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

**Proof**

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. Thedegreeof 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 polynomialG-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

## Main Theorem

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.

**Proof**

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. ♦

## Summary

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!)