scholarly journals Primitive normalisers in quasipolynomial time

2021 ◽  
Author(s):  
Mun See Chang ◽  
Colva M. Roney-Dougal
Keyword(s):  
Symmetric Group ◽  
Generating Set ◽  
Subexponential Time ◽  
Time Solution

AbstractThe normaliser problem has as input two subgroups H and K of the symmetric group $$\mathrm {S}_n$$ S n , and asks for a generating set for $$N_K(H)$$ N K ( H ) : it is not known to have a subexponential time solution. It is proved in Roney-Dougal and Siccha (Bull Lond Math Soc 52(2):358–366, 2020) that if H is primitive, then the normaliser problem can be solved in quasipolynomial time. We show that for all subgroups H and K of $$\mathrm {S}_n$$ S n , in quasipolynomial time, we can decide whether $$N_{\mathrm {S}_n}(H)$$ N S n ( H ) is primitive, and if so, compute $$N_K(H)$$ N K ( H ) . Hence we reduce the question of whether one can solve the normaliser problem in quasipolynomial time to the case where the normaliser in $$\mathrm {S}_n$$ S n is known not to be primitive.

Integral Cayley Graphs over Finite Groups

2020 ◽  
Vol 27 (01) ◽  
pp. 131-136
Author(s):  
Elena V. Konstantinova ◽  
Daria Lytkina
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Cyclic Group ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Generating Set ◽  
A Finite Group

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

scholarly journals A Combinatorial Model for the Decomposition of Multivariate Polynomial Rings as $S_n$-Modules

10.37236/8935 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Rosa Orellana ◽  
Michael Zabrocki
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Polynomial Rings ◽  
Invariant Polynomials ◽  
Multivariate Polynomial ◽  
Generating Set ◽  
Combinatorial Model

We consider the symmetric group $S_n$-module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition $\lambda$ (a partition of $n$) is the number of multiset tableaux of shape $\lambda$ satisfying certain column and row strict conditions.  We also present a finite generating set for the ring of $S_n$ invariant polynomials of this ring. 

The girth of Cayley graphs over Sylow 2-subgroups of the symmetric groups S2n with diagonal bases

2019 ◽  
Vol 18 (12) ◽  
pp. 1950237
Author(s):  
Bartłomiej Pawlik
Keyword(s):  
Symmetric Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Symmetric Groups ◽  
Polynomial Representation ◽  
Generating Set ◽  
Minimal Generating Set

A diagonal base of a Sylow 2-subgroup [Formula: see text] of symmetric group [Formula: see text] is a minimal generating set of this subgroup consisting of elements with only one nonzero coordinate in the polynomial representation. For different diagonal bases, Cayley graphs over [Formula: see text] may have different girths (i.e. minimal lengths of cycles). In this paper, all possible values of girths of Cayley graphs over [Formula: see text] with diagonal bases are calculated. A criterion for whenever such Cayley graph has girth equal to 4 is presented.

scholarly journals Symmetric polynomials in the variety generated by Grassmann algebras

2021 ◽  
Author(s):  
Nazan Akdoğan ◽  
Şehmus Fındık
Keyword(s):  
Symmetric Group ◽  
Free Algebra ◽  
Module Structure ◽  
Symmetric Polynomials ◽  
Associative Algebras ◽  
Generating Set ◽  
Commutator Ideal ◽  
Infinite Dimensional ◽  
Grassmann Algebras ◽  
Polynomial Formula

Let [Formula: see text] denote the variety generated by infinite-dimensional Grassmann algebras, i.e. the collection of all unitary associative algebras satisfying the identity [Formula: see text], where [Formula: see text]. Consider the free algebra [Formula: see text] in [Formula: see text] generated by [Formula: see text]. We call a polynomial [Formula: see text] symmetric if it is preserved under the action of the symmetric group [Formula: see text] on generators, i.e. [Formula: see text] for each permutation [Formula: see text]. The set of symmetric polynomials forms the subalgebra [Formula: see text] of invariants of the group [Formula: see text] in [Formula: see text]. The commutator ideal [Formula: see text] of the algebra [Formula: see text] has a natural left [Formula: see text]-module structure, and [Formula: see text] is a left [Formula: see text]-module. We give a finite free generating set for the [Formula: see text]-module [Formula: see text].

A SIMPLE GENERALIZED CONSTRUCTION OF RESOLVABLE BALANCED INCOMPLETE BLOCK DESIGNS WITH PRIME BLOCK SIZES

2021 ◽  
Vol 10 (6) ◽  
pp. 2767-2784
Author(s):  
A.J. Saka ◽  
R.A. Adetona ◽  
T.G. Jaiyéolá
Keyword(s):  
Symmetric Group ◽  
Cyclic Subgroup ◽  
Dihedral Group ◽  
Block Designs ◽  
Incomplete Block ◽  
Generating Set ◽  
Balanced Incomplete Block Designs ◽  
Balanced Incomplete Block ◽  
Block Sizes ◽  
Design Construction

This paper presents a Simple Generalized Construction of Resolvable Balanced Incomplete Block Designs whose parameter combination is of the form $v=k^2, ~r=k+1, ~\lambda=k^0=1$, where $k$ is prime. The design construction was achieved by using the cyclic subgroup of the symmetric group $S_k$ whose generator is one of the permutations of the $2$-permutation generating set of the Dihedral group $D_k$ and $2$-permutation generating set of the presentation of $S_k$. The method is efficient, sufficient and also mitigate against the tediousness encountered in other methods of construction when $v$ is large.

scholarly journals ON THE SINGULAR PART OF THE PARTITION MONOID

2011 ◽  
Vol 21 (01n02) ◽  
pp. 147-178 ◽  
Author(s):  
JAMES EAST
Keyword(s):  
Symmetric Group ◽  
Singular Part ◽  
Minimal Cardinality ◽  
Generating Set ◽  
Generators And Relations ◽  
Ideal Formula ◽  
Idempotent Rank ◽  
The Ideal

We study the singular part of the partition monoid [Formula: see text]; that is, the ideal [Formula: see text], where [Formula: see text] is the symmetric group. Our main results are presentations in terms of generators and relations. We also show that [Formula: see text] is idempotent generated, and that its rank and idempotent-rank are both equal to [Formula: see text]. One of our presentations uses an idempotent generating set of this minimal cardinality.

scholarly journals On Certain Integral Schreier Graphs of the Symmetric Group

10.37236/961 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Paul E. Gunnells ◽  
Richard A. Scott ◽  
Byron L. Walden
Keyword(s):  
Symmetric Group ◽  
Positive Eigenvalue ◽  
Generating Set ◽  
Schreier Graphs

We compute the spectrum of the Schreier graph of the symmetric group $S_n$ corresponding to the Young subgroup $S_2\times S_{n-2}$ and the generating set consisting of initial reversals. In particular, we show that this spectrum is integral and for $n\geq 8$ consists precisely of the integers $\{0,1,\ldots,n\}$. A consequence is that the first positive eigenvalue of the Laplacian is always $1$ for this family of graphs.

scholarly journals Infinite partition monoids

2014 ◽  
Vol 24 (04) ◽  
pp. 429-460 ◽  
Author(s):  
James East
Keyword(s):  
Symmetric Group ◽  
Length Function ◽  
Countable Subset ◽  
Generating Set ◽  
Infinite Set

Let 𝒫X and 𝒮X be the partition monoid and symmetric group on an infinite set X. We show that 𝒫X may be generated by 𝒮X together with two (but no fewer) additional partitions, and we classify the pairs α, β ∈ 𝒫X for which 𝒫X is generated by 𝒮X ∪ {α, β}. We also show that 𝒫X may be generated by the set ℰX of all idempotent partitions together with two (but no fewer) additional partitions. In fact, 𝒫X is generated by ℰX ∪ {α, β} if and only if it is generated by ℰX ∪ 𝒮X ∪ {α, β}. We also classify the pairs α, β ∈ 𝒫X for which 𝒫X is generated by ℰX ∪ {α, β}. Among other results, we show that any countable subset of 𝒫X is contained in a 4-generated subsemigroup of 𝒫X, and that the length function on 𝒫X is bounded with respect to any generating set.

scholarly journals A generating set for the canonical ideal of HKG-curves

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Aristides Kontogeorgis ◽  
Ioannis Tsouknidas
Keyword(s):  
Generating Set ◽  
Canonical Ideal
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