K-Representations and G-Representations
As mentioned at the end of the previous article, we shall attempt to construct analytic representations of from continuous representations of
Let . Consider , where is the group of diagonal matrices in K so
as a topological group. From our study of representations of the n-torus, we know that is a direct sum of 1-dimensional irreps of the form:
where . E.g. if then the representation is trivial; if the representation is obtained by taking the determinant.
Hence, the character of is a Laurent polynomial with non-negative integer coefficients, i.e.
Definition. For a continuous finite-dimensional representation , we will write for expressed as a Laurent polynomial in .
The same holds for a complex analytic representation of G.
- If is the identity map, its Laurent polynomial is
- For , its Laurent polynomial is
The following are clear for any continuous representations V, W of K.
In summary, so far we have the following:
Main Examples: Sym and Alt
For any vector space V, the group acts on by permuting the components. We denote:
where is the sign of w. Let be a fixed basis of V. The case d=2 is quite easy to describe for the above spaces, for we can just take the following bases:
Denote these two types of elements by and respectively, so that and Note that ; this is not true for higher values of d.
Similarly, in general, we can pick the following as bases:
where the components commute in Sym and anticommute in Alt (e.g. ).
Now suppose and acts on it; G and act on in the following manner:
These two actions commute, from which one easily shows that and are G-invariant subspaces of .
Suppose we have acting on . This takes:
To compute the Laurent polynomials of these spaces, we let the diagonal matrix act on them, giving:
Hence we have:
Lemma 1. For a representation V of K, the Laurent polynomial is symmetric.
For any , take the corresponding permutation matrix ; we have:
Thus for any ♦
Lemma 2. Given K-representations V, W, if , then
Hence by the previous article, the same holds for analytic G-representations V, W.
Any , being unitary, is diagonalizable by a unitary matrix, i.e. there exists such that . Hence the given condition implies:
By character theory of compact groups, as K-reps. ♦
Now for the final piece of the puzzle.
Lemma 3. Let be a symmetric polynomial. There are polynomial representations:
Taking homogeneous parts, let us assume for some degree Write f as a linear sum of elementary symmetric polynomials with integer coefficients; separating terms we have , where g, h are both linear combinations of with non-negative integer coefficients. Hence, it suffices to show that for some polynomial representation
Since , we can just pick:
from the above. ♦
Immediately we have:
Corollary 1. For any symmetric Laurent polynomial , there exist rational representations:
Indeed, for some symmetric polynomial g in Applying lemma 3, we can find such that ; now we take and ♦
Finally we have:
Corollary 2. Any irreducible K-module V can be lifted to a rational irreducible G-module.
By corollary 1, where W, W’ are rational G-modules. Then so as K-modules by lemma 2 above. By corollary 3 here, is a rational irreducible G-module. ♦
Summary of Results
Thus we obtain:
Note the following.
- When we consider virtual representations (recall that these are formal differences of two representations), this corresponds to all symmetric Laurent polynomials with integer coefficients.
- Any rational G-representation is of the form where is a polynomial G-representation.
Our next task is to identify the irreducible rational G-modules V. Tensoring by some power of det, we may assume V is polynomial, so is a symmetric polynomial in Studying these will lead us back to Schur polynomials.