Real mutually unbiased bases and representations of groups of odd order by real scaled Hadamard matrices of 2-power size

We prove the following two results relating real mutually unbiased bases and representations of finite groups of odd order. Let $q$ be a power of 2 and $r$ a positive integer. Then we can find a $q^{2r}\times q^{2r}$ real orthogonal matrix $D$, say, of multiplicative order $q^{2r-1}+1$, whose $q^{2r-1}+1$ powers $D$, \dots, $D^{q^{2r-1}+1}=I$ define $q^{2r-1}+1$ mutually unbiased bases in $\mathbb{R}^{q^{2r}}$. Thus the scaled matrices $q^rD$, \dots, $q^rD^{q^{2r-1}}$ are $q^{2r-1}$ different Hadamard matrices. When we take $q=2$, we achieve the maximum number of real mutually unbiased bases in dimension $2^{2r}$ using the elements of a cyclic group. We also prove the following. Let $G$ be an arbitrary finite group of odd order $2k+1$, where $k\geq 3$. Then $G$ has a real representation $R$, say, of degree $2^{2^{k-1}}$ such that the elements $R(\sigma)$, $\sigma\in G$, define $|G|$ mutually unbiased bases in $\mathbb{R}^{d}$, where $d= 2^{2^{k-1}}$. In addition, a group of order 5 defines five real mutually unbiased bases in $\mathbb{R}^{16}$ and a group of order 3 defines three real mutually unbiased bases in $\mathbb{R}^{4}$. Thus, an arbitrary group of odd order has a faithful representation by real scaled Hadamard matrices of 2-power size.Comment: 17 pages. Replaces previous version with more comprehensive result

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.Comment: 9 pages, some earlier questions resolve

Rank-related dimension bounds for subspaces of bilinear forms over finite fields

Let q be a power of a prime and let V be a vector space of finite dimension n over the field of order q. Let Bil(V) denote the set of all bilinear forms defined on V x V, let Symm(V) denote the subspace of Bil(V) consisting of symmetric bilinear forms, and Alt(V) denote the subspace of alternating bilinear forms. Let M denote a subspace of any of the spaces Bil(V), Symm(V), or Alt(V). In this paper we investigate hypotheses on the rank of the non-zero elements of M which lead to reasonable bounds for dim M. Typically, we look at the case where exactly two or three non-zero ranks occur, one of which is usually n. In the case that M achieves the maximal dimension predicted by the dimension bound, we try to enumerate the number of forms of a given rank in M and describe geometric properties of the radicals of the degenerate elements of M.Comment: 25 page

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.Comment: 11 page

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.Comment: 4 page

Partial orthogonal spreads over F2\mathbb{F}_2 invariant under the symmetric and alternating groups

Let m be an integer greater than 2 and let V be a vector space of dimension 2^m over F_2. Let Q be a non-degenerate quadratic form of maximal Witt index defined on V. We show that the symmetric group S_{2m+1} acts on V as a group of isometries of Q and permutes the members of a partial orthogonal spread of size 2m+1. This implies that any group of even order 2m or odd order 2m+1 acts transitively and regularly on a partial orthogonal spread in V. We also show that the alternating group A_9 acts in a natural manner on a complete spread of size 9 defined on a vector space of dimension 8 over F_2.Comment: 9 pages, Corollary 1 extended to all finite groups of order at least

Dimension bounds for constant rank subspaces of symmetric bilinear forms over a finite field

Let V be a vector space of dimension n over the finite field F_q, where q is odd, and let Symm(V) denote the space of symmetric bilinear forms defined on V x V. We investigate constant rank r subspaces of Symm(V) in this paper. We have proved elsewhere that such a subspace has dimension at most n when q is larger than r but in this paper we provide generally improved upper bounds. Our investigations yield information about common isotropic points for such constant rank subspaces, and also how the radicals of the elements in the subspace are distributed throughout V.Comment: 14 pages; fixed simple typo

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.Comment: 13 page

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.Comment: 27 page

Connections between rank and dimension for subspaces of bilinear forms

Let $K$ be a field and let $V$ be a vector space of dimension $n$ over $K$. Let $M$ be a subspace of bilinear forms defined on $V\times V$. Let $r$ be the number of different non-zero ranks that occur among the elements of $M$. Our aim is to obtain an upper bound for $\dim M$ in terms of $r$ and $n$ under various hypotheses. As a sample of what we prove, we mention the following. Suppose that $m$ is the largest integer that occurs as the rank of an element of $M$. Then if $m\leq \lceil n/2\rceil$ and $|K|\geq m+1$, we have $\dim M\leq rn$. The case $r=1$ corresponds to a constant rank space and it is conjectured that $\dim M\leq n$ when $M$ is a constant rank $m$ space and $|K|\geq m+1$. We prove that the dimension bound for a constant rank $m$ space $M$ holds provided $|K|\geq m+1$ and either $K$ is finite or $K$ has characteristic different from 2 and $M$ consists of symmetric forms. In general, we show that if $M$ is a constant rank $m$ subspace and $|K|\geq m+1$, then $\dim M\leq \max\,(n,2m-1)$. We also provide more detailed results about constant rank subspaces over finite fields, especially subspaces of alternating or symmetric bilinear forms.Comment: 24 page