Universal subspaces for compact Lie groups

For a representation of a connected compact Lie group G in a finite dimensional real vector space U and a subspace V of U, invariant under a maximal torus of G, we obtain a sufficient condition for V to meet all G-orbits in U, which is also necessary in certain cases. The proof makes use of the cohomology of flag manifolds and the invariant theory of Weyl groups. Then we apply our condition to the conjugation representations of U(n), Sp(n), and SO(n) in the space of $n\times n$ matrices over C, H, and R, respectively. In particular, we obtain an interesting generalization of Schur's triangularization theorem.Comment: 20 page

On singular values and similarity classes of matrices

AbstractLet s be the map Mn → Rn, where Mn is the space of n × n complex matrices, which assigns to each matrix A the n-tuple of its singular values arranged in decreasing order. We are interested in describing the image s(Δ) of a similarity class of matrices Δ ⊂ Mn. The problem is completely solved when Δ consists of cyclic matrices and in a number of other cases. We tabulate descriptions of s(Δ) for all similarity classes Δ ⊂ Mn when 2⩽n⩽4

Classification of Base Sequences (+1,)

Base sequences BS(+1,) are quadruples of {±1}-sequences (;;;), with A and B of length +1 and C and D of length n, such that the sum of their nonperiodic autocor-relation functions is a -function. The base sequence conjecture, asserting that BS(+1,) exist for all n, is stronger than the famous Hadamard matrix conjecture. We introduce a new definition of equivalence for base sequences BS(+1,) and construct a canonical form. By using this canonical form, we have enumerated the equivalence classes of BS(+1,) for ≤30. As the number of equivalence classes grows rapidly (but not monotonically) with n, the tables in the paper cover only the cases ≤13

Computational methods for difference families in finite abelian groups

Our main objective is to show that the computational methods, developed previously to search for difference families in cyclic groups, can be fully extended to the more general case of arbitrary finite abelian groups. In particular the power spectral density test and the method of compression can be used to help the search

Symmetric Hadamard matrices of order 116 and 172 exist

We construct new symmetric Hadamard matrices of orders 92, 116, and 172. While the existence of those of order 92 was known since 1978, the orders 116 and 172 are new. Our construction is based on a recent new combinatorial array (GP array) discovered by N. A. Balonin and J. Seberry. For order 116 we used an adaptation of an algorithm for parallel collision search. The adaptation pertains to the modification of some aspects of the algorithm to make it suitable to solve a 3-way matching problem. We also point out that a new infinite series of symmetric Hadamard matrices arises by plugging into the GP array the matrices constructed by Xia, Xia, Seberry, and Wu in 2005