Jet bundles on projective space II

Let G=SL(E) be the special linear algebraic group on E where E is a finite dimensional vector space over a field K of characteristic zero. In this paper we study the canonical filtration of the dual G-module of global sections of a G-linearized invertible sheaf L on the grassmannian G/P where P in G is the parabolic subgroup stabilizing a subspace W in E. We classify the canonical filtration as P-module and as a consequence we recover known formulas on the P-module structure of the jet bundle J(L) on projective space. We study the incidence complex for the invertible sheaf O(d) on the projective line and prove it gives a resolution of the incidence scheme I(O(d)) of O(d). The aim of this study is to apply it to the study of resolutions of ideal sheaves of discriminants of invertible sheaves on grassmannians and flag varieties. We also give an elementary proof of the Cauchy formula. Hence the paper introduce the canonical filtration of an arbitrary irreducible SL(E)-module and initiates a study of the canonical filtration as P-module where P in SL(E) is a parabolic subgroup.


Introduction
In a series of papers (see [14], [15], [16] and [17]) the structure of the jet bundle as P -module has been studied using different techniques. In this paper we continue this study using techniques from algebraic groups, sheaf theory, generalized Verma modules, canonical filtrations of irreducible SL(V )-modules and annihilator ideals of highest weight vectors and study the canonical filtration U l (g)L d of the SL(V )module H 0 (X, O X (d)) * where X = G(m, m + n) is the grassmannian of m-planes in an m + n-dimensional vector space . Using results obtained in [14] we classify U l (g)L d and as a corollary we recover a well known result on the structure of the jet bundle P l (O(d)) on P(V * ) as P -module. As a consequence we get formulas on the splitting type of the jet bundle on projective space as abstract locally free sheaf. We also classify the P -module of the first order jet bundle P 1 X (O X (d)) on any grassmannian X = G(m, m + n) (see Corollary 3.10).
In the first section of the paper we study the jet bundle P l G/H (E) of any locally free G-linearized sheaf E on any quotient G/H. Here G is an affine algebraic group of finite type over an algebraically closed field K of characteristic zero and H ⊆ G is a closed subgroup. There is an equivalence of categories between the category of finite dimensional H-modules and the category of finite rank locally free O G/Hmodules with a G-linearization. The main result of this section is Theorem 2.3 where we give a classification of the L-modules structure of the fiber P l G/H (E)(x) * where L ⊆ H is a Levi subgroup. Here x ∈ G/H is the distinguished K-rational point defined by the identity e ∈ G. We also study the structure of P l X (O X (d))(x) * as L-module where X = G(m, m + n) is the grassmannian of m-planes in an m + ndimensional vector space (see Corollary 2.5 and 2.8).
In the second section we study the canonical filtration U l (g)L d for the irreducible SL(V )-module H 0 (G, O G (d)) * . Here G = G(m, m + n). We prove in Theorem 3.5 there is an isomorphism of P -modules when G = G(1, n + 1) = P n is projective n-space. As a result we recover in Corollary 3.6 the structure of the fiber P l G (O G (d))(x) * as P -module. This result was proved in another paper (see [11]) using different techniques. We also recover in Corollary 3.8 a known formula on the structure of the jet bundle on projective space as abstract locally free sheaf (see [15], [17], [19], [20], [21] and [22]).
In the third section we study the incidence complex In Appendix A and B we study SL(V )-modules, automorphisms of SL(V )modules and give an elementary proof of the Cauchy formula.
The study of the jet bundle P l X (O X (d)) of a line bundle O X (d) on the grassmannian X = G(m, m + n) is motivated partly by its relationship with the discriminant D l (O X (d)) of the line bundle O X (d). There is by [12] for all 1 ≤ l < d an exact sequence of locally free O X -modules (s, x) with the property that T l (x)(s) = 0 in P l X (O X (d))(x). The scheme P(Q * ) is the incidence scheme of the l'th Taylor morphism ). The map π is a surjective generically finite morphism between irreducible schemes. There is by [12] a Koszul complex of locally free sheaves on Y = P(W * ) × X which is a resolution of the ideal sheaf of P(Q * ) when it is locally generated by a regular sequence. The complex 1.0.1 might give information on a resolution of the ideal sheaf of D l (O X (d)). A resolution of the ideal sheaf of D l (O X (d)) will give information on its syzygies. By [12] the first discriminant D 1 (O P (d)) on the projective line P = P 1 is the classical discriminant of degree d polynomials, hence it is a determinantal scheme. By the results of [9] we get an approach to the study of the syzygies of D 1 (O P (d)). Hence we get two approaches to the study of syzygies of discriminants of line bundles on projective space and grassmannians: One using Taylor maps, incidence schemes, jet bundles and generalized Verma modules. Another one using determinantal schemes.

Jetbundles on quotients
In this section we study the jet bundle of any finite rank G-linearized locally free sheaf E on the grassmannian G/P = G(m, m + n) as L-module, where L ⊆ P is a maximal linearly reductive subgroup.
Let K be an algebraically closed field of characteristic zero and let V be a Kvector space of dimension n. Let H ⊆ G ⊆ GL(V ) be closed subgroups. The following holds: There is a quotient morphism and G/H is a smooth quasi projective scheme of finite type over K. Moreover For a proof see [7]. Let X = G/H and let mod G (O G/H ) be the category of locally free O G/H -modules with a G-linearization. Let mod(H) be the category of finite dimensional H-modules. It follows from [7] there is an exact equivalence of categories Proof. By [17] sequence 2.2.1 is an exact sequence of locally free O G/H -modules. The scheme Y is equipped with the diagonal G-action. It follows p * and q * preserve G-linearizations. We get a diagram of exact sequences of O Y -modules with a Glinearization Since p * preserves G-linearization we get a morphism φ : P l G/H (E) → P l−1 G/H (E) preserving the G-linearization, and the Proposition is proved.
Let g = Lie(G) and h = Lie(H). Let L ⊆ H be a Levi subgroup of H. It follows L is a maximal linearly reductive subgroup of H. The group L is not unique but all such groups are conjugate under automorphisms of H. Let x ∈ G/H be the K-rational point defined by the identity e ∈ G. Theorem 2.3. There is for all l ≥ 1 an isomorphism Proof. Dualize the sequence 2.2.1 and take the fiber at x to get the exact sequence 0 → P l−1 X (E)(x) * → P l X (E)(x) * → E(x) * ⊗ Sym l (g/h) → 0 of H-modules (and L-modules). This sequence splits since L is linearly reductive and the Theorem follows by induction on l.
Hence the study P l X (E)(x) * as L-module is reduced to the study of E(x) * and Sym l (g/h).
Let W ⊆ V be K-vector spaces of dimension m and m+n and let G = SL(V ) and P ⊆ G the subgroup fixing W . It follows G/P = G(m, m + n) is the grassmannian of m-planes in V . Let g = Lie(G) and p = Lie(P ). Fix a basis e 1 , .., e m for W and e 1 , .., e m , e m+1 , .., e m+n for V . It follows the K-rational points of P are matrices M on the form where det(A)det(B) = 1, A an m × m-matrix and B an n × n-matrix. Let L ⊆ P be the subgroup defined as follows: The K-rational points of L are matrices M on the form where det(A)det(B) = 1 and similarly A an m×m-matrix and B an n×n-matrix. It follows L is a Levi subgroup of P , hence it is a maximal linearly reductive subgroup.
Proposition 2.4. There is a canonical isomorphism Proof. By definition g = sl(V ), hence φ ∈ g is a map with tr(φ) = 0. Let i : W → V be the inclusion map and p : V → V /W the projection map. Define the following map: It follows j(p) = 0 hence we get a well defined map One checks g/p and Hom(W, V /W ) are P -modules and j a morphism of P -modules. It is an isomorphism and the Proposition follows.
Corollary 2.5. On X = G(m, m + n) there is an isomorphism Proof. The proof follows from Theorem 2.3 and Proposition 2.4.
There is an isomorphism of P -modules hence the decomposition into irreducible components of the module Sym i (W * ⊗ V /W ) as L-module may be done using the Cauchy formula (see Appendix B).
Let λ − |i| denote λ is a partition of the integer i If λ = {λ 1 , .., λ d } is a partition of an integer l, let µ(λ) denote the following partition: Let for any partition λ of an integer l and any vector space W , S λ (W ) denote the Schur-Weyl module of λ.
Proof. By Corollary 2.5 there is an isomorphism the result follows from the Cauchy formula (see Appendix B or [5]).
In the following we use the notation introduced in [7]. Let P semi = SL(m) × SL(n) ⊆ P be the semi simplification of P . We get a vector bundle π : G/P semi → G/P = G(m, m + n).
Let X = G/P and Y = G/P semi Given any finite dimensional P -module W , let L X (W ) denote its corresponding O X -module. Let W semi denote the restriction of W to P semi . By the results of [7] it follows there is an isomorphism of locally free sheaves. This will help calculating the higher cohomology group H i (X, L X (W )) since P semi is semi simple and π is a locally trivial fibration. If W is the P -module corresponding to the dual of the j'th exterior power of the jet bundle ∧ j P l X (O X (d)) * we can use this construction to calculate the cohomology group Such a calculation will by the results of [12], Example 5.12 give information on resolutions of the ideal sheaf of D l (O X (d)) since the push down of the Koszul complex 1.0.1 is the locally trivial sheaf To describe the locally trivial sheaf O(−j) ⊗ H i (X, ∧ j P l X (O X (d)) * ) for all i, j we need to calculate the dimension h i (X, ∧ j P l X (O X (d)) * ) and this calculation may be done using the approach indicated above.
Proof. This follows from Corollary 2.5 and Proposition 5.1.

On canonical filtrations and jet bundles on projective space
In this section we study the canonical filtration for the dual of the SL(V )-module of global sections of an invertible sheaf on the grassmannian. We classify the canonical filtration on projective space and as a result recover known formulas on the splitting type of the jet bundle as abstract locally free sheaf.
Let W ⊆ V be vector spaces over K of dimension m and m + n. Let W have basis e 1 , .., e m and V have basis e 1 , .., e m+n . Let V * have basis x 1 , .., x m+n . Let G = SL(V ) and P ⊆ G the parabolic subgroup of elements fixing W . It follows there is a quotient morphism Let g = Lie(G) and p = Lie(P ). Let U(g) be the universal enveloping algebra og g and let U l (g) be the l'th term ot its canonical filtration.
By the Corollary 3.11 in [13] there is for all 1 ≤ l ≤ d an exact sequence of P -modules Since the grassmannian is projectively normal in the Plucker embedding we get an inclusion It is the ideal generated by elements x ∈ U(g) with the property x(L d ) = 0. Let ann l (L d ) be its canonical filtration. We get an exact sequence of G-modules and an exact sequence of P -modules The following holds: Proof. The proof is obvious.
The Lie algebra p is the sub Lie algebra of g = sl(V ) given by matrices M of the following type: where A is an m × m-matrix, B and n × n-matrix and tr(A) + tr(B) = 0. Let p L be the sub Lie algebra of p consisting of matrices M ∈ p of the following type: The sub Lie algebra p L ⊆ p is a sub P -module of p.
There is an exact sequence of P -modules and p/p L is the trivial P -module.
The following holds: There is a filtration of P -modules Assume dim K (W ) = 1 and let W = L. There is an exact sequence of P -modules giving an isomorphism of P -modules g/p L ⊗ L ∼ = V .
Proof. We prove 3.3.1: In the following A, a are square matrices of size m and b, B square matrices of size n. The K-rational points of the group P are matrices g on the form where det(A)det(B) = 1. Assume x ∈ p is the following element: and tr(aAa −1 ) = tr(aa −1 A) = tr(A) = 0 hence g(x) ∈ p L and 3.3.1 is proved.
We prove 3.3.2: By 3.3.1 it follows p L ⊆ p is a sub P -module. One checks p/p L is a trivial P -module. We clearly get an exact sequence of P -modules and 3.3.2 is proved.
We prove 3.3.3: Since We prove 3.3.4: Since p/p L is a trivial P -module there are isomorphisms of P -modules The injection j gives rise to an injection gives rise to a filtration of P -modules There is an isomorphism and claim 3.3.4 is proved. We prove 3.3.5: Let V = K{e 0 , .., e n } and L = W = e 0 . It follows P ⊆ G = SL(V ) is the group whose K-rational points are the following: . Also B is an n × n-matrix with coefficients in K. By definition the maps in the sequence are maps of P -modules. It follows p = Lie(P ) is the Lie algebra whose elements x are matrices on the following form: where B is any n × n-matrix with coefficients in K. The sub Lie algebra p L ⊆ p is the Lie algebra of matrixes x ∈ p on the following form: where B is any n × n-matrix with tr(B) = 0. Let x i ∈ g be the following element: Let the first column vector of x i be the vector e i and let the rest of the entries be such that tr(x i ) = 0. It follows x i ⊗ e 0 ∈ g ⊗ L and x i (e 0 ) = e i hence the vertical map is surjective. One easily checks the sequence is exact and 3.3.5 is proved.
We get two P -modules: There is an injection of P -modules Assume in the following m = 1 and L = W . It follows G = P(V * ) = P is projective n-space.
The following formula holds: Proof. we prove the result by induction on k. Assume k = 1 and let and the claim holds for k = 1. Assume the result is true for k. Hence We get The Proposition is proved.
Proof. There are embeddings of P -modules Recall from [14] the Theorem follows.
Corollary 3.6. There is for all 1 ≤ l < d an isomorphism Proof. There is by [14], Theorem 3.10 an isomorphism P l P (O P (d))(x) * ∼ = U l (g)L d of P -modules. From this isomorphism and Theorem 3.5 the Corollary follows since as P -modules.
Note: Corollary 3.6 is proved in [11] Theorem 2.4 using more elementary techniques.
Let Y = Spec(K) and π : P(V * ) → Y be the structure morphism. Let P = P(V * ). Since Sym l (V * ) is a finite dimensional SL(V )-module it follows it is a free O Y -module with an SL(V )-linearization. It follows π * Sym l (V * ) is a locally free O P -module with an SL(V )-linearization since π * preserves the SL(V )-linearization.
Proposition 3.7. There is for all 1 ≤ l < d an isomorphism Proof. Let P ⊆ SL(V ) be the subgroup fixing the line L ∈ V There is an exact equivalence of categories The P -module corresponding to O P (d − l) ⊗ π * Sym l (V * ) is (L) d−l ⊗ Sym l (V * ). By the equivalence 3.7.1 and Corollary 3.6 we get an isomorphism P l P (O P (d)) ∼ = O P (d − l) ⊗ π * Sym l (V * ) of locally free sheaves with SL(V )-linearization and the Proposition is proved.
We get a formula for the splitting type of P l P (O P (d)) on projective space: Corollary 3.8. There is for all 1 ≤ l < d an isomorphism of locally free sheaves.
Proof. The P -modules Sym l (V * ) corresponds to the free O P -module ⊕ ( n+l n ) O P . The Corollary now follows from Proposition 3.7.
Let X = G(m, m + n) and consider the P -modules Proposition 3.9. There is an isomorphism hence there is an inclusion L d−1 ⊗Sym 1 (g/p L ⊗L) and the Proposition is proved.
Corollary 3.10. There is an isomorphism Proof. There is by [14], Theorem 3.10 an isomorphism  )) and X = G(m, m + n). This double complex might give rise to a resolution of the ideal sheaf of the l'th discriminant D l (O X (d)) ⊆ P(W * ) of the line bundle O X (d). By [12], Theorem 5.2 it follows knowledge on the P -module structure of P l X (O X (d)) gives information on the SL(V )-module structure of the higher cohomology groups H i (X, ∧ j P l X (O X (d)) * ) for all i ≥ 0. This again gives information on the dimension h i (X, ∧ j P l X (O X (d)) * ). We get a description of the locally free sheaf Example 3.11. Canonical filtration for the grassmannian G(2, 4).
Consider the example where m = n = 2 and X = G(2, 4). We get two inclusions We may choose a basis for p ⊆ g on the following form: where L x is the line spanned by the following vector x: Let n ⊆ g be the sub Lie algebra spanned by the following vectors: Letñ be the vector space spanned by the vectors x 1 , x 2 , x 4 , x 4 and x. It follows . The vector space V has a basus e 1 , e 2 , e 3 and e 4 . The vector space W has basis e 1 , e 2 . It follows ∧ 2 W has a basis given by e 1 ∧ e 2 = e [12] and ∧ 2 V has basis given by e [12], e [13], e [14], e[23], e[24], e [34]. By definition L = e [12]. We get the following calculation: Let a = d(d − 1). A basis for the P -module U 2 (g)L d = U 2 (ñ)L d are the following vectors: There is no equality of P -modules as submodules of Sym d (∧ m V ) in general as Example 3.11 shows. Since U l (g)L d and L d−l ⊗ Sym l (g/p L ⊗ L) by Theorem 3.5 and Proposition 3.3 are isomorphic when m = 1 and 1 ≤ l < d, have the same dimension over K and both have natural filtrations of P -modules we may conjecture they are isomorphic as P -modules for all m, n ≥ 1. Note: There is a canonical line L d ∈ U l (g)L d for all l. There is similarly a canonical line Hence the two P -modules U l (g)L d and L d−l ⊗ Sym l (g/p L ⊗ L) look similar.
In general the SL(V )-module Sym d (∧ m V ) decompose where V λi are irreducible SL(V )-modules and a i ≥ 1 are integers (see Proposition 5.4 for the situation of G (2, 4)). One may ask if there is a non-trivial automorphism with the property that the morphism

Jet bundles and incidence complexes on the projective line
In this section we construct a resolution by locally free sheaves of the ideal sheaf of the l'th incidence scheme I l (O P (d)) ⊆ P(W * ) × P. Here O P (d) is an invertible sheaf on the projective line P = P 1 and W = H 0 (P, O p (d)). There is on Y = P(W * ) × P 1 a morphism φ(O(d)) of locally free sheaves -is a resolution of the ideal sheaf of I l (O(d)). This follows from the fact that the ideal sheaf of I l (O(d)) is locally generated by a regular sequence. We also calculate the higher direct images of the terms The aim of the construction is to use it to construct a resolution of the ideal sheaf of the discriminant D l (O(d)) where O(d) is a line bundle on projective space or a grassmannian. Let A be an arbitrary commutative ring with unit and let φ : E → F be a map of finite rank locally free A-modules. Define the following map: Let I φ ⊆ A be the image of φ * . We let I φ be the ideal of φ. Define the following map Lemma 4.2. The following holds for all p ≥ 1: and the claim of the Lemma follows.
Let r = rk(E ⊗ F * ). We get a complex of locally free A-modules ., x n } be a regular sequence of elements in A and let E = Ae be the free A-module on the element e. Let F = A{e 1 , .., e n } be a free rank n module on e 1 , .., e n . Let y i = e * i . Define φ : E → F by φ(e) = x 1 e 1 + · · · + x n e n .
Let e ⊗ y i = z i . It follows Hence the complex ∧ • E ⊗ F * equals the Koszul complex K • (x) of the regular sequence x. It is an exact complex since x is a regular sequence. The construction of the differential in the Koszul complex is intrinsic, hence we may generalize to morphisms of locally free sheaves. Let Y be an arbitrary scheme and let φ : E → F be a map of locally free O Y -modules. Let Since I φ is quasi coherent sheaf of ideals it follows the ideal sheaf I φ corresponds to a subscheme Z(φ) ⊆ Ythe zero scheme of φ. Let U ⊆ Y be an open subset and define the following map: This gives a well defined map of locally free sheaves since we have not chosen a basis for the module ∧ p (E ⊗ F * )(U ) to give a definition. By Lemma 4.2 it follows d p • d p+1 = 0 for all p ≥ 0 hence we get a complex of locally free sheaves. The sequence of maps of locally free sheaves (r = rk(E ⊗ F * ) ) is a resolution of the ideal sheaf I Z(φ) of Z(φ) ⊆ Y since it is locally isomorphic to the Koszul complex K • (x i ) for all i.
Since the ideal I i is generated by a regular sequence of lenght r it follows Example 4.6. The incidence complex of O(d) on the projective line.
Let P = P 1 K where K is a field of characteristic zero and let O(d) ∈ Pic(P) = Z be a line bundle where d ∈ Z. Let Let Y = P(W * ) × P and consider the following diagram There is a sequence of locally free O Y -modules Restrict the map 4.6.1 to the open set U i0 = D(y i ) × D(x 0 ) ⊆ Y . We get the following two maps of modules: We get the map Restrict the map 4.6.1 to the open set U i1 We get the following two maps of modules: We get the map It follows the ideal sheaf be an arbitrary degree d polynomial with a d = 0. Let f (i) (t) denote the formal derivative with respect to t. It follows Proof. The proof is by induction. It is clearly true for l = 1. Assume it is true for l > 1. Consider k = l + 1. We get and the claim of the Lemma follows. which is a domain, hence w l−1 is a non zero divisor in A[t]/w l . By induction it follows z l , .., z 0 is a regular sequence in A[t]. Assume i ≤ l. It follows the sequence z l , .., z i+1 is a regular sequence in A[t]. We see from Lemma 4.7 z i is non zero in and K[u 0 , .., u i , u l+1 , .., u d , t] is a domain. It follows z i is a non zero divisor in A[t]/(z l , .., z i+1 ). It follows z l , .., z 0 is a regular sequence in A[t] and the claim follows. A similar argument proves w l , .., w 0 is a regular sequence in A[s] and the Lemma is proved.
It follows the ideal sheaf I I l (O(d)) is locally generated by a regular sequence.
gives by Example 4.3 rise to a Koszul complex Definition 4.9. Let the complex Since the ideal sheaf of I l (O(d)) by the discussion above is locally generated by a regular sequence it follows from Example 4.3 the complex 4.9.1 is a resolution.
In [11], Theorem 5.10 one calculates the higer direct images We get the following calculations: Let V = K{e 0 , e 1 } and P = P(V * ). Let W = H 0 (P, O(d)) = Sym d (V * ) and consider the diagram .
By the results of this paper it follows there is an isomorphism ⊗ π * Sym l (V * ) a sheaves with an SL(V )-linearization. We get . By the equivariant projection formula for higher direct images we get We get the following Theorem: Theorem 4.10. The following holds: Proof. The proof follows from the calculation of the equivariant cohomology of line bundles on projective space (see [7]).
Hence we have complete control on the sheaf Y ) on the projective line and projective space for all i, j. Using the techniques introduced in this paper one may describe resolutions of incidence schemes I l (O(d)) on more general grassmannians and flag varieties. The hope is we may be able to construct resolutions of the ideal sheaf of D l (O(d)) using indicence resolutions in a more general situation.
Note: In [9] resolutions of ideal sheaves of determinantal schemes are studied and much is known on such resolutions. In [12] it is proved D 1 (O(d)) is a determinantal scheme for any d ≥ 2 on the projective line P 1 . Assume L ∈ Pic G (G/P ) is a Glinearized linebundle, G a semi simple linear algebraic group and P a parabolic subgroup. If one can prove D l (L) is a determinantal scheme we get two approaches to the study of resolutions of ideal sheaves of discriminants: One using jet bundles and incidence schemes, another one using determinantal schemes.

Appendix A: Automorphisms of representations
Let W ⊆ V be vectorspaces of dimension two and four over the field K. Consider the subgroup P ⊆ G = SL(V ) where P is the parabolic subgroup of elements fixing W . It follows π : G → G/P = G(2, 4) is a principal P -bundle. Let g = Lie(G) and p = Lie(P ) be the Lie algebras of G and P . In this section we study the decomposition into irreducibles and automorphisms of some G-modules. We also study some P semi -modules where P semi is the semi-simplification of P . It follows P semi equals SL(2) × SL(2). Since p ⊆ g is a P -sub module it follows the quotient g/p is a P -module hence a P semi module. We may apply the theory of highest weights since P semi = SL(2) × SL(2) is a semi simple algebraic group.
and its decomposition into irreducible SL(2) × SL(2)-modules can be done using well known formulas (see [5]). Alternatively one may compute its highest weight vectors and highest weights explicitly using the construction from Section 5. One checks f is a highest weight vector for SL(V ) with highest weight 0, hence it defines the unique trivial character of SL(V ). Its dual is the defining equation for G = G/P as closed subscheme of P(∧ 2 V * ).
Proposition 5.2. The following hold: there is an isomorphism of SL(V )-modules Proof. The result is proved using the theory of highest weights. There is a split exact sequence of SL(V )-modules Dualize this sequence to get the split exact sequence of SL(V )-modules. By the Borel-Weil-Bott Theorem it follows Q d is an irreducible SL(V )-module. If d = 2k we get by induction the equality and the claim of the Proposition is proved in the case where d = 2k. The claim when d = 2k + 1 follows by a similar argument and the Proposition is proved.
Proof. We get by Proposition 5.4 isomorphisms of SL(V )-modules Proof. This follows from Proposition 5.4 and the Borel-Weil-Bott theorem (BWB). From the BWB theorem it follows H 0 (G, O G (d)) * is an irreducible SL(V )-module for all d ≥ 1. From this and Proposition 5.4 the claim of the Corollary follows.
Hence the SL(V )-module Sym d (∧ 2 V ) is a multiplicity free SL(V )-module for all d ≥ 1. This is not true in general for Sym d (∧ m K m+n ) when m, n > 2.
In general if S λ and S µ are two Schur-Weyl modules (see [5]) there is a decomposition where V λi is an irreducible SL(V )-module for all i.
It is an open problem to calculate this decomposition for two arbitrary partitions λ and µ.

Appendix B: The Cauchy formula
We include in this section an elementary discussion of the Cauchy formula using multilinear algebra. Let W ⊆ V be vector spaces of dimension m and m + n over K and let P ⊆ SL(V ) be the subgroup fixing W . Let g = Lie(G) and p = Lie(P ). There is a canonical isomorphism g/p ∼ = Hom(W, V /W ) of P -modules, hence the elements of g/p may be interpreted as linear maps. The symmetric power Sym k (g/p) = Sym k (Hom(W, V /W )) is a P -module hence a P semi = SL(m) × SL(n)-module and we want to give an explicit construction of its highest weight vectors as P semi -module. defined by x 1 ∧ · · · ∧ x m ⊗ e 1 ∧ · · · ∧ e m → x 1 ⊗ e 1 x 1 ⊗ e 2 · · · x 1 ⊗ e m x 2 ⊗ e 1 x 2 ⊗ e 2 · · · x 2 ⊗ e m x m ⊗ e 1 x m ⊗ e 2 · · · x m ⊗ e m .
Here e 1 , .., e m is a basis for U and x 1 , .., x m is a basis for U * .
Proof. The proof is left to the reader as an exercise.
Note: in Proposition 6.1 the element x i ⊗e j is an element of U * ⊗U = Hom(U, U ). Hence the determinant x 1 ⊗ e 1 x 1 ⊗ e 2 · · · x 1 ⊗ e m x 2 ⊗ e 1 x 2 ⊗ e 2 · · · x 2 ⊗ e m x m ⊗ e 1 x m ⊗ e 2 · · · x m ⊗ e m .
may be interpreted as a polynomial of degree m in the elements x i ⊗ e j , hence it is an element of Sym m (Hom(U, U )). Let B ⊆ SL(m, K) × SL(n, K) ⊆ SL(V ) = SL(m + n, K) be the following subgroup: B consists of matrices with determinant one of the form 0 · · · 0 a 21 a 22 · · · 0 . . . . . . · · · . . . a m1 a m2 · · · a mm      and Let T be a B-module and v ∈ T a vector with the property that for all x ∈ B it follows xv = λ(x)v where λ ∈ Hom(B, K * ) is a character of B. It follows v is a highest weight vector for The i ′ th wedge product Here λ(g) = b1···bi am−i+1···am is a character λ ∈ Hom(B, K * ). Proof. The proof is left to the reader as an exercise.
Hence the i'th determinant |x i | ∈ Sym i (Hom(W, V /W )) is a highest weight vector for the SL(m) × SL(n)-module Sym i (Hom(W, V /W )). By the results of [2] it follows the vectors x d0 0 x d1 1 · · · x di i with id i = k are all highest weight vectors for the module Sym k (Hom(W, V /W )) ∼ = Sym k (W * ⊗ V /W ).