I am trying to get a handle on tensors. Specifically, 'irreducible
tensors'. 'Spherical irreducible tensors' even.

My understanding is that a tensor is a function from V^p (V*)^q into a
field F, where V is a vector space, V* is the dual space (i.e. the space
of linear maps from V into F), and V^n = VxVxVx...xV, where 'x' is the
tensor product. The tensor product is like the cartesian product except
there are some additional equivalence relations on the resulting space,
such as (0,y) = (x,0) = 0. Assuming that (VxV)* = (V*xV*) then you can
say that a tensor is an element of the space (V^p (V*)^q)* which equals
(V*)^p V^q. How's that?

A tensor T is irreducible if it cannot be written as a sum of tensors of
the same rank. This confuses me very much. Given a rank-n tensor on a
d-dimensional space, you can write it out with respect to a basis as an
array of d^n numbers, much as you can write out a representation of a
linear operator as a d-by-d matrix. Every d-by-d matrix represents a
linear operator on a d-dimensional space; similarly, every array of d^n
numbers represents a tensor in a given basis. So what's to stop me from
decomposing this tensor indefinitely into sums -- I can certainly well
write a matrix as a sum of as many other matrices as I desire. There
must be some invariant property that the summed tensors must have in
order for a reduction to be valid -- but I am not sure what this
property is.

Is there such a thing as 'irreducible vectors' or 'irreducible matrices'?

I'm told that "A tensor is a representation of a linearly acting group:
GL(n), O(n), U(n), ... depending on what the basic vector space you're
starting with, and whether you have an inner product defined on it," but
I'm not sure what that means. I know what a group representation is: a
homomorphism from the group into GL(n). I suppose 'linearly acting'
refers to a group action.

The usual demonstration of reducibility of tensors starts by taking a
dyadic tensor T_ij = U_i V_j and showing that it can be written as a sum
of a scalar (U dot V delta_ij/3), an antisymmetric part, and a symmetric
traceless part. The demonstration is clear enough, but I am not certain
what is so special about being traceless or about being symmetric or
anti-symmetric. Sakurai then shows that these components have 1, 3, and
5 degrees of freedom, and then makes the leap: "This suggests that the
dyadic has been decomposed into tensors that can transform like
spherical harmonics with l=0, l=1, and 2." This 'suggestion' leaves me
really unconvinced and dissatisfied.

The demonstration that a 'spherical irreducible tensor' can be written
in terms of Wigner function is clear enough, but I don't see the
connection between 'spherical irreducible tensors' and general tensors.

Another big of confusion I have is in the two viewpoints of a tensor
both as 'data' and as an operator. The stress tensor is something I
think of as 'data', and even, say, angular momentum eigenstates seem
like 'data'. Whereas a matrix I usually consider an 'operator'. I
assume this is related to dual spaces -- i.e. a vector is usually
'data', but its dual is an operator from the vector space into a field.
Eh?

Is there such a thing as 'irreducible vectors' or 'irreducible matrices'?

I'm told that "A tensor is a representation of a linearly acting group:
GL(n), O(n), U(n), ... depending on what the basic vector space you're
starting with, and whether you have an inner product defined on it," but
I'm not sure what that means. I know what a group representation is: a
homomorphism from the group into GL(n). I suppose 'linearly acting'
refers to a group action.