scholarly journals The center of the wreath product of symmetric group algebras

2021 ◽  
Vol 31 (2) ◽  
pp. 302-322
Author(s):  
O. Tout ◽  
Keyword(s):  
Symmetric Group ◽  
Recent Result ◽  
Wreath Product ◽  
Group Algebra ◽  
Symmetric Groups ◽  
Conjugacy Classes ◽  
Group Algebras ◽  
Hyperoctahedral Group ◽  
Symmetric Group Algebras

We consider the wreath product of two symmetric groups as a group of blocks permutations and we study its conjugacy classes. We give a polynomiality property for the structure coefficients of the center of the wreath product of symmetric group algebras. This allows us to recover an old result of Farahat and Higman about the polynomiality of the structure coefficients of the center of the symmetric group algebra and to generalize our recent result about the polynomiality property of the structure coefficients of the center of the hyperoctahedral group algebra.

scholarly journals Generalized Pattern Avoidance Condition for the Wreath Product of Cyclic Groups with Symmetric Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-17
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Cyclic Groups ◽  
Bijective Proof ◽  
Avoidance Condition

We continue the study of the generalized pattern avoidance condition for Ck≀Sn, the wreath product of the cyclic group Ck with the symmetric group Sn, initiated in the work by Kitaev et al., In press. Among our results, there are a number of (multivariable) generating functions both for consecutive and nonconsecutive patterns, as well as a bijective proof for a new sequence counted by the Catalan numbers.

The Poset of Conjugacy Classes and Decomposition of Products in the Symmetric Group

1992 ◽  
Vol 35 (2) ◽  
pp. 152-160 ◽  
Author(s):  
François Bédard ◽  
Alain Goupil
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Conjugacy Classes ◽  
The Other ◽  
Linear Combinations ◽  
Graded Poset

AbstractThe action by multiplication of the class of transpositions of the symmetric group on the other conjugacy classes defines a graded poset as described by Birkhoff ([2]). In this paper, the edges of this poset are given a weight and the structure obtained is called the poset of conjugacy classes of the symmetric group. We use weights of chains in the posets to obtain new formulas for the decomposition of products of conjugacy classes of the symmetric group in its group algebra as linear combinations of conjugacy classes and we derive a new identity involving partitions of n.

scholarly journals Defect 3 Blocks of Symmetric Group Algebras, I

2001 ◽  
Vol 237 (1) ◽  
pp. 95-120 ◽  
Author(s):  
Stuart Martin ◽  
Kai Meng Tan
Keyword(s):  
Symmetric Group ◽  
Group Algebras ◽  
Symmetric Group Algebras

scholarly journals On the orders of conjugacy classes in group algebras of p-groups

2004 ◽  
Vol 77 (2) ◽  
pp. 185-190 ◽  
Author(s):  
A. Bovdi ◽  
L. G. Kovács ◽  
S. Mihovski
Keyword(s):  
Conjugacy Class ◽  
Group Algebra ◽  
Commutator Subgroup ◽  
Conjugacy Classes ◽  
Group Algebras ◽  
Normalized Units

AbstractLet p be a prime, a field of pn elements, and G a finite p-group. It is shown here that if G has a quotient whose commutator subgroup is of order p and whose centre has index pk, then the group of normalized units in the group algebra has a conjugacy class of pnk elements. This was first proved by A. Bovdi and C. Polcino Milies for the case k = 2; their argument is now generalized and simplified. It remains an intriguing question whether the cardinality of the smallest noncentral conjugacy class can always be recognized from this test.

scholarly journals Gelfand Pairs Involving the Wreath Product of Finite Abelian Groups with Symmetric Groups

2020 ◽  
pp. 1-7
Author(s):  
Omar Tout
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Wreath Product ◽  
Abelian Groups ◽  
Symmetric Groups ◽  
Gelfand Pair ◽  
Gelfand Pairs ◽  
Finite Abelian Groups ◽  
A Finite Group

Abstract It is well known that the pair $(\mathcal {S}_n,\mathcal {S}_{n-1})$ is a Gelfand pair where $\mathcal {S}_n$ is the symmetric group on n elements. In this paper, we prove that if G is a finite group then $(G\wr \mathcal {S}_n, G\wr \mathcal {S}_{n-1}),$ where $G\wr \mathcal {S}_n$ is the wreath product of G by $\mathcal {S}_n,$ is a Gelfand pair if and only if G is abelian.

scholarly journals $\delta$-Connectivity in Random Lifts of Graphs

10.37236/6639 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Shashwat Silas
Keyword(s):  
Lower Bound ◽  
Symmetric Group ◽  
Wreath Product ◽  
Connected Graph ◽  
Minimum Degree ◽  
Symmetric Groups ◽  
Gamma Delta ◽  
Random Generators

Amit and Linial have shown that a random lift of a connected graph with minimum degree $\delta\ge3$ is asymptotically almost surely (a.a.s.) $\delta$-connected and mentioned the problem of estimating this probability as a function of the degree of the lift. Using a connection between a random $n$-lift of a graph and a randomly generated subgroup of the symmetric group on $n$-elements, we show that this probability is at least  $1 - O\left(\frac{1}{n^{\gamma(\delta)}}\right)$ where $\gamma(\delta)>0$ for $\delta\ge 5$ and it is strictly increasing with $\delta$. We extend this to show that one may allow $\delta$ to grow slowly as a function of the degree of the lift and the number of vertices and still obtain that random lifts are a.a.s. $\delta$-connected. We also simplify a later result showing a lower bound on the edge expansion of random lifts. On a related note, we calculate the probability that a subgroup of a wreath product of symmetric groups generated by random generators is transitive, extending a well known result of Dixon which covers the case for subgroups of the symmetric group.

scholarly journals On the Ext-Quiver of Blocks of Defect 3 of Symmetric Group Algebras

1996 ◽  
Vol 185 (2) ◽  
pp. 440-480 ◽  
Author(s):  
Stuart Martin ◽  
Lee Russell
Keyword(s):  
Symmetric Group ◽  
Group Algebras ◽  
Symmetric Group Algebras
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