The most general form of one-matrix model correlator (modulo linearity) is
By –invariance of one can immediately notice that
is an invariant tensor
so it cannot be something other than a linear combination of products of –symbols
-s, in turn, should be some functions of matrix invariants integrated over our distibution, which are spanned more-or-less by
where is a Young diagram. This is the quick and dirty reason why we focus on calculating these quantities everywhere else. They are our geniune building blocks. If you aren't convinced, keep reading.
To get a precise answer, we need a bit more sophisticated arguments. By the same -invariance of we insert and write
Next expand each factor:
Hence
By the Weingarten formula (for factors)
one obtains
Finally the remaining matrix–integral depends only on the cycle–structure of : if has cycles , then
Putting everything together gives the general expansion