Communicated by I. P. ShestakovThe integral subsets of octonions are an analog of integers in real numbers and related to many interesting topics in geometry and physics via E 8 -lattices. In this paper, we study the properties of the multiplication of the integral subsets of octonions by studying configuration of Fano plane via blocks and operations on them. And we show that the integral subsets are integral indeed by introducing new and elementary methods.

We study the toric degeneration of Weyl group translated Schubert divisors of a partial flag variety F n 1 ,...,n k ;n via Gelfand-Cetlin polytopes. We propose a conjecture that Schubert varieties of appropriate dimensions intersect transversally up to translation by Weyl group elements, and verify it in various cases, including the complex Grassmannian Gr(2, n) and the complete flag variety F 1,2,3;4 .