But - again - that is a mathematical construction, which a physical particle is not.

I think I'm getting a glimmer of understanding on this now

From what I gather, a set of possible transformations is a group in group theory

Physical space is a type of group, i.e. a Poincare group, and is described by the set of all transformations on objects, or something (i.e. motion or lack thereof)

An irreducible poincare group is a tinest example of physical space, i.e. a 'particle'

So although it has no physical space, yet the irreducible Poincare group is intrinsically (but not practically) capable of those same types of transformations within itself as in the larger Poincare group within itself

E.g. a larger object (many particles) can undergo shears and strains, i.e. internal motion. In theory an infinitesimal particle can, it just doesn't have the space to undergo those

I'm inferring from this that a subset of physical space is also a Poincare group?

The Poincare group is the set of symmetries of spacetime (in special relativity), not spacetime itself. It characterizes the basic geometry of spacetime, so the dynamics of any physical system must be invariant under the action of this group, e.g. a physical system has the same dynamics as a rotated version of itself (as long as you rotate the whole physical system), and a physical system has the same dynamics as a version of itself with the velocities of all its components boosted uniformly in some direction (this is the principle of relativity). So the important things, like the dynamics, are invariant under the Poincare group, and the things that change are just dependent on the perspective of an observer.

Where the irreducible representations come in is in quantum mechanics, where physical systems are described by Hilbert spaces. These are spaces in the abstract mathematical sense of a vector space. They're used in quantum mechanics as a way to mathematically describe the fact that there are multiple possible outcomes of an experiment or interaction. The way this is done is by describing a quantum state as a unit vector in Hilbert space, each possible outcome corresponds to a dimension (a basis vector) in this space, and the probability of each outcome corresponds to the projection of the unit vector onto one of these dimensions. You can picture this as an arrow of unit length with its tail at the origin, and evolution of the state makes the arrow rotate about the origin, consequently changing its projection onto the basis vectors.

A representation of a group on a vector space is a way of describing the action of that group on the vectors in the space. Any vector space can be decomposed into subspaces which are invariant under the group action, meaning that if a vector starts off in that subspace, then the action of the group will not move it out of that subspace - these subspaces are the irreducible representations of the group.

So the irreducible representations of the Poincare group correspond to the components or properties of a physical system that are invariant under the basic symmetries of spacetime, i.e. that are independent of one's perspective, and in that sense they're considered basic or fundamental.

Yes, mathematical constructions are merely a model of reality

It is interesting to note that (correct me if I'm wrong) perhaps our mathematical constructs are based on a classical intuition and perception of the universe?

And here we are trying to fit that classical intuition into the quantum realm

Perhaps that can be related to why shit gets complicated with particle physics?

That's certainly part of it. But part of it is we're not really terribly good at talking about the universe even with classical intuition.

That said, model-theoretic semantics go back to, at least, William of Ockham, who developed a formalized subset of Latin (which kind of reminds me of the Google C++ style guide in a way). We're clearly not utterly incompetent in this space, but it helps to stand on the shoulders of the giants to see clearly.

> classical intuition

I am not sure if the modern mix of group theory and differential geometry which forms the foundation of theoretical physics today can qualify as such, even though the word "geometry" does bring the expectation that we should be able to "see" the things we are talking about.