Group Theory and Representation Theory

Group representation theory and quantum physics∗

This is a basic tutorial on the use of group representation theory in quantum physics, in particular for such systems as molecules and crystals, which forms the basis of spectroscopic studies in physics and chemistry. Note that, despite the now venerable character of the aforementioned scientific endeavors, group representation theory still finds interesting and novel applications, such as the ...

Voting, the Symmetric Group, and Representation Theory

We show how voting may be viewed naturally from an algebraic perspective by viewing voting profiles as elements of certain well-studied QSn-modules. By using only a handful of simple combinatorial objects (e.g., tabloids) and some basic ideas from representation theory (e.g., Schur's Lemma), this allows us to recast and extend some well-known results in the field of voting theory.

Černý’s conjecture and group representation theory

Let us say that a Cayley graph of a group G of order n is a Černý Cayley graph if every synchronizing automaton containing as a subgraph with the same vertex set admits a synchronizing word of length at most (n− 1)2. In this paper we use the representation theory of groups over the rational numbers to obtain a number of new infinite families of Černý Cayley graphs.

Representation Theory of the Symmetric Group

Remark 1.2. The definition of an FG-module is more technical than the definition of a representation of G, but, as the exercise shows, the two notions are equivalent. Module can be more convenient to work with, because there is less notation, and we can use results from ring theory without any translation. The language of representations is preferable if we want to have an explicit map ρ : G→ G...

A Course in Finite Group Representation Theory