Noether's Bound in the Invariant Theory of Finite Groups and Vector Invariants of Iterated Wreath Products of Symmetric Groups

2000 ◽  
Vol 50 (1) ◽  
pp. 93-105
Author(s):  
L. Smith
Keyword(s):  
Finite Groups ◽  
Invariant Theory ◽  
Symmetric Groups ◽  
Wreath Products

scholarly journals ON THE REGULARITY CONJECTURE FOR THE COHOMOLOGY OF FINITE GROUPS

2008 ◽  
Vol 51 (2) ◽  
pp. 273-284 ◽  
Author(s):  
David J. Benson
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Finite Groups ◽  
Cohomology Ring ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Characteristic P ◽  
Residue Classes ◽  
The Difference ◽  
A Finite Group

AbstractLet $K$ be a field of characteristic $p$ and let $G$ be a finite group of order divisible by $p$. The regularity conjecture states that the Castelnuovo–Mumford regularity of the cohomology ring $H^*(G,K)$ is always equal to 0. We prove that if the regularity conjecture holds for a finite group $H$, then it holds for the wreath product $H\wr\mathbb{Z}/p$. As a corollary, we prove the regularity conjecture for the symmetric groups $\varSigma_n$. The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

scholarly journals Dual Graded Graphs and Bratteli Diagrams of Towers of Groups

10.37236/7790 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Christian Gaetz
Keyword(s):  
Finite Groups ◽  
Symmetric Groups ◽  
Wreath Products ◽  
Critical Groups ◽  
Bratteli Diagrams ◽  
Graded Graph ◽  
Differential Poset ◽  
Dual Graded Graphs ◽  
Nested Sequence ◽  
Special Case

An $r$-dual tower of groups is a nested sequence of finite groups, like the symmetric groups, whose Bratteli diagram forms an $r$-dual graded graph.  Miller and Reiner introduced a special case of these towers in order to study the Smith forms of the up and down maps in a differential poset.  Agarwal and the author have also used these towers to compute critical groups of representations of groups appearing in the tower.  In this paper I prove that when $r=1$ or $r$ is prime, wreath products of a fixed group with the symmetric groups are the only $r$-dual tower of groups, and conjecture that this is the case for general values of $r$.  This implies that these wreath products are the only groups for which one can define an analog of the Robinson-Schensted bijection in terms of a growth rule in a dual graded graph.

Representation Theory and Harmonic Analysis of Wreath Products of Finite Groups

2009 ◽  
Author(s):  
Tullio Ceccherini-Silberstein ◽  
Fabio Scarabotti ◽  
Filippo Tolli
Keyword(s):  
Representation Theory ◽  
Harmonic Analysis ◽  
Finite Groups ◽  
Wreath Products

Invariant Theory of Finite Groups

1997 ◽  
pp. 311-348
Author(s):  
David Cox ◽  
John Little ◽  
Donal O’Shea
Keyword(s):  
Finite Groups ◽  
Invariant Theory

Invariant Theory of Finite Groups

1992 ◽  
pp. 306-344
Author(s):  
David Cox ◽  
John Little ◽  
Donal O’Shea
Keyword(s):  
Finite Groups ◽  
Invariant Theory
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