arXiv subjects

Rod Gow

Publications and source records attributed to Rod Gow.

25 records · Page 2Linked to original sources

arXiv2016

Rank-related dimension bounds for subspaces of symmetric bilinear forms

Let V be a vector space of dimension n over a field K and let Symm(V) denote the space of symmetric bilinear forms defined on V x V. Let M be a subspace of Symm(V). We investigate a variety of hypotheses concerning the rank of elements in M that lead to reasonable bounds for dim M. For example, if every non-zero element of M has odd rank, and r is the maximum rank of the elements of M, then dim M is at most r(r+1)/2 (thus dim M is bounded independently of n). This should be contrasted with the simple observation that Symm(V) contains a subspace of dimension n-1 in which each non-zero element has rank 2. The bound r(r+1)/2 is almost certainly too large, and a bound r seems plausible, this being true when K is finite. We also show that dim M is at most r$ when K is any field of characteristic 2. Finally, suppose that n=2r, where r is an odd integer, and the rank of each non-zero element of M is either r or n. We show that if K has characteristic 2, then dim M is at most 3r. Furthermore, if dim M=3r, we obtain interesting subspace decompositions of M and V related to spreads, pseudo-arcs and pseudo-ovals. Examples of such subspaces M exist if K has an extension field of degree r.

math.RA↗

arXiv2015

A dimension bound for constant rank subspaces of matrices over a finite field

K be a field and let m and n be positive integers, where m does not exceed n. We say that a non-zero subspace of m x n matrices over K is a constant rank r subspace if each non-zero element of the subspace has rank r, where r is a positive integer that does not exceed m. We show in this paper that if K is a finite field containing at least r+1 elements, any constant rank r subspace of m x n matrices over K has dimension at most n.

math.RA↗

arXiv2013

Constant rank-distance sets of hermitian matrices and partial spreads in hermitian polar spaces

In this paper we investigate partial spreads of $H(2n-1,q^2)$ through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of $H(2n-1,q^2)$. We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of $H(3,q^2)$ for a range of sizes.

math.CO↗

arXiv2007

Galois extensions and subspaces of bilinear forms with special rank properties

Let K be a field admitting a cyclic Galois extension of degree n. The main result of this paper is a decomposition theorem for the space of alternating bilinear forms defined on a vector space of odd dimension n over K. We show that this space of forms is the direct sum of (n-1)/2 subspaces, each of dimension n, and the non-zero elements in each subspace have constant rank defined in terms of the orders of the Galois automorphisms. Furthermore, if ordered correctly, for each integer k lying between 1 and (n-1)/2, the rank of any non-zero element in the sum of the first k subspaces is at most n-2k+1. Slightly less sharp similar results hold for cyclic extensions of even degree.

math.AC↗

arXiv2007

Parity properties of Costas arrays defined via finite fields

A Costas array of order $n$ is an arrangement of dots and blanks into $n$ rows and $n$ columns, with exactly one dot in each row and each column, the arrangement satisfying certain specified conditions. A dot occurring in such an array is even/even if it occurs in the $i$-th row and $j$-th column, where $i$ and $j$ are both even integers, and there are similar definitions of odd/odd, even/odd and odd/even dots. Two types of Costas arrays, known as Golomb-Costas and Welch-Costas arrays, can be defined using finite fields. When $q$ is a power of an odd prime, we enumerate the number of even/even odd/odd, even/odd and odd/even dots in a Golomb-Costas array. We show that three of these numbers are equal and they differ by $\pm 1$ from the fourth. For a Welch-Costas array of order $p-1$, where $p$ is an odd prime, the four numbers above are all equal to $(p-1)/4$ when $p\equiv 1\pmod{4}$, but when $p\equiv 3\pmod{4}$, we show that the four numbers are defined in terms of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$, and thus behave in a much less predictable manner.

math.CO↗

arXiv2007

Extending real-valued characters of finite general linear and unitary groups on elements related to regular unipotents

When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the values of irreducible characters of the extended groups on these elements. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way.

math.RT↗

arXiv2007

Generation of mutually unbiased bases as powers of a unitary matrix in 2-power dimensions

Let q be a power of 2. We show by representation theory that there exists a q x q unitary matrix of multiplicative order q+1 whose powers generate q+1 pairwise mutually unbiased base in C^q. When q is a power of an odd prime, there is a q x q unitary matrix of multiplicative order q+1 whose first (q+1)/2 powers generate (q+1)/2 pairwise mutually unbiased bases. We also show how the existence of these matrices implies the existence of a special type of orthogonal decomposition with respect to the Killing form of the special linear and symplectic Lie algebras.

math.RT↗