This insufferable phrase is copypasted and crossreferenced across every website and likely some research papers by beaten-down scared physicists. What does it mean? I think its not even technically right because SPINORS are also representations of groups that "correspond" to particles and they are not irreducible.
Can you explain what this mean in a pedagogical sense? It strikes me that this correspondence is dome kind of classification scheme where "particles have same number as the mathematical object so you can tag/index/classify them by the same way as the math object". If you can present an non-physical toy example with simpler symmetries like SO(2) in R3 or something along these lines, with the understanding that "real physics" has more dimensions and complexity. That's ok. You see, you can study linear algebra in 2 dimensions to get a taste of it, before generalizing ro N dimensions.