Table of Contents
Statement
For the simplest case of Gaussian Hermitian matrix model the average of Schur functions is some combo of Schur functions evaluated at some specific points.
This property generalizes to many more sophisticated cases as well in the symbolic form
Motivation and overview
To solve matrix model means to calculate all the correlators in this model. This is exactly what we will do in this note! For instance, in the Gaussian case the answer will turn out to be
where is a partition, sum is taken over partitions and are irreducible characters of symmetric group.
But from the point of view of symmetric functions this answer can be further simplified. The quantity, that is calculated in (3) is actually a correlator of power sum
But the Frobenius map then enables us to write
This form is clearly more consice and transparent and hence got its own name.
We can obtain (3) using different methods [MO22] by
- Wick's theorem
- using determinant formula for generating function
- using Itzykson-Zuber as a source in generating function
- solving Virasoro constraint explicilty
The last two are known to survive -deformation [MO22; Oko97]. Some of these methods are described below.
Wick's theorem
Wick's theorem is the most straightforward way to calculate all the correlators of Gaussian integrals. But this is also the only case when it's simpler to start not with Gaussian Hermitian matrix model, but rather with rectangular complex one (which is also just a product of regular Gaussian integrals, though less symmetric, what will actually simplify some formulae!). See [MM17] and references therein.
Rectangular complex matrix model
Elementary correlators in the rectangular complex matrix model are of form
where here is not only Young diagram, which characterizes the cycle type of permutation, but rather permutation itself, so the action is defined. and operations should be understood as operations on the permutation's cycle type. is a so-called power power sum.
Wick's theorem for these correlators gives
we have
where is an identity matrix. Now,
We can cook up orthogonality relation for the characters above from Schur orthogonality relations
Let's multiply them by . We get
Next, set , and sum over , . We know that , so
This is exactly the expression we see in (10). Substituting it we obtain
So we've actually found all the correlators
By the inverse Frobenius map we have
and term-by-term comparison of the last two expressions gives
Gaussian Hermitian matrix model
Wick's theorem is a bit more complicated in this case
So, all the correlators of this model are then
Permutations of cycle type are of order 2, i.e. , or , thus in the last product we can sum over with limits . So,
There are no permutations of order 2 with , therefore
and we have
As in complex model we have
and
Now, let's get our hands dirty and simplify the last sum. By definition
so
Class-sum
commutes with all elements of since for any we have
For representation it means the same
This is exactly the statement of Schur's lemma for , so
where is identity matrix and is just some number to be found. We proudly substitute this result in (26)
What remains to do is to find . From (30) we have
In the last line Frobenius characteristic map was used. Dirty work is done, now back to (24)
What a beautiful result! Inverse Frobenius characteristic map allows us to obtain
Itzykson-Zuber source
The starting point is known character expansion for the Itzykson-Zuber integral
This integral can be used as a source instead of
so the generating function becomes
Note, however, that is actually a symmetric function of , so it has the same number of degrees of freedom as our default generating function. The benefit of this kind of source is that can be evaluated independently. For this you need to complete the square
take Gaussian integral over and trivial integral over (in the calculation of Itzykson-Zuber integral, as well as everywhere else in this notes by default Haar measure is properly normalized)
The last equality is due to Cauchy identity. From the other hand via character expansion of the Itzykson-Zuber integral we have
Term-by-term comparison of (39) and (40) gives
Moreover, by inverse Frobenius characteristic map we have