335 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
Counting Coxeter's friezes over a finite field via moduli spaces
We count the number of Coxeter's friezes over a finite field. Our method uses
geometric realizations of the spaces of friezes in a certain completion of the
classical moduli space allowing repeated points in the
configurations. Counting points in the completed moduli space over a finite
field is related to the enumeration problem of counting partitions of
cyclically ordered set of points into subsets containing no consecutive points.
In Appendix we provide an elementary solution for this enumeration problem
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