characteristic two.

We prove that the invariant subfield of two cases non-singular, and singular quadratic space over a finite field of characteristic two is purely transcendental.     

Generation of classical groups over finite fields

Let U_n(V) and Sp_n(V) denote the unitary group and the symplectic group of the n dimensional vector space V over a finite field of characteristic not 2, respectively. Assume that the hyperbolic rank of U_n(V) is at least one. Then U_n(V) is generated by 4 elements and Sp_n(V) by 3 elements. Further, U_2m+1(V) is generated by 3 elements and Sp_4m(V) by 2 elements.     

Rational irreducible characters and rational conjugacy classes in finite groups

We prove that a finite group has two rational-valued irreducible characters if and only if it has two rational conjugacy classes, and determine the structure of any such group. Along the way we also prove a conjecture of Gow stating that any finite group of even order has a non-trivial rational-valued irreducible character of odd degree.

     

Symplectic invariants and covariants of matrices

This paper deals with symplectic algebraic invariants and covariants of matrices. We adapt the Aronhold symbolic method to characterize and compute the generators of algebras of symplectic invariants and covariants. New symplectic identities are obtained and applied to construct minimal generator systems of a matrix of order 4 and a Hamiltonian matrix of order 2n.     

Generalized trigonometry and Chebyshev functions in finite fields

Trigonometry in finite fields was introduced by de Souza et al. and further developed by Lima and Panario and others, giving functions with many properties similar to trigonometric functions over the reals. Those explorations used a degree-2 extension of a base field. While this corresponds most closely to trigonometry over the reals, in finite fields we can have extensions of other degrees. In this paper we generalize the definitions of trigonometric functions and their related Chebyshev polynomials to arbitrary degrees and explore their properties. Many familiar results carry over into the generalized setting.     

Moment subset sums over finite fields

The k-subset sum problem over finite fields is a classical NP-complete problem. Motivated by coding theory applications, a more complex problem is the higher m-th moment k-subset sum problem over finite fields. We show that there is a deterministic polynomial time algorithm for the m-th moment k-subset sum problem over finite fields for each fixed m when the evaluation set is the image set of a monomial or Dickson polynomial of any degree n. In the classical case $m=1$, this recovers previous results of Nguyen-Wang (the case $m=1,p>2$) [22] and the results of Choe-Choe (the case $m=1,p=2$) [3].     

Explicit factorizations of generalized Dickson polynomials of order 2m via generalized cyclotomic polynomials over finite fields

We derive explicit factorizations of generalized cyclotomic polynomials of order $2^m$ and generalized Dickson polynomials of the first kind of order $2^m$ over finite field $F_q$.     

Some classes of monomial complete permutation polynomials over finite fields of characteristic two

In this paper, four classes of complete permutation polynomials over finite fields of characteristic two are presented. To consider the permutation property of the first three classes, Dickson polynomials play a key role. The fourth class is a generalization of a known result. In addition, we also calculate the inverses of these bijective monomials.     

Finite group action and Gromov-Witten invariants in symplectic four-manifolds

Let a finite group act semi-freely on a closed symplectic four-manifold with a 2-dimensional fixed point set. Then we show that the relative Gromov-Witten invariants are the same as the invariants on the quotient set-up with respect to the fixed point set.