Full Record

Author | Reese, Randall Dean |

Title | Topics Pertaining to the Group Matrix: k-Characters and Random Walks |

URL | https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=6569&context=etd |

Publication Date | 2015 |

Degree | MS |

Degree Level | masters |

University/Publisher | Brigham Young University |

Abstract | Linear characters of finite groups can be extended to take k operands. The basics of such a k-fold extension are detailed. We then examine a proposition by Johnson and Sehgal pertaining to these k-characters and disprove its converse. Probabilistic models can be applied to random walks on the Cayley groups of finite order. We examine random walks on dihedral groups which converge after a finite number of steps to the random walk induced by the uniform distribution. We present both sufficient and necessary conditions for such convergence and analyze aspects of algebraic geometry related to this subject. |

Subjects/Keywords | k-characters; group determinant; random walks; branched covering; Mathematics |

Language | en |

Rights | License: http://lib.byu.edu/about/copyright/ |

Country of Publication | us |

Format | application:pdf |

Record ID | oai:scholarsarchive.byu.edu:etd-6569 |

Repository | byu |

Date Retrieved | 2019-12-27 |

Date Indexed | 2019-12-30 |

Sample Search Hits | Sample Images

…element of GL(C).
Definition 1.6. Another important representation is the regular representation. The regular
representation is given by the natural action of G on the *group* algebra CG. Explicitly,
G = {g1 , g2 , . . . , gn } (…

…the proof of
Table 1.1: Regular Character for G
Ï€
g = id
0
id
n
which employs module theory and will not be discussed herein).
Theorem 1.9. For a finite *group* G, let Irr(G) = {Ï‡1 , Ï‡2 , . . . , Ï‡k }. The regular character
Ï€…

…characters of a *group* G is equal to the
number of conjugacy classes of G. That is, |Irr(G)| = k, where k is the number of conjugacy
classes of G.
3
Characters for arbitrary finite groups were first defined by Frobenius in his 1896 work
¨
Uber…

…Gruppencharaktere [12] (as cited in [3]) for use in factoring the *group* determinant.
Chapter 2 will further explicate this topic. Character values for different characters of G can
be summarized in a character table.
Definition 1.12…

…x5D; The character table of a *group* G is an invertible matrix.
Example. A *group* of relative importance in this thesis is the dihedral *group* of order 10,
denoted herein as D10 = r, s | r5 = s2 = id, sâˆ’1 rs = râˆ’1 . D10 has four conjugacy classes:
{…

…*group* determinant provided
a major impetus for the development of character theory. We begin this chapter with a
collection of definitions and theorems relevant to this pursuit.
2.1
The *Group* Matrix
Definition 2.1. [26] Let G = {g1 , g2…

…gn } be a finite *group* of order n. We define the
âˆ’1 , where the Î¾g
are indeterminates in the ring
*group* matrix as the n Ã— n matrix Î¾gi gj
k
C[Î¾g1 , Î¾g2 , . . . , Î¾gn ] corresponding to the gk âˆˆ G, and the rows and columns of the matrix…

…are
indexed by the elements of G. We denote the *group* matrix by XG .
Example. [27] Let G = S3 . We label the elements of S3 :
id = g1 ,
(123) = g2 ,
(132) = g3 ,
(12) = g4 ,
(13) = g5 ,
(23…