scholarly journals The signed permutation group on Feynman graphs

2016 ◽  
Vol 57 (8) ◽  
pp. 082302
Author(s):  
Julian Purkart
Keyword(s):  
Permutation Group ◽  
Signed Permutation ◽  
Feynman Graphs

scholarly journals Generating All 36,864 Four-Color Adinkras via Signed Permutations and Organizing into ℓ- and ℓ˜-Equivalence Classes

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 120 ◽  
Author(s):  
S. Gates ◽  
Kevin Iga ◽  
Lucas Kang ◽  
Vadim Korotkikh ◽  
Kory Stiffler
Keyword(s):  
Permutation Group ◽  
Quaternion Algebra ◽  
Equivalence Classes ◽  
Inner Product ◽  
Complete Analysis ◽  
Signed Permutation ◽  
Signed Permutations ◽  
Non Linear

Recently, all 1,358,954,496 values of the gadget between the 36,864 adinkras with four colors, four bosons, and four fermions have been computed. In this paper, we further analyze these results in terms of B C 3 , the signed permutation group of three elements, and B C 4 , the signed permutation group of four elements. It is shown how all 36,864 adinkras can be generated via B C 4 boson × B C 3 color transformations of two quaternion adinkras that satisfy the quaternion algebra. An adinkra inner product has been used for some time, known as the gadget, which is used to distinguish adinkras. We show how 96 equivalence classes of adinkras that are based on the gadget emerge in terms of B C 3 and B C 4 . We also comment on the importance of the gadget as it relates to separating out dynamics in terms of Kähler-like potentials. Thus, on the basis of the complete analysis of the supersymmetrical representations achieved in the preparatory first four sections, the final comprehensive achievement of this work is the construction of the universal B C 4 non-linear σ -model.

B7as a supergroup of crystal and quasicrystal symmetries

2017 ◽  
Vol 73 (2) ◽  
pp. 135-139
Author(s):  
Kazimierz Stróż
Keyword(s):  
Finite Group ◽  
Permutation Group ◽  
Three Dimensional ◽  
Symmetry Groups ◽  
Signed Permutation ◽  
Acta Cryst ◽  
Matrix Groups ◽  
Minimal Dimension ◽  
A Finite Group ◽  
Basis Vectors

In sharp contrast to the generation of a finite group that includes all the 14 types of Bravais lattices as its subgroups [Hosoya (2000).Acta Cryst.A56, 259–263; Hosoya (2002).Acta Cryst.A58, 208], it was proved that a signed permutation groupBkmay be interpreted as the supergroup of both crystal and quasicrystal symmetries. Minimal dimensionk= 6 is adequate for lattices referred to their three non-coplanar shortest vectors, or for symmetry groups of most quasicrystal types. If one prefers complete, well defined semi-reduced lattice descriptions or needs a dodecagonal group, theB7supergroup is necessary. All considered matrix groups correspond to isometric transformations in extendedk-bases and may be easily derived fromB7and projected onto three-dimensional crystallographic space. Three models of extended bases are proposed: semi-reduced, cyclic and axial. In all cases additional basis vectors are strictly (functionally) related to three original basis vectors.

The complements in an affine group

2013 ◽  
Vol 50 (2) ◽  
pp. 258-265
Author(s):  
Pál Hegedűs
Keyword(s):  
Permutation Group ◽  
Structural Similarity ◽  
Affine Group ◽  
Permutation Module ◽  
Regular Module ◽  
Brauer Characters

In this paper we analyse the natural permutation module of an affine permutation group. For this the regular module of an elementary Abelian p-group is described in detail. We consider the inequivalent permutation modules coming from nonconjugate complements. We prove their strong structural similarity well exceeding the fact that they have equal Brauer characters.

scholarly journals ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

2021 ◽  
pp. 1-40
Author(s):  
NICK GILL ◽  
BIANCA LODÀ ◽  
PABLO SPIGA
Keyword(s):  
Permutation Group ◽  
Model Theory ◽  
Independent Set ◽  
Maximum Size ◽  
Proper Subset ◽  
Permutation Groups ◽  
Finite Permutation Group ◽  
Pointwise Stabilizer ◽  
Relational Complexity

Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

Some generalized characters associated to a transitive permutation group

2020 ◽  
Vol 23 (3) ◽  
pp. 393-397
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Frobenius Group ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Transitive Permutation Group ◽  
Point Stabilizer ◽  
Necessary And Sufficient ◽  
Regular Character ◽  
Permutation Character

AbstractLet G be a finite transitive permutation group of degree n, with point stabilizer {H\neq 1} and permutation character π. For every positive integer t, we consider the generalized character {\psi_{t}=\rho_{G}-t(\pi-1_{G})}, where {\rho_{G}} is the regular character of G and {1_{G}} the 1-character. We give necessary and sufficient conditions on t (and G) which guarantee that {\psi_{t}} is a character of G. A necessary condition is that {t\leq\min\{n-1,\lvert H\rvert\}}, and it turns out that {\psi_{t}} is a character of G for {t=n-1} resp. {t=\lvert H\rvert} precisely when G is 2-transitive resp. a Frobenius group.

scholarly journals The Maximal Number of Orbits of a Permutation Group with Bounded Movement

1999 ◽  
Vol 214 (2) ◽  
pp. 625-630 ◽  
Author(s):  
Jung R. Cho ◽  
Pan Soo Kim ◽  
Cheryl E. Praeger
Keyword(s):  
Permutation Group ◽  
Bounded Movement
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