Polynomial Representations

We assume that K is an infinite field, so that there is no difference between formal polynomials and the functions that they give rise to.

Definition 6.1.1 (Polynomial representation). A polynomial representation of GLm(K) is a representation (ρ, V) of GLm(K), where V is a finite-dimensional vector space over K such that for each v ∈ V and ξ ∈ V′, the function

g ↦ 〈ξ, ρ(g)v〉

is a polynomial function in the entries of the matrix g. If, for all ν ∈ V and ξ ∈ V′, this polynomial is homogeneous of degree n, then (ρ, V) is said to be a homogeneous polynomial representation of degree n.

Here, as usual, V′ denotes the space of K-linear maps V → K.

[1] Exercise 6.1.2. A representation (ρ, V) is a polynomial representation if and only if, for any basis e1, …, em of V, taking ξ1, …, ξm to be the dual basis of V′, the function

g ↦ 〈ξi, ρ(g)ej〉

is a polynomial function of the entries of g for all i, j ∈ {1, …, m}. Furthermore, ρ(g) is homogeneous of degree d if and only if the polynomial in (6.1) is homogeneous of degree d for all i, j ∈ {1, …, m}.

Example 6.1.3 (The defining representation of GLm(K)). Let V = Km. View the elements of V as column vectors and let g ∈ GLm(K) act on ν ∈ V by ρ1(g)ν = gν (matrix multiplication). Taking ei to be the basis of Kn given by the coordinate vectors,

〈ξi, ρ1(g)ej〉 = gij,

where gij is the (i, j)th entry of the matrix g ∈ GLm(K). Therefore, (ρ1, Kn) is a homogeneous polynomial representation of GLm(K) of degree 1. It is often called the defining representation of GLm(K).

Example 6.1.4 (The determinant). The determinant function GLm (K) → K* is a multiplicative character which is also a polynomial of degree m in the entries.