Exercise set 2.2

Throughout, K denotes a field and V a vector space over K of finite dimension d ≥ 1. We fix an integer m ≥ 1 and denote by G the group GL_K(m). We denote by AK(m) the ring of polynomial functions on G and by SK(m) and SK(m,n) the appropriate Schur algebras (as defined in the lecture).

(1)

(Difficulty level 1) Let U and W be finite dimensional K-vector spaces. Convince yourself that the following are equivalent for a (set) map φ : U → W:

(a)

There exists a basis {w_i} of W such that the K-valued functions φ_i on U defined by φ(u) = ∑ _iφ_i(u)w_i are all polynomial.

(b)

For any basis {w_i} of W, the K-valued functions φ_i on U defined by φ(u) = ∑ _iφ_i(u)w_i are all polynomial.

(c)

For any linear functional ζ on W, the K-valued function ζ ∘ φ on U is polynomial.

(d)

For any polynomial function f on W, the K-valued function f ∘ φ on U is polynomial.

If any of these holds, then φ is said to be a polynomial map from U to W.

(2)

(Difficulty level 2) Let ρ : G → GL(V ) be a polynomial representation and ρ′ : G → GL(V ^*) the contragredient representation of ρ. Observe that gρ′(g)^-1 is a polynomial map from G to End_KV ^* (although not a group homomorphism, but only an anti-homomorphism).

(3)

(Difficulty level 1) Verify that S_K(m,1) is isomorphic to the matrix algebra M_m(K).

(4)

(Difficulty level 2) For α in SK(m), let α_n denote its projection to SK(m). (Recall that this projection is induced from the inclusion of homogeneous polynomials of degree n in the ring AK(m).) We think of αα_n as a map from SK(m) to SK(m), considering α_n to be an element of SK(m) under the inclusion of SK(m,n) ⊆ SK(m). (Recall that this inclusion is induced by the projection of AK(m) onto its homogeneous component of degree n.) Show that δ_I,n (where I stands for the identity element of G, the m×m identity matrix), as n varies over the non-negative integers, are pairwise orthogonal central idempotents. (Caution: they are not primitive central idempotents, except in very special cases, as we will see.)

(5)

(Difficulty level 3) We now outline a proof of the fact that a (finite dimensional) polynomial SK(m)-module arises from a polynomial representation of G. More precisely, we show that any polynomial SK(m)-module : SK(m) → End_K(V ) that is homogeneous of degree n arises from a homogeneous polynomial representation ρ : G → GL(V ) of G of degree n. The candidate for ρ is clear: ρ(x) := (δ_x) for x in G. It is also clear that ρ is a representation since δ_xy = δ_x ⋆ δ_y). It remains only to prove that ⟨ξ,ρ(g)v⟩ is a polynomial function of g, as g varies over G (for any fixed v in V and ξ in V ^*).

Since SK(m,n) is the dual of the finite dimensional K-vector space AK(m,n), it is clear that there exists a homogeneous polynomial c_ξ,v in AK(m,n) such that

Since v = (δ_I,n)v, it follows that

On the other hand, since c_ξ,v is homogeneous of degree n, we have

From the equations in the last three displays, we conclude that

Putting α = δ_g, we obtain ⟨ξ,ρ(g)v⟩ = c_ξ,v(g). □