-
Recent Posts
Archives
- March 2023
- January 2023
- May 2020
- April 2020
- March 2020
- June 2018
- July 2016
- June 2016
- May 2016
- March 2015
- February 2015
- January 2015
- December 2014
- December 2013
- November 2013
- July 2013
- June 2013
- May 2013
- March 2013
- February 2013
- January 2013
- December 2012
- November 2012
- October 2012
- September 2012
- August 2012
- April 2012
- March 2012
- February 2012
- January 2012
- December 2011
- November 2011
- October 2011
Categories
Meta
Pages
Tag Archives: group actions
Solving Permutation-Based Puzzles
Introduction In the previous article, we described the Schreier-Sims algorithm. Given a small subset which generates the permutation group G, the algorithm constructs a sequence such that for: we have a small generating set for each Specifically, via the Sims … Continue reading
Posted in Uncategorized
Tagged group actions, group theory, permutations, rubik's cube, schreier-sims, symmetric group
2 Comments
Schreier-Sims Algorithm
Introduction Throughout this article, we let G be a subgroup of generated by a subset We wish to consider the following questions. Given A, how do we compute the order of G? How do we determine if an element lies in G? Assuming , how … Continue reading
Posted in Uncategorized
Tagged group actions, group theory, permutations, programming, rubik's cube, schreier-sims, symmetries
Leave a comment
Polynomials and Representations XXIV
Specht Modules Till now, our description of the irreps of are rather abstract. It would be helpful to have a more concrete construction of these representations – one way is via Specht modules. First write Thus if , the only common irrep between … Continue reading
Posted in Uncategorized
Tagged group actions, representation theory, symmetric group, young symmetrizer, young tableaux
Leave a comment
Polynomials and Representations XIX
Representations of the Symmetric Group Let [d] be the set {1,…,d}, and Sd be the group of bijections From here on, we shall look at the representations of Note that this requires a good understanding of representation theory (character theory) of finite groups. To start, let … Continue reading
Posted in Uncategorized
Tagged character theory, group actions, representation theory, symmetric group
Leave a comment
The Group Algebra (I)
[ Note: the contents of this article overlap with a previous series on character theory. ] Let K be a field and G a finite group. The group algebra K[G] is defined to be a vector space over K with basis , where “g” here is … Continue reading
Posted in Notes
Tagged character theory, group actions, group algebras, modules, representation theory, semisimple rings, simple modules
Leave a comment
Burnside’s Lemma and Polya Enumeration Theorem (1)
[ Note: this article assumes you know some rudimentary theory of group actions. ] Let’s consider the following combinatorial problem. Problem. ABC is a given equilateral triangle. We wish to colour each of the three vertices A, B and C by … Continue reading
Intermediate Group Theory (3)
Automorphisms and Conjugations of G We’ve seen how groups can act on sets via bijections. If the underlying set were endowed with a group structure, we can restrict our attention to bijections which preserve the group operation. Definition. An automorphism of … Continue reading
Posted in Notes
Tagged advanced, automorphisms, conjugate, group actions, group theory, semidirect products
Leave a comment
Intermediate Group Theory (2)
This is a continuation from the previous post. Let G act on set X, but now we assume that both G and X are finite. Since X is a disjoint union of transitive G-sets, and each transitive G-set is isomorphic to G/H for some subgroup H ≤ G, it follows that … Continue reading
Posted in Notes
Tagged advanced, cauchy's theorem, group actions, group theory, normaliser, sylow theorems
Leave a comment
Mathematics and Such 