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 ...

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.

We prove that every group G of order n has t,log n/log 2+0dog log n) elements x, x, such that every group element is a product of the form x,'-• X,'% e ; s 1,0, 11. The result is true more generally for quasigroups. As a corollary we obtain that for n even, every one-factorization of the complete graph on n vertices contains at most r one-factors whose union is connected. The aim of this note i...

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...

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.

This report briefly summarizes the discussions carried out in the working group on classification, representation and modeling of uncertain data. The discussion was divided into two subgroups: the first subgroup studied how different representation and modeling alternatives currently proposed can fit in a bigger picture of theory and technology interaction, while the second subgroup focused on ...

si def = jfj; xj = igj. Let s be a composition, that is to say a q-tuple which is the composition of some x in X; we set Xs = fx 2 X; s(x) = sg. We de ne and study a notion of designs for subsets of F q . In the case q = 2, our de nition coincides with the usual notion of designs and in fact our de nition is equivalent to the notion of strong colored designs of A. Bonnecaze, P. Sol e, P. Udaya ...