Symmetry representations
Let's start with the basics:

A representation is a set of matrices that fulfill the multiplication table for the point group.

So I might be able to use a set of 3x3 matrices, since of course I can define any of the symmetry operations over Cartesian (x,y,z) space. e.g.:

$$E = \left(
\begin{array}{ccc}
1 &amp; 0 &amp; 0 \\
0 &amp; 1 &amp; 0 \\
0 &amp; 0 &amp; 1 \end{array}
 \right) \;
i = 
\left(
\begin{array}{ccc}
-1 &amp; 0 &amp; 0 \\
0 &amp; -1 &amp; 0 \\
0 &amp; 0 &amp; -1 \end{array}
 \right) 
$$

But usually, I can fulfill the multiplication table with much simpler (smaller) matrices.


An irreducible representation is a set of matrices that are as small as possible.
A reducible representation is a set of matrices that can be decomposed into a linear combination of irreducible representations. This can occur by multiple techniques, including block-diagonalization.


The distinction exists, because for chemical point groups, there are usually a finite number of irreducible representations (given in character tables), but of course there are an infinite number of reducible representations.

The power of symmetry and group theory is that any function of a molecule, like orbitals, vibrations, etc. can be described using a representation. Thus, we usually label orbitals and vibrations by their corresponding irreducible representation.


I'll have to hunt down my copy of Ogden, but usually people use the Cartesian axes or $p_z$, $p_x$, and $p_y$ to illustrate the different irreducible representations. That is, a basis is some function (like the Cartesian axes or an atomic orbital) used to generate a representation. We look at how the basis is transformed under the different symmetry operations and write down the matrix or characters from those transformations.


(I try not to use that term because students find it confusing and different guides and texts use subtly different definitions for it.)


The reduction formula (or algorithm) is a way of decomposing the reducible representation (which is really a big matrix) into a combination of the different irreducible representations, but only needing to act on the characters of the representations.


That is, I don't need to actually use the reducible matrices themselves. Using the reduction formula, I can get the linear combination of irreducible representations only using multiplication of characters of the representations. It's much easier to do the reduction formula than to figure out how to transform a large reducible matrix.