Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties

In the sl\_n case, A. Berenstein and A. Zelevinsky studied the Sch\"{u}tzenberger involution in terms of Lusztig's canonical basis, [3]. We generalize their construction and formulas for any semisimple Lie algebra. We use for this the geometric lifting of the canonical basis, on which an analogue of the Sch\"{u}tzenberger involution can be given. As an application, we construct semitoric degenerations of Richardson varieties, following a method of P. Caldero, [6]Comment: 22 pages, 3 figure

Well, Papa, can you multiply triplets?

We show that the classical algebra of quaternions is a commutative $\Z_2\times\Z_2\times\Z_2$-graded algebra. A similar interpretation of the algebra of octonions is impossible.Comment: 3 page

Orthogonal Designs and a Cubic Binary Function

Orthogonal designs are fundamental mathematical notions used in the construction of space time block codes for wireless transmissions. Designs have two important parameters, the rate and the decoding delay; the main problem of the theory is to construct designs maximizing the rate and minimizing the decoding delay. All known constructions of CODs are inductive or algorithmic. In this paper, we present an explicit construction of optimal CODs. We do not apply recurrent procedures and do calculate the matrix elements directly. Our formula is based on a cubic function in two binary n-vectors. In our previous work (Comm. Math. Phys., 2010, and J. Pure and Appl. Algebra, 2011), we used this function to define a series of non-associative algebras generalizing the classical algebra of octonions and to obtain sum of squares identities of Hurwitz-Radon type

Introducing supersymmetric frieze patterns and linear difference operators

We introduce a supersymmetric analog of the classical Coxeter frieze patterns. Our approach is based on the relation with linear difference operators. We define supersymmetric analogs of linear difference operators called Hill's operators. The space of these "superfriezes" is an algebraic supervariety, which is isomorphic to the space of supersymmetric second order difference equations, called Hill's equations.Comment: Appendix 2 on Supercontinuants is written by Alexey Ustino