An Aspect of Icosahedral Symmetry

2002 ◽  
Vol 45 (4) ◽  
pp. 686-696 ◽  
Author(s):  
Jan Rauschning ◽  
Peter Slodowy
Keyword(s):  
Symmetric Group ◽  
Moduli Space ◽  
Irreducible Module ◽  
Projective Line ◽  
Icosahedral Symmetry ◽  
3 Dimensional ◽  
Icosahedral Group ◽  
Regular Icosahedron

AbstractWe embed the moduli space Q of 5 points on the projective line S5-equivariantly into (V), where V is the 6-dimensional irreducible module of the symmetric group S5. This module splits with respect to the icosahedral group A5 into the two standard 3-dimensional representations. The resulting linear projections of Q relate the action of A5 on Q to those on the regular icosahedron.

scholarly journals The Moduli Space of 3-Dimensional Associative Algebras

2009 ◽  
Vol 37 (10) ◽  
pp. 3666-3685 ◽  
Author(s):  
Alice Fialowski ◽  
Michael Penkava
Keyword(s):  
Moduli Space ◽  
Associative Algebras ◽  
3 Dimensional

scholarly journals Principal co-Higgs bundles on ℙ1

2020 ◽  
Vol 63 (2) ◽  
pp. 512-530 ◽  
Author(s):  
Indranil Biswas ◽  
Oscar García-Prada ◽  
Jacques Hurtubise ◽  
Steven Rayan
Keyword(s):  
Moduli Space ◽  
Borel Subgroup ◽  
Algebraic Groups ◽  
Projective Line ◽  
Higgs Bundles ◽  
Complex Projective Line ◽  
Theoretic Characterization ◽  
Complex Projective

AbstractFor complex connected, reductive, affine, algebraic groups G, we give a Lie-theoretic characterization of the semistability of principal G-co-Higgs bundles on the complex projective line ℙ1 in terms of the simple roots of a Borel subgroup of G. We describe a stratification of the moduli space in terms of the Harder–Narasimhan type of the underlying bundle.

scholarly journals Cross-ratio Dynamics on Ideal Polygons

2020 ◽  
Author(s):  
Maxim Arnold ◽  
Dmitry Fuchs ◽  
Ivan Izmestiev ◽  
Serge Tabachnikov
Keyword(s):  
Hyperbolic Space ◽  
Moduli Space ◽  
Hyperbolic Plane ◽  
Projective Line ◽  
Cross Ratio ◽  
Poisson Structures ◽  
Completely Integrable ◽  
The Cross

Abstract Two ideal polygons, $(p_1,\ldots ,p_n)$ and $(q_1,\ldots ,q_n)$, in the hyperbolic plane or in hyperbolic space are said to be $\alpha $-related if the cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = \alpha $ for all $i$ (the vertices lie on the projective line, real or complex, respectively). For example, if $\alpha = -1$, the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of Möbius-equivalent polygons. We prove that this relation, which is generically a 2-2 map, is completely integrable in the sense of Liouville. We describe integrals and invariant Poisson structures and show that these relations, with different values of the constants $\alpha $, commute, in an appropriate sense. We investigate the case of small-gons and describe the exceptional ideal pentagons and hexagons that possess infinitely many $\alpha $-related polygons.

VERSAL DEFORMATIONS OF THREE-DIMENSIONAL LIE ALGEBRAS AS L∞ ALGEBRAS

2005 ◽  
Vol 07 (02) ◽  
pp. 145-165 ◽  
Author(s):  
ALICE FIALOWSKI ◽  
MICHAEL PENKAVA
Keyword(s):  
Vector Space ◽  
Lie Algebras ◽  
Moduli Space ◽  
Deformation Theory ◽  
Three Dimensional ◽  
3 Dimensional ◽  
Exterior Degree ◽  
Versal Deformations

We consider versal deformations of 0|3-dimensional L∞ algebras, also called strongly homotopy Lie algebras, which correspond precisely to ordinary (non-graded) three-dimensional Lie algebras. The classification of such algebras is well-known, although we shall give a derivation of this classification using an approach of treating them as L∞ algebras. Because the symmetric algebra of a three-dimensional odd vector space contains terms only of exterior degree less than or equal to three, the construction of versal deformations can be carried out completely. We give a characterization of the moduli space of Lie algebras using deformation theory as a guide to understanding the picture.

scholarly journals The uniformization of the moduli space of principally polarized abelian 6-folds

2020 ◽  
Vol 2020 (761) ◽  
pp. 163-217
Author(s):  
Valery Alexeev ◽  
Ron Donagi ◽  
Gavril Farkas ◽  
Elham Izadi ◽  
Angela Ortega
Keyword(s):  
Weyl Group ◽  
Abelian Variety ◽  
Moduli Space ◽  
Projective Line ◽  
Geometric Description ◽  
Canonical Class ◽  
Hurwitz Space ◽  
The Way

AbstractStarting from a beautiful idea of Kanev, we construct a uniformization of the moduli space \mathcal{A}_{6} of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general principally polarized abelian variety of dimension 6 is a Prym–Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E_{6} lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E_{6}-covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge–Hurwitz eigenbundles with respect to the Kanev correspondence and (3) a description of the ramification divisor of the Prym–Tyurin map from the Hurwitz space to \mathcal{A}_{6} in the terms of syzygies of the Abel–Prym–Tyurin curve.

scholarly journals A MATHEMATICAL APPROACH TO STRUCTURAL TRANSITIONS IN VIRAL CAPSIDS

2012 ◽  
Vol 09 ◽  
pp. 11-23
Author(s):  
GIULIANA INDELICATO ◽  
REIDUN TWAROCK
Keyword(s):  
Phase Transitions ◽  
Icosahedral Symmetry ◽  
Mathematical Approach ◽  
Structural Transitions ◽  
Viral Capsids ◽  
3 Dimensional ◽  
Crystallographic Theory

The majority of viruses have protein containers, called capsids, in which proteins are arranged with icosahedral symmetry. The capsids of these viruses follow structural blueprints that can be modelled as subsets of 3-dimensional aperiodic lattices called quasicrystals. These, in turn, can be obained by projection from 6-dimensional Bravais lattices. In this work we apply the crystallographic theory of phase transitions to these 6D lattices, and via projection of these results derive information on the structural transitions of viruses important for infection.

HOW SINGULAR A SPACE CAN SUPERSTRINGS THREAD?

1991 ◽  
Vol 06 (03) ◽  
pp. 207-216 ◽  
Author(s):  
TRISTAN HÜBSCH
Keyword(s):  
Minkowski Space ◽  
Moduli Space ◽  
Space Time ◽  
3 Dimensional ◽  
Ricci Flat ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space

Many superstring models with N=1 supergravity in 4-dimensional Minkowski space-time involve σ-models with complex 3-dimensional, Ricci-flat target manifolds. In general, inclusion of singular target spaces probes the boundary of the moduli space and completes it. Studying suitably singular σ-models, the author found certain criteria for the severity of admissible singularizations.

scholarly journals Finite Generation of the Algebra of Type A Conformal Blocks via Birational Geometry

2018 ◽  
Author(s):  
Han-Bom Moon ◽  
Sang-Bum Yoo
Keyword(s):  
Moduli Space ◽  
Type A ◽  
Projective Line ◽  
Birational Geometry ◽  
Finite Generation ◽  
Conformal Blocks ◽  
Parabolic Bundles ◽  
Mori Dream Space ◽  
Birational Model ◽  
Effective Cone

Abstract We study the birational geometry of the moduli space of parabolic bundles over a projective line, in the framework of Mori’s program. We show that the moduli space is a Mori dream space. As a consequence, we obtain the finite generation of the algebra of type A conformal blocks. Furthermore, we compute the H-representation of the effective cone that was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles.

Dodecahedral slices and polyhedral pieces

2010 ◽  
Vol 94 (529) ◽  
pp. 5-17
Author(s):  
Doug French ◽  
David Jordan
Keyword(s):  
Icosahedral Symmetry ◽  
Surprising Result ◽  
Regular Pentagon ◽  
Regular Dodecahedron ◽  
Regular Icosahedron ◽  
Image Position

Figure 1 shows how a regular dodecahedron can be dissected into three slices by two planes through the two sets of vertices, each set defining a regular pentagon parallel to the top and bottom faces. A surprising result emerges if we calculate the ratio of the volumes of the three slices. We first prove this result directly and then show it by a dissection argument using simple polyhedral pieces of five types. These pieces can be used to build many polyhedra, including the regular dodecahedron, the regular icosahedron, the great dodecahedron, the small and great stellated dodecahedra and all the Archimedean polyhedra which have icosahedral symmetry.

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