Compact cellular algebras and permutation groups

1999 ◽  
Vol 197-198 (1-3) ◽  
pp. 247-267 ◽  
Author(s):  
S Evdokimov
Keyword(s):  
Permutation Groups ◽  
Cellular Algebras

scholarly journals On Highly Closed Cellular Algebras and Highly Closed Isomorphisms

10.37236/1450 ◽  
1998 ◽  
Vol 6 (1) ◽  
Author(s):  
Sergei Evdokimov ◽  
Ilia Ponomarenko
Keyword(s):  
Matrix Algebra ◽  
Point Of View ◽  
The Other ◽  
Permutation Groups ◽  
Colored Graph ◽  
Algebraic Point ◽  
Cellular Algebras ◽  
Other Hand ◽  
Adjacency Algebra ◽  
Centralizer Algebras

We define and study $m$-closed cellular algebras (coherent configurations) and $m$-isomorphisms of cellular algebras which can be regarded as $m$th approximations of Schurian algebras (i.e. the centralizer algebras of permutation groups) and of strong isomorphisms (i.e. bijections of the point sets taking one algebra to the other) respectively. If $m=1$ we come to arbitrary cellular algebras and their weak isomorphisms (i.e. matrix algebra isomorphisms preserving the Hadamard multiplication). On the other hand, the algebras which are $m$-closed for all $m\ge 1$ are exactly Schurian ones whereas the weak isomorphisms which are $m$-isomorphisms for all $m\ge 1$ are exactly ones induced by strong isomorphisms. We show that for any $m$ there exist $m$-closed algebras on $O(m)$ points which are not Schurian and $m$-isomorphisms of cellular algebras on $O(m)$ points which are not induced by strong isomorphisms. This enables us to find for any $m$ an edge colored graph with $O(m)$ vertices satisfying the $m$-vertex condition and having non-Schurian adjacency algebra. On the other hand, we rediscover and explain from the algebraic point of view the Cai-Fürer-Immerman phenomenon that the $m$-dimensional Weisfeiler-Lehman method fails to recognize the isomorphism of graphs in an efficient way.

scholarly journals Compact cellular algebras and permutation groups

1999 ◽  
Vol 197-198 ◽  
pp. 247-267 ◽  
Author(s):  
Sergei Evdokimov ◽  
Marek Karpinski ◽  
Ilia Ponomarenko
Keyword(s):  
Permutation Groups ◽  
Cellular Algebras

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).

scholarly journals Permutation groups with small orbit growth

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.

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

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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