Teoria de representações: grupo simétrico e grupo unitário

Abstract

In this monograph we briefly study representations of groups in their general form and representations of the symmetrical group and the unitary group, due to their great importance for theoretical physics. In these studies we explore the Schur-Weyl duality that arises from the commutation between these two groups to study unitary random matrices, more specifically the average product of unitary matrix elements with respect to the haar measure, this study is called the Weingarten calculation. Finally, we use the results of the random unitary matrix studies to study the quantity distribution involving the switch in the unitary group, which is given by: C = U V U †V†.