Representation Theory and Integrable Probability Fundamentals

Modern probability models often reveal deep algebraic patterns beneath their apparent randomness. This text-based course introduces how representation theory provides exact analytical tools to study stochastic systems, random partitions, and interacting particle processes. You will build a solid mathematical bridge between abstract algebra and probabilistic models, learning how symmetric functions and group characters translate into solvable probability measures. What you'll learn: - Understand core concepts of representation theory, including symmetric groups and unitary group representations - Work with Schur and Macdonald symmetric polynomials as algebraic tools for measure generation - Analyze discrete Markov chains and particle systems through Schur generating functions - Explore infinite-dimensional group limits and asymptotic distributions in random matrices - Apply character theory to compute explicit probability distributions and correlation kernels - Connect classical integrable models with modern stochastic systems and asymptotic analysis The course starts with essential terminology and fundamental definitions before progressing through algebraic machinery and concrete probabilistic applications through detailed written explanations and worked mathematical steps. This guide is designed for learners with an interest in mathematical probability and algebra seeking a clear conceptual entry point without advanced prerequisites. Start reading today to uncover the algebraic foundations governing modern integrable probability.