Melissa J. Banister

Harvey Mudd College Mathematics 2004

Thesis Proposal: Separating Sets for The Alternating and Dihedral Groups
Thesis Advisor: Prof. Michael Orrison
Second Reader: Prof. Weiqing Gu
Current Draft of My Thesis: Final Thesis
Presentation Days Slides Slides

Separating Sets for The Alternating and Dihedral Groups

We present the results of an investigation into the representation theory of the alternating and dihedral groups and explore how their irreducible representations can be distinguished with the use of class sums. We call a set of class sums a separating set if each pair of irreducible representations is assigned a different eigenvalue by at least one of the class sums.

The symmetric group has been examined in great detail. The four class sums corresponding to the 2-cycles, 3-cycles, 4-cycles, and 5-cycles are sufficient to distinguish all of the irreducible representations of the symmetric group of order 41. A natural question to ask is whether the irreducible representations of other finite groups can be distinguished with a small number of class sums relative to the order of the group.

For the dihedral group, there is a separating set of size 2. The nature of this separating set raises the question: When do the class sums corresponding to a set of generators of a group form a separating set?

For the alternating group, the answer is not as simple. Certain class sums must be used in any separating set, yet do not alone suffice. We present a conjecture on how to construct a separating set of minimal size, and have verified this conjecture computationally for the alternating group of order up to 25 with the use of a computer program.