Explore the intriguing world of non-unitary physical systems in this 44-minute conference talk by David Ridout at the Centre de recherches mathématiques (CRM). Delve into the concept of representations that are reducible but indecomposable, a crucial aspect of modern mathematical physics. Learn about the basics of these complex representations, with references to statistical loop models and Conformal Field Theory (CFT). Gain insights into how these challenging concepts are essential for a comprehensive understanding of representation theory and its applications in physics. Discover the connections between representation theory and mathematical physics, focusing on examples that highlight the importance of reducible but indecomposable representations in non-unitary systems.