Exercise set 2.1

Throughout K denotes a field and V a vector space over K of finite dimension d ≥ 1.

(1)

(Difficulty level 1) Convince yourself of the following facts:

(a)

The notion of a polynomial function on V is independent of the choice of a basis for V .

(b)

Any invariant subspace and any quotient of a polynomial representation of GL(V ) is a polynomial representation.

(c)

Any invariant subspace and any quotient of a homogeneous polynomial representation of GL(V ) is a homogeneous polynomial representation of the same degree.

(d)

If U and W are polynomial representations of GL_K(V ), then so are U⊕W and U⊗W.

(e)

If U and W are homogeneous polynomial representations of GL_K(V ) of degrees p and q respectively, then U ⊗W is a homogeneous polynomial representation of GL(V ) of degree p + q.

(f)

gdet^eg for e an integer e ≥ 0 defines a homogeneous polynomial representation of GL_K(V ) of degree de.

(2)

(Difficulty level 2) Show the following: K is infinite if and only if there is a unique polynomial (with respect to any fixed basis) that represents any fixed polynomial function on V .

(3)

(Difficulty level 2) Suppose that K is finite of cardinality q. Determine the kernel of the surjective K-algebra morphism from K[X₁,…,X_d] to the ring K[V ] of polynomial functions on V . What is the dimension (as a K-vector space) of K[V ]?

(4)

(Difficulty level 2) Suppose that K is infinite. Show that the natural restriction map from polynomial functions on End_K(V ) to functions on GL_K(V ) is an injection. (We may thus identify polynomial functions on GL_K(V ) with those on End_KV .)

(5)

(Difficulty level 2) Write a compact expression (as a rational function in t) for the formal power series ∑ _n≥0tⁿ dim_KK[X₁,…,X_d]_n, where K[X₁,…,X_d] is the polynomial ring in d variables and K[X₁,…,X_d]_n is its subspace of polynomials of degree n.