63 research outputs found
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
-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
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