Now that quantum information theory (QIT) reaches the point that group theory have deeply combined with its applications and theoretical understandings, such as random benchmarking, quantum scrambling, and measurement induced phase transition (MITP). The celebrated Schur-Wely duality and tools like Pauli twirling have found ubiquitous use in the area. So, I am wondering are there some references recommended for systematically learn the group theory related to the topics mentioned above? Especially the Schur-Wely duality because I am interested in the MITP, which closely related to this topics. p.s. It is feverable that the math is not too heavy, and QIT-oriented.
Short answer: there are no shortcuts. Things like Schur–Weyl duality are from representation theory. This generally assumes you understand group theory to begin with. Most physics-oriented texts I've seen which try to introduce group theory go through it so fast it's unclear if the reader can actually get anything out of it before moving on. That makes it very difficult by the time they get to the "higher" subjects like representation theory. I would absolutely recommend actually taking the time to read the first several chapters of Fraleigh (easiest), Herstein's Topics in Algebra (more theoretical), or Dummit & Foote (upper undergraduate/beginning graduate level textbook). I'd claim the last one is required just to read the wikipedia article on Schur–Weyl duality, although there are plenty of better explanations of it once you get a firm grasp on groups or modules. From there you can try representation theory.