The square root rank of a nonnegative matrix A is the minimum rank of a matrix B such that A = B ◦B, where ◦ denotes entrywise product. We show that the square root rank of the slack matrix of the correlation polytope is exponential. Our main technique is a way to lower bound the rank of certain matrices under arbitrary sign changes of the entries using properties of the roots of polynomials in...

Let G = GLr+1 over a nonarchimedean local field F . The Kashiwara crystal B(∞) is the quantized enveloping algebra of the lower triangular maximal unipotent subgroup N−. Examples are given where an integral over N−(F ) may be replaced by a sum over B(∞). Thus the Gindikin-Karpelevich formula evaluates the integral of the standard spherical vector in the induced model of a principal series repre...

Let G be a semisimple, simply connected algebraic group over an algebraically closed field of prime characteristic p > 0. Let U be the unipotent part of a Borel subgroup B ⊂ G and u the Lie algebra of U . Springer [15] has shown for good primes, that there is a B-equivariant isomorphism U → u, where B acts through conjugation on U and through the adjoint action on u (for G = SLn one has the wel...

Functional relations and nested Bethe ansatz for sl(3) chiral Potts model at q 2 = −1. Abstract We obtain the functional relations for the eigenvalues of the transfer matrix of the sl(3) chiral Potts model for q 2 = −1. For the homogeneous model in both directions a solution of these functional relations can be written in terms of roots of Bethe ansatz-like equations. In addition , a direct nes...

In Part I we proposed a multilinear algebra framework to solve 0-dimensional systems of polynomial equations with simple roots. We extend this incorporate multiple roots: block term decomposition (BTD) the null space Macaulay matrix reveals dual (sub)space disjoint root in each term. The BTD is joint triangularization multiplication tables and three-way generalization Jordan canonical form case...

The highest possible minimal norm of a unimodular lattice is determined in dimensions n ≤ 33. There are precisely five odd 32-dimensional lattices with the highest possible minimal norm (compared with more than 8.1020 in dimension 33). Unimodular lattices with no roots exist if and only if n ≥ 23, n 6= 25.

Communication is a field of study that has its roots in the study of persuasion. From Aristotle to presentday scholars, we have centered our attention on the requisite skills and abilities necessary to change attitudes and behaviors. While arguments too numerous to review here have ensued during the development of competing positions, what has been sorely lacking is evidence of whether or not t...

Suppose Γ is a Lie incidence geometry defined over some field F having a Lie incidence geometry Γ0 of the same type but defined over a subfield F0 ≤ F as a subgeometry. We investigate the following question: how many points (if any at all) do we have to add to the point-set of Γ0 in order to obtain a generating set for Γ? We note that if Γ is generated by the points of an apartment, then no add...

We give a simple proof of Chebotarëv’s theorem: Let p be a prime and ω a primitive pth root of unity. Then all minors of the matrix

We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group Eτ,η(sl2) at the discrete coupling constants: 2Nη = m1 + im2τ , where N,m1 and m2 are integers. Then we show that...