We are back to the convention and We wish to focus on irreducible polynomial representations of G.
The weak Peter-Weyl theorem gives:
Theorem. Restricting the RHS to only polynomial irreducible V gives us on the LHS, where each polynomial in restricts to a function
Since is a matrix coefficient, we have and this is clearly a -submodule. Furthermore, as functions , the are algebraically independent over by the main lemma here.
Since each is irreducible as a -module, corresponds to a direct sum over some set S of G-irreps V. It remains to show that S is precisely the set of polynomial irreps.
Next, as -representations, we have a decomposition
into homogeneous components; each is a finite-dimensional representations of By considering the action of , each component is a polynomial representation of G. Hence every irrep in S must be polynomial.
Conversely, if W is any polynomial irrep of G of dimension m, upon taking a basis the action of every can be written as an matrix with entries in Hence contains W; since W is irreducible, contains W. ♦
Decomposing By Degrees
Thus we have as isomorphism of -reps (*):
Definition. The degree of a polynomial irrep V is the unique for which contains it.
Let us pick one particular component We get a G-rep by taking the action of Hence its Laurent polynomial can be computed by considering the action of on it. Since the space is spanned by monomials of degree d, we have:
Property 1. For a polynomial G-irrep V, we have
Recall that if G acts on V, then acts on via:
Hence, taking the dual gives:
Property 2. The action of on gives the character
The pair of diagonal matrices takes . Hence, taking the basis of monomials of degree d, we have:
Property 3. The action of on has character:
Finally, from lemma 3 here and property 1 above, we see that:
Property 4. For each d, the number of polynomial G-irreps of degree d is exactly the cardinality of
Now we are ready to prove:
Theorem. A polynomial representation V of G of degree d is irreducible if and only if is a Schur polynomial in of of degree d.
For each d, let be the polynomial irreps of degree d; let a homogeneous symmetric polynomial in of degree d. By property 2, the character of on is:
By property 3 and (*), summing this over all d and j gives the power series:
Finally by property 4, for each d, the number of is exactly the size of Thus by the criterion for orthonormal basis proven (much) earlier, the forms an orthonormal basis of Hence, each is, up to sign, a Schur polynomial of degree d. Since the coefficients of are non-negative, they are the Schur polynomials. ♦
We have a correspondence between:
which takes Under this correspondence, for partition ,
Indeed, the first two correspondences are obvious; the third is what we just proved. The fourth is immediate from the definition of . The final correspondence follows from the third one. Denote for the corresponding irrep of ; we can now port over everything we know about symmetric polynomials, such as:
Setting for gives:
where is the number of SYT of shape
Unfortunately, the involution map does not have a nice interpretation in our context. (No it does not take the polynomial irrep to its dual!)