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

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

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

A time-varying markov-switching model for economic growth

This paper investigates economic growth’s pattern of variation across and within countries usinga Time-Varying Transition Matrix Markov-Switching Approach. The model developed follows theapproach of Pritchett (2003) and explains the dynamics of growth based on a collection of differentstates, each of which has a sub-model and a growth pattern, by which countries oscillate over time. Thetransition matrix among the different states varies over time, depending on the conditioning variablesof each country, with a linear dynamic for each state. We develop a generalization of the Diebold’sEM Algorithm and estimate an example model in a panel with a transition matrix conditioned onthe quality of the institutions and the level of investment. We found three states of growth: stablegrowth, miraculous growth, and stagnation. The results show that the quality of the institutions is animportant determinant of long-term growth, whereas the level of investment has varying roles in thatit contributes positively in countries with high-quality institutions but is of little relevance in countrieswith medium- or poor-quality institutions.