10 research outputs found
Majorana representations of the symmetric group of degree 4
AbstractThe Monster group M acts on a real vector space VM of dimension 196,884 which is the sum of a trivial 1-dimensional module and a minimal faithful M-module. There is an M-invariant scalar product (,) on VM, an M-invariant bilinear commutative non-associative algebra product â
on VM (commonly known as the ConwayâGriessâNorton algebra), and a subset A of VMâ{0} indexed by the 2A-involutions in M. Certain properties of the quintetM=(M,VM,A,(,),â
) have been axiomatized in Chapter 8 of Ivanov (2009) [Iv09] under the name of Majorana representation of M. The axiomatization enables one to study Majorana representations of an arbitrary group G (generated by its involutions). A representation might or might not exist, but it always exists whenever G is a subgroup in M generated by the 2A-involutions contained in G. We say that thus obtained representation is based on an embedding of G in the Monster. The essential motivation for introducing the Majorana terminology was the most remarkable result by S. Sakuma (2007) [Sak07] which gave a classification of the Majorana representations of the dihedral groups. There are nine such representations and every single one is based on an embedding in the Monster of the relevant dihedral group. It is a fundamental property of the Monster that its 2A-involutions form a class of 6-transpositions and that there are precisely nine M-orbits on the pairs of 2A-involutions (and also on the set of 2A-generated dihedral subgroups in M). In the present paper we are making a further step in building up the Majorana theory by classifying the Majorana representations of the symmetric group S4 of degree 4. We prove that S4 possesses precisely four Majorana representations. The Monster is known to contain four classes of 2A-generated S4-subgroups, so each of the four representations is based on an embedding of S4 in the Monster. The classification of 2A-generated S4-subgroups in the Monster relies on calculations with the character table of the Monster. Our elementary treatment shows that there are (at most) four isomorphism types of subalgebras in the ConwayâGriess-Norton algebra of the Monster generated by six Majorana axial vectors canonically indexed by the transpositions of S4. Two of these subalgebras are 13-dimensional, the other two have dimensions 9 and 6. These dimensions, not to mention the isomorphism type of the subalgebras, were not known before
An efficient sum of squares nonnegativity certificate for quaternary quartic
We give an embarrassingly simple construction of a representation of a nonnegative 4-ary homogeneous polynomial (i.e. a form) of degree 4 as a ratio of a sum of squares forms of degrees 8 and 4, respectively. This allows an efficient check of nonnegativity of using semidefinite optimization. No similar results were known for forms of degree greater than 2 in more than 3 variables
Splitting fields of real irreducible representations of finite groups
We show that any irreducible representation ρ of a finite group G of exponent n, realisable over R, is realisable over the field E:= Q(ζn) ∩ R of real cyclotomic numbers of order n, and describe an algorithmic procedure transforming a realisation of ρ over Q(ζn) to one over E.</p
Anisotropic shear viscosity in nematic liquid crystals: new optical measurement method
We propose a new optical method and the experimental set-up for measuring the anisotropic shear viscosities of nematic liquid crystals (LCs). LC shear viscosities can be optimized to improve liquid crystal display (LCD) response times, e. g. in vertical aligned nematic (VAN) or bistable nematic displays (BND). In this case a strong back-flow effect essentially determines the LCD dynamic characteristics. A number of shear viscosity coefficients defines the LCD response time. The proposed method is based on the special type of a shear flow, namely, the decay flow, in the LC cell with suitably treated substrates instead of magnetic or electric field application. A linear regime of a quasi-stationary director motion induced by a pressure difference and a proper configuration of a LC cell produces decay flow conditions in the LC cell. We determine three principal shear viscosity coefficients by measuring relative time variations of the intensity of the light passed through LC cells. The shear viscosity coefficient measurements provide a new opportunity for the development of new LC mixtures with fast response times in VAN, BND and other important LCD types
Author Correction: SciPy 1.0: fundamental algorithms for scientific computing in Python (Nature Methods, (2020), 17, 3, (261-272), 10.1038/s41592-019-0686-2)
An amendment to this paper has been published and can be accessed via a link at the top of the paper