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The closure diagram for nilpotent orbits of the split real form of E8

Dragomir Đoković — 2003

Open Mathematics

Let 𝒪 1 and 𝒪 2 be adjoint nilpotent orbits in a real semisimple Lie algebra. Write 𝒪 1 𝒪 2 if 𝒪 2 is contained in the closure of 𝒪 1 . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E 8. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible...

A class of cyclic (v; k1, k2, k3; λ) difference families with v ≡ 3 (mod 4) a prime

Dragomir Ž. ÐokovicIlias S. Kotsireas — 2016

Special Matrices

We construct several new cyclic (v; k1, k2, k3; λ) difference families, with v ≡ 3 (mod 4) a prime and λ = k1 + k2 + k3 − (3v − 1)/4. Such families can be used in conjunction with the well-known Paley-Todd difference sets to construct skew-Hadamard matrices of order 4v. Our main result is that we have constructed for the first time the examples of skew Hadamard matrices of orders 4 · 239 = 956 and 4 · 331 = 1324.

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....

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