Courses

1. Groups (12-June) We begin by introducing the definition of a group, along with its axioms, properties, and structure. Next, we focus on the symmetric group and the general linear group, exploring their specific structures. Finally, we introduce the concepts of homomorphism and isomorphism, which delve into the abstract idealization of a group

2. Representations (13-June) We define what a representation is, introduce the representation space, and explore some important representations of Sn. We then focus on the task of reducing each representation as much as possible into its irreducible components, i.e., irreducible representations. We conclude the session by introducing Schur’s Lemma and Character theory, which play a major role in reducing representations and identifying irreducible representations.

3. Irreducible representations of Sn and GL(d) (18-June) We introduce Young diagrams to find meaningful labeling of the basis of irreducible representations of Sn and GL(d) and delve into the task of building such representations. For this, we introduce Schur-Weyl duality and exploit the knowledge of GL(d) representations to build Snrepresentations. We explore the connection between these groups through the Clebsch-Gordan coefficients and compare this method with the more conventional Young-Yamanouchi algorithm.

4. Applications and Insights from representation theory to quantum information theory (19-June) We explore some interesting observations from representation theory in quantum theory, such as the connection with the quantum marginal problem, the ‘simple’ proof of entropy inequalities, and the simplifications of Schur-Weyl duality in IID scenarios, which are very common in quantum information applications.