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On generating sets for ideals defining S -varieties
Evgeny Smirnov
Abstract Let G be a semisimple algebraic group. Kostant's theorem describes the ideal defining the G-orbit closure of the sum of highestweight vectors in a (reducible) G-module, such that the corresponding highest weights are linearly independent. This ideal is generated by quadratic polynomials. In this paper we generalize this result, assuming that the highest weights can be linearly dependent. In this case the equations defining these varieties are not necessary quadratic.

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Intro duction

Let G be a semisimple algebraic group. We consider a (possibly reducible) finite-dimensional G-module, fix a highest weight vector in each of its irreducible submodules and consider their sum. The closure of the G-orbit of such a vector is said to be an S -variety. A classical example of an S -variety is provided by a Grassmann cone Gr(m, n), obtained as the closure of the highest weight vector orbit in the representation of SLn acting on m k n . The Grassmann cone can be defined by a set of equations, known as Plucker relations ; more¨ over, there equations generate its ideal. In the present paper, we obtain analogous relations for an arbitrary S -variety. Earlier it was done in the case of linearly independent highest weights (see [1]). The author is grateful to Ernest B. Vinberg for the attention to this work, and to Dmitry Timashev for many useful remarks.

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Notation
k G B X(T ) X+ (T ) V + v ground field, algebraically closed and of zero characteristics; simply connected semisimple algebraic group; a fixed Borel subgroup in G; the character group for a fixed maximal torus T B ; semigroup formed by the dominant characters for T ; -the irreducible G-module with a highest weight X+ (T ) highest weight vector in V .

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H V -varieties
a particular case: the closures of the highest weight Such varieties are often called H V -varieties. In [2] it the closure X of the highest weight vector orbit in an dule V is defined by the following system of equations: (v v ) = (2 + 2, 2)(v v ), (1)

First consider vector orbits. is proved that irreducible mo

where stands for the Casimir operator, and is equal to the half-sum of the positive roots. (1) is an equality of elements from S 2 V , so it can be considered as a system of d (d + 1)/2 quadratic equations on the coordinates of v , where d = dim V . If G = SLn , and = k is the k -th fundamental weight, what we get is exactly the set of Plucker relations. ¨ Here is a streghtening of the main result from [2]: Prop osition 1. The relations (1) generate the ideal of X . Proof. This follows from the Kostant theorem (Theorem 3 of this paper). Its proof will be given in the next section.

3 S -varieties: Linearly indep endent weight case
Consider the S -variety which corresponds to dominant weights 1 , . . . , k X+ (T ), that is, the closure of the G-orbit of the sum of highest vectors + + v1 + · · · + vk V1 · · · Vk = V . Denote it by X = X (1 , . . . , k ). Also denote the set 1 , . . . , k by E0 , and let E be the semigroup in X+ (T ) generated by the weights

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From [3], one knows that the coordinate ring k [X ] considered as a G-module is equal to k [X ] =
E

S
X+ (T )

S = k [G/U ],

(2)

where S {f k [G] : f (g b) = (b)f (g ) g G, b B }, =
(so, S equals V as a G-module), and the embedding k [X ] k [G/U ] is G-equivariant. From the definition of S we see that the rings k [X ] and k [G/U ] are graded by the elements of E, e.g., S Sµ = S+µ . In [1], the following theorem, due to Kostant, is stated. It is a generalization of Prop. 1. Since our main result is based on its proof, we give this theorem with a sketch of the proof.

Theorem 2 (Kostant). Let the semigroup E X+ (T ) be freely generated by a finite set of weights E0 = {1 , . . . , k } E. Let I be an ideal in k [V ] = k [V1 · · · Vk ], generated by the coordinates of tensors (v vµ ) - ( + µ + 2, + µ)(v vµ ), (3)

where v V , vµ Vµ , and and µ run over the set E0 . Then the coordinate algebra of the S -variety X (1 , . . . , k ) is equal to k [V ]/I . Proof. The relations (v vµ ) - ( + µ + 2, + µ)(v vµ ) = 0 hold + + + + + for the sum of highest vectors v1 + · · · + vk , because v vµ = v+µ . Further, they are G-invariant. This means that they hold along the whole orbit. This proves the inclusion I I (X ). Now let us prove the reverse inclusion. Let n = dim G, and {x1 , . . . , xn } and {x , . . . , x } be two bases of the Lie algebra g = n 1 Lie(G), dual with respect to the Cartan-Killing form. Consider the Casimir operator
n

=
i=1

xi x U (g). i

acts is a simple [] >

on V as a scalar ( + 2, ). Denote this scalar by []. If dominant weight, such that = - ki i , where i are roots, and ki 0, with a strictly positive ki among them, then [ ]. Indeed, ki ( + 2, i ) = ( + 2, )+

( + 2, ) = ( + 2, ) +

ki ((i , ) + ( + 2, i )) > ( + 2, ). (4)

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Take , µ E0 . Now let us rewrite the operator - ( + µ)[]Id acting on the tensor product V Vµ , in a slightly different form.
n

(-(+µ)[]Id)(v vµ ) =
i=1

(xi x + x xi ) - 2(, µ) (v vµ ). i i

(5) By assumption, the coordinates of all these expressions belong to I . From the presentation (2) of the coordinate algebra k [X ] it follows that the ideal I is linearly generated by the kernels of pro jections onto the highest components S n1 V
1

· · · S nk V

k

V

n1 1 +···+nk k

for n1 , . . . , nm Z+ . Each such kernel is the image of a pro jection P = - []Id EndS n1 V1 · · · S nk Vk , where = n1 1 + · · · + nk k . Let v V , and let vj denote the pro jection of v onto Vj . Then n n the coordinates of the tensor P (v1 1 · · · vk k ) can be considered as elements of k [V ]. A simple calculation shows that
n n P (v1 1 · · · vk k ) = 1r
(6) where
n

T

n1 r s (v1

···

n vk k

)=
i=1 n1 v1

n n n (v1 1 · · · xi vr r · · · x vs s · · · vk + i n n · · · x vr r · · · xi vs s · · · vk ). (7) i

+

According to (5), the coordinates of each summand from (6) belong n n to I . So, the coordinates of P (v1 1 · · · vk k ) belong to I as well. From the proof, we get the following corollary, also stated in [1]. Corollary 3. Let E0 be arbitrary (probably without the linear independency condition). Then the ideal I contains the coordinates of al l the expressions of the form ( - (i1 + · · · + im )[])(vi1 · · · vim ), where vip Vip , ip E0 .

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4 S -varieties: Linearly dep endent weight case
Now let us find equations defining an S -variety in the case when the highest vectors i are linearly dependent, i.e., when the semigroup E admits relations on elements of E0 . It is clear that in this case the relations analogous to (3) hold. But the quotient of k [V ] over these relations is larger than the coordinate ring k [X ]. The former ring may include more that one G-invariant components of the form V ; their number equals to the number of presentations of as Z+ -linear combinations of elements from E0 . In this case we add an additional set of relations to the Kostant's one. To do this, let us scale the highest vectors of Vi in the following way. Considet the algebra k [G/U ] = S V . Let f be the = highest vector of S satisfying the condition f (w0 ) = 1, where w0 is the longest element of the Weyl group of G. So, the vectors f form a multiplicative semigroup, that is isomorphic to X+ (T ). Now consider a G-module embegging V = V1 · · · Vk + k [G/U ] mapping each vi to f . Each such embedding will be called i a canonical one. Now we state the main result of this paper. Theorem 4. Let E X+ be a semigroup generated by a finite set of weights E0 = {1 , . . . , k } E with defining relations of the form n1 i1 + · · · + nr ir = m1 j1 + · · · + ms js , where i1 , . . . , ir , µj1 , . . . , µjr E0 , {i1 , . . . , ir } {j1 , . . . , js } = , ni , mj Z+ . Also suppose that the G-module V = V1 · · · Vk is embedded into k [G/U ] canonical ly. Then the ideal I (X ) k [V ] of X = X (1 , . . . , k ) is generated by the coordinates of tensors (v vµ ) - ( + µ + 2, + µ)(v vµ ), where v V , vµ Vµ , , µ E0 , and
n m1 n m 1 (vi1 · · · vir ) - 2 (vm · · · vjs ),
1 r 1 s

(8)

(9)

n m1 n m where vi1 · · · vir S n1 Vi1 · · · S nr Vir , vm · · · vjs r s 1 1 S m1 Vj1 · · · S ms Vjs , and 1 2 are the G-equivariant linear maps from S n1 Vi1 · · · S nr Vir S m1 Vj1 · · · S ms Vjs to V ,

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+ = n1 i1 + · · · + nr ir = m1 j1 + · · · + ms js , such that 1 ((vi1 )n1 + + + · · · (vi )nr ) = 2 ((vj1 )m1 · · · (vj )ms ).
r s

Proof. Denote the ideal generated by the coordinates of (8) and (9), by I . These coordinates vanish on the sum of the highest vectors (this holds for (9), since the scaling of the highest vectors is canonical). They generate a G-invariant ideal. This means that I I (X ). The coordinate ring k [V ] = k [V1 · · · Vk ] is isomorphic to S (V ) = S m (V ), and
mZ+

S m (V

1

· · · Vk ) = i1 +···+ik =m

S i1 V

1

· · · S ik Vk .

Each G-module S i1 V1 · · · S ik Vk equals to the direct sum of the highest irreducible component Vi1 1 +···+ik k and the remaining (lower) components of the decomposition. According to Corollary 3, all the lower components are contained in the ideal I for each (i1 , . . . , ik ). Roughly speaking, the generators of I that come from the relations (9), identify the highest components of the spaces S n1 Vi · · · 1 S nr Vi and S m1 Vj · · · S ms Vj . These relations vanish on the s r 1 kernel Ker F of the operator F End (S n1 Vi1 · · · S nr Vir S m1 Vj1 · · · S ms Vjs ), F : v1 + v2 1 (v1 ) - 2 (v2 ), where v1 S n1 Vi1 · · · S nr Vir , v2 S m1 Vj1 · · · S ms Vjs ). The space Ann Ker F of the linear functions that vanish on Ker F is G-invariant. As a G-module, it is isomorphic to the (indecom posable) representation Vn1 i +···+nr i . So, since the intersection of r 1 Ann Ker F with I is nontrivial, Ann Ker F I . Each relation in the semigroup E is obtained as the sum of certain defining relations, i.e., can be presented in form k (nk1 ki1 + · · · + nkr kir ) = k (mk1 kj1 + · · · + mks kjs ). For each k , the highest + + + ~ vectors vk = (v )nk1 · · · (v )nkr and vk = (vµ )nkj · · · + (v )nkr of the highest components of S nk1 Vki · · · S nkr Vki r kjs 1 S mk1 Vkj · · · S mks Vkj are equal modulo I . This means that s 1 vk and the highest vectors ~ k k vk of the highest comp onents of mk1 V the modules k (S nk1 Vki · · · S nkr Vki ) k (S kj1 · · · r 1 mks V ) are also equal mo dulo I . S kjs Thus, the quotient of the ring k [V ] over the ideal I can be embedded into S = k [X ]. Since I I (X ), this ring equals k [X ]. So, E
ki1 kir kj1

I = I (X ).

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References
[1] G. Lancaster, J. Towber, Representation-functors and flagalgebras for the classical groups I, J.Algebra 59 (1979), 16­38. [2] W. Lichtenstein, A system of quadrics describing the orbit of the highest weight vector, Proc. AMS 84 (1982), No. 4, 605­608. ` [3] E. B. Vinberg, V. L. Popov, On a class of quasihomogeneous affine varieties, Izv. USSR Math., 6 (1972), 743­758.
Independent University of Moscow, Bolshoi Vlasievskii per., 11, 119002 Moscow, Russia E-mail address: smirnoff@mccme.ru

Submitted: November 12, 2003.

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