Marks of permutation groups and isomer enumeration

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).

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Permutation groups with small orbit growth

Journal of Group Theory ◽

10.1515/jgth-2018-0220 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Manuel Bodirsky ◽

Bertalan Bodor

Keyword(s):

Automorphism Group ◽

Constraint Satisfaction ◽

Constraint Satisfaction Problem ◽

Permutation Groups ◽

First Order ◽

Finite Covers

Abstract Let K exp + \mathcal{K}_{{\operatorname{exp}}{+}} be the class of all structures 𝔄 such that the automorphism group of 𝔄 has at most c ⁢ n d ⁢ n cn^{dn} orbits in its componentwise action on the set of 𝑛-tuples with pairwise distinct entries, for some constants c , d c,d with d < 1 d<1 . We show that K exp + \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of finite covers of first-order reducts of unary structures, and also that K exp + \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from K exp + \mathcal{K}_{{\operatorname{exp}}{+}} . We also show that Thomas’ conjecture holds for K exp + \mathcal{K}_{{\operatorname{exp}}{+}} : all structures in K exp + \mathcal{K}_{{\operatorname{exp}}{+}} have finitely many first-order reducts up to first-order interdefinability.

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The 300 “correlators” suggests 4D, $$ \mathcal{N} $$ = 1 SUSY is a solution to a set of Sudoku puzzles

Journal of High Energy Physics ◽

10.1007/jhep05(2021)077 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Aleksander J. Cianciara ◽

S. James Gates ◽

Yangrui Hu ◽

Renée Kirk

Keyword(s):

Coxeter Groups ◽

Auxiliary Field ◽

Permutation Groups ◽

Weight Space ◽

Field Problem ◽

Truncated Octahedron ◽

Color Problem ◽

Minimal Representations

Abstract A conjecture is made that the weight space for 4D, $$ \mathcal{N} $$ N -extended supersymmetrical representations is embedded within the permutahedra associated with permutation groups 𝕊d. Adinkras and Coxeter Groups associated with minimal representations of 4D, $$ \mathcal{N} $$ N = 1 supersymmetry provide evidence supporting this conjecture. It is shown that the appearance of the mathematics of 4D, $$ \mathcal{N} $$ N = 1 minimal off-shell supersymmetry representations is equivalent to solving a four color problem on the truncated octahedron. This observation suggest an entirely new way to approach the off-shell SUSY auxiliary field problem based on IT algorithms probing the properties of 𝕊d.

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Closures of finite permutation groups and relation algebras

Groups, Combinatorics & Geometry ◽

10.1017/cbo9780511629259.033 ◽

2010 ◽

pp. 368-379

Author(s):

Cheryl E. Praegei

Keyword(s):

Permutation Groups ◽

Relation Algebras

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L'application des representations au calcul explicite sur certaines chaînes de Markov finies

Advances in Applied Probability ◽

10.2307/1427292 ◽

1984 ◽

Vol 16 (3) ◽

pp. 656-666 ◽

Cited By ~ 2

Author(s):

Bernard Ycart

Keyword(s):

Finite Group ◽

Random Walk ◽

Irreducible Representations ◽

Permutation Groups ◽

A Finite Group

We give here concrete formulas relating the transition generatrix functions of any random walk on a finite group to the irreducible representations of this group. Some examples of such explicit calculations for the permutation groups A4, S4, and A5 are included.

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