The last time we talked about symmetric functions, I asked whether the vector space could be turned into an algebra, i.e. equipped with a nice product. It turns out that the induced representation allows us to construct such a product as follows:
Given representations of , their tensor product is a representation of the direct product in a natural way. Now, injects naturally into , which gives a new representation
The character of this representation is called the induction product of the characters of , and with this product becomes a graded commutative algebra. (Commutativity and associativity are fairly straightforward to verify.) It now remains to answer the first question: does there exist an algebra homomorphism ? And can we describe the inner product on coming from the inner product on ?
To answer this question we’ll introduce perhaps the most important class of symmetric functions, the Schur functions .
N.B. I’ll be skipping even more proofs than usual today, partly because they require the development of a lot of machinery I haven’t described and partly because I don’t understand them all that well. Again, good references are Sagan or EC2.
One way that the Schur functions naturally occur is in the description of Schur-Weyl duality. To the extent that I understand this, it goes as follows. On there are two commuting representations: one of which acts on all the factors simultaneously and one of which permutes the factors. An important property of these representations is that they are mutual centralizers, i.e. each is the largest group of transformations which commutes with the other. It then turns out that the above representation decomposes as
where runs through all partitions with boxes and at most rows, is the irreducible representation of associated to the partition , and is a corresponding irreducible representation of .
We already know of two of these representations, the symmetric algebra and the exterior algebra . It turns out that the symmetric algebra corresponds to the partition (one row) and the exterior algebra corresponds to the partition (one column), and it’s not hard to see that the former is always there and the latter is only there as long as . As we saw previously, the characters of these representations can be described in terms of eigenvalues. This turns out to be true more generally.
Proposition: Suppose is a representation which is homogeneous, i.e. it respects scalar multiplication, and polynomial, i.e. every entry of the output is a polynomial function of the entries of the input . Then the character of is a symmetric function of the eigenvalues of .
The restriction to homogeneous polynomial representations is to avoid discontinuity and powers of the representation . Anyway, the above result is not too hard to see: since the diagonalizable matrices are dense and characters are both continuous and invariant under conjugation, the character must depend only on the multiset of eigenvalues, and since the representation is homogeneous and polynomial, the character must be as well.
I am not familiar with the explicit description of the representations , but the important point is that these representations are well-defined with respect to Schur-Weyl duality.
Definition 1: The Schur polynomial is the value of the character of evaluated at the diagonal matrix with entries .
For example, (the symmetric algebra) and (the exterior algebra). Note that if has more than rows then is not defined.
Of course, since I haven’t told you how to write these functions down, this is a rather unsatisfying definition, but at least it has a concrete tie to an important representation-theoretic concept. Now, it turns out that , so the Schur functions are well-defined as symmetric functions.
Schur-Weyl duality turns out to imply a strong relationship between these characters and the characters of the symmetric group which generalizes the relationship the complete homogeneous symmetric functions and elementary symmetric functions have to the trivial and sign representations.
Definition 2: Let denote the character of the representation of corresponding to evaluated at a permutation . Then
The Weyl character formulas imply the following determinantal formula for . First, some notation. Define
In particular is the Vandermonde determinant , and for any other partition the Vandermonde determinant divides it. (One could imagine these determinants appearing in the discussion of polynomial interpolation when certain coefficients are restricted to be zero, but I have never heard anyone talk about Schur functions this way.)
Definition 3: .
This definition has the advantage that it does not refer directly to representation theory, and it is also relatively straightforward to do computations with for small cases.
There are two other determinantal formulas for the Schur functions which are useful in studying, for example, the cohomology ring of a Grassmannian, but they can wait.
The characteristic map
We now define the characteristic map as follows: if is a class function on , then define
and extend by linearity. As we saw above, , so this is a natural definition from that perspective.
Proposition: . In other words, defines a ring homomorphism from with the induction product to .
The main technical detail of this proof is Frobenius reciprocity, but the point is that we have now found the relationship between and that we were looking for. With a little more work one can show that is bijective, and it follows that the form a basis of the symmetric functions.
The characteristic map takes the usual inner product on to the unique inner product on symmetric functions satisfying
The study of this inner product was initiated by Philip Hall and clarifies a number of results in symmetric function theory; see EC2, for example. But I haven’t digested this point enough to say anything meaningful.