Generators of supersymmetric polynomials in positive characteristic
aa r X i v : . [ m a t h . R T ] J u l GENERATORS OF SUPERSYMMETRICPOLYNOMIALS IN POSITIVE CHARACTERISTIC
A.N. GRISHKOV, F. MARKO, AND A.N. ZUBKOV
Abstract.
In [1], Kantor and Trishin described the algebra ofpolynomial invariants of the adjoint representation of the Lie su-pergalgebra gl ( m | n ) and a related algebra A s of what they calledpseudosymmetric polynomials over an algebraically closed field K of characteristic zero. The algebra A s was investigated earlier byStembridge who in [4] called the elements of A s supersymmetricpolynomials and determined generators of A s .The case of positive characteristic p has been recently investi-gated by La Scala and Zubkov in [3]. They formulated two conjec-tures describing generators of polynomial invariants of the adjointaction of the general linear supergroup GL ( m | n ) and generators of A s , respectively. In the present paper we prove both conjectures. Introduction and notation
Let K be an algebraically closed field K of positive characteristic p = 2. The following notation is related to general linear supergroup G = GL ( m | n ). Let K [ c ij ] be a commutative superalgebra freely gen-erated by elements c ij for 1 ≤ i, j ≤ m + n , where c ij is even if either1 ≤ i, j ≤ m or m + 1 ≤ i, j ≤ m + n , and c ij is odd otherwise.Denote by C the generic matrix ( c ij ) ≤ i,j ≤ m + n and write C as a blockmatrix (cid:18) C C C C (cid:19) , where entries of C and C are even and en-tries of C and C are odd. The localization of K [ c ij ] by elements det ( C ) and det ( C ) is the coordinate superalgebra K [ G ] of the gen-eral linear supergroup G = GL ( m | n ). The general linear supergroup G = GL ( m | n ) is a group functor from the category SAlg K of com-mutative superalgebras over K to the category of groups, representedby its coordinate ring K [ G ], that is G ( A ) = Hom
SAlg K ( K [ G ] , A ) for A ∈ SAlg K . Here, for g ∈ G ( A ) and f ∈ K [ G ] we define f ( g ) = g ( f ).Denote by Ber ( C ) = det ( C − C C − C ) det ( C ) − the Berezinianelement.Let T be the standard maximal torus in G and X ( T ) be a set ofcharacters. Let V be a G -supermodule with weight decomposition V = ⊕ λ ∈ X ( T ) V λ , where λ = ( λ , . . . , λ m + n ) and each V λ splits into a sum of its even subspace ( V λ ) and odd subspace ( V λ ) . The (formal)supercharacter χ sup ( V ) of V is defined as χ sup ( V ) = X λ ∈ X ( T ) ( dim ( V λ ) − dim ( V λ ) ) x λ . . . x λ m m y λ m +1 . . . y λ m + n n . If V is a G -supermodule with a homogeneous basis { v , . . . , v a , v a +1 ,. . . , v a + b } such that v i is even for 1 ≤ i ≤ a and v i is odd for a + 1 ≤ i ≤ a + b , and the image ρ V ( v i ) of a basis element v i under a comultiplication ρ V is given as ρ V ( v i ) = P ≤ j ≤ a + b v j ⊗ f ji , then the supertrace T r ( V )is defined as P ≤ i ≤ a f ii − P a +1 ≤ i ≤ a + b f ii .Let E be a standard G -supermodule given by basis elements e , . . . ,e m that are even and e m +1 , . . . , e m + n that are odd and by comultipli-cation ρ E ( e i ) = P ≤ j ≤ m + n e j ⊗ c ji . Denote by Λ r ( E ) the r -th exteriorpower of E and by C r the supertrace of Λ r ( E ).The algebra R of invariants with respect to adjoint action of G isa set of functions f ∈ K [ G ] satisfying f ( g − g g ) = f ( g ) for any g , g ∈ G ( A ) and any commutative supergalgebra A over K . Thealgebra R pol of polynomial invariants is a subalgebra of R consisting ofpolynomial functions.As in the characteristic zero case, the description of polynomial in-variants R pol can be reduced to that of the algebra A s of supersym-metric polynomials. The algebra A s consists of polynomials f ( x | y ) = f ( x , . . . , x m , y , . . . y n ) that are symmetric in variables x , . . . x m and y , . . . , y n separately and such that ddT f ( x | y ) x = y = T = 0. The main toolfor this reduction is the Chevalley epimorphism φ : K [ G ] → A , where A = K [ x ± , . . . , x ± m , y ± . . . y ± n ], and φ ( c ij ) = δ ij x i for 1 ≤ i ≤ m and φ ( c ij ) = δ ij y i − m for m + 1 ≤ i ≤ m + n . Then for any G -supermodule V we have φ ( T r ( V )) = χ sup ( V ). In particular, for 0 ≤ r we have φ ( C r ) = c r = X ≤ i ≤ min ( r,m ) ( − r − i σ i ( x , . . . , x m ) p r − i ( y , . . . , y n ) , where σ i is the i -th elementary symmetric function and p j is the j -thcomplete symmetric function.A homogeneous polynomial f ( x | y ) = P a λ x λ . . . x λ m m y λ m +1 . . . y λ m + n n is called p -balanced if p | ( λ i + λ j ) whenever 1 ≤ i ≤ m < j ≤ m + n and a λ = 0. Denote by A s ( p ) the subalgebra of A s generated by p -balancedpolynomials.The following theorem is the main result of this paper. Theorem 1. (Conjecture 5.2 of [3] ) The algebra A s is generated over A s ( p ) by elements c r for r ≥ . ENERATORS OF SUPERSYMMETRIC POLYNOMIALS 3
For a matrix M , denote by σ i ( M ) the i -th coefficient of the char-acteristic polynomial of M . Then all elements C r , σ i ( C ) p , σ j ( C ) p , σ n ( C ) p Ber ( C ) k ∈ R pol . As a consequence of previous theorem weobtain that these are generators of algebra R pol . Theorem 2. (Conjecture 5.1 of [3] ) The algebra R pol is generated byelements C r , σ i ( C ) p , σ j ( C ) p , σ n ( C ) p Ber ( C ) k , where ≤ r, ≤ i ≤ m, ≤ j ≤ n, < k < p .Proof. Theorem 5.2 of [3] states that the restriction of φ on R is amonomorphism. It was also noticed there that φ ( R pol ) ⊂ A s which is aconsequence of arguments analogous to Theorem 1.1 of [1]. Proposition5.1 of [3] states that σ i ( x , . . . , x m ) p for 1 ≤ i ≤ m , σ j ( y , . . . , y n ) p for1 ≤ j ≤ n and u k ( x | y ) = σ m ( x , . . . , x m ) k σ m ( y , . . . , y n ) p − k for 0 Algebra R is equaled to R pol [ σ m ( C ) ± p , σ n ( C ) ± p ] .Proof. If f ∈ R , then its multiple by a sufficiently large power of σ m ( C ) p σ n ( C ) p is a polynomial invariant. (cid:3) Nice supersymmetric polynomials In this section we will compare algebras corresponding to differentvalues of m, n and apply the Schur functor. Therefore we adjust thenotation slightly to reflect the dependence on m, n . For example, wewill write R ( m | n ) instead of R and A s ( m | n ) instead of A s .Denote by A ns ( m | n ) the algebra of ”nice” supersymmetric polyno-mials. It is a subalgebra of A s ( m | n ) generated by polynomials c r ( m | n ) = X ≤ i ≤ r ( − r − i σ i ( x , . . . , x m ) p r − i ( y , . . . , y n ) ,σ i ( x, m ) p = σ i ( x , . . . , x m ) p , σ j ( y, n ) p = σ j ( y , . . . , y n ) p and u k ( m | n ) = σ m ( x, m ) k σ n ( y, n ) p − k for 1 ≤ i ≤ m , 1 ≤ j ≤ n and 0 < k < p . Since φ ( C r ( m | n )) = c r ( m | n ),using Proposition 5.1 of [3] we conclude that A ns ( m | n ) ⊂ φ ( R pol ( m | n )).We will see below that this inclusion is actually an equality. Denote by A ns ( m | n, t ) the homogeneous component of A ns ( m | n ) of degree t .For any integers M ≥ m, N ≥ n there is a graded superalgebra mor-phism p e : K [ x , . . . , x M , y , . . . , y N ] → K [ x , . . . , x m , y , . . . , y n ] that A.N. GRISHKOV, F. MARKO, AND A.N. ZUBKOV maps x i x i for i ≤ m , y j y j for j ≤ n and the remaining gener-ators x i , y j to zero. Clearly p e restricts to a map from A s ( M | N ) into A s ( m | n ). Lemma 1.1. The map p e takes A ns ( M | N ) to A ns ( m | n ) .Proof. Assume M > m or N > n . It follows from p e ( σ i ( x, M )) = σ i ( x, m ) if i ≤ m and p e ( σ i ( x, M )) = 0 otherwise; p e ( σ j ( y, N )) = σ j ( y, n ) if j ≤ n and p e ( σ j ( y, N )) = 0 otherwise; p e ( c r ( M, N )) = c r ( m | n ) if r ≤ m, n and p e ( c r ( M, N )) = 0 otherwise, and p e ( u k ( M | N ))= 0. (cid:3) For the integers M ≥ m, N ≥ n consider the Schur superalgebra S ( M | N, r ) and its idempotent e = P µ ξ µ , where the sum runs over allweights µ for which µ i = 0 whenever m < i ≤ M or M + n < i ≤ M + N . Then S ( m | n, r ) ≃ eS ( M | N, r ) e and there is a natural Schurfunctor S ( M | N, r ) − mod → S ( m | n, r ) − mod given by V eV . If V is a S ( M | N, r )-supermodule, then eV is a supersubspace of V andtherefore, eV has the canonical S ( m | n, r )-supermodule structure. Lemma 1.2. The map p e induces an epimorphism of graded algebras φ ( R pol ( M | N )) → φ ( R pol ( m | n )) .Proof. Applying the Chevalley map φ to the collection of simple poly-nomial G -supermodules L and using Theorem 5.3 of [3] we obtain thatthe algebra φ ( R pol ) is spanned by the supercharacters χ sup ( L ). If λ isa highest weight of L , then χ sup ( L ) is a homogeneous polynomial ofdegree r = | λ | = P ≤ i ≤ m + n λ i . By the standard property of a Schurfunctor there is a simple S ( M | N, r )-supermodule L ′ such that eL ′ ≃ L .Since p e ( χ sup ( L ′ )) = χ sup ( L ), the claim follows. (cid:3) Proposition 1.1. φ ( R pol ( m | n )) = A ns ( m | n ) .Proof. Fix a homogeneous element f ∈ φ ( R pol ( m | n )) of degree r andchoose M ≥ m strictly greater than r . By Lemma 1.2 there is a homo-geneous polynomial f ′ ∈ φ ( R pol ( M | n )) of degree r such that p e ( f ′ ) = f .Using Chevalley map and applying Theorem 5.3 of [3] to the collectionof costandard polynomial modules ∇ ( µ ) we obtain that f ′ is a linearcombination of supercharacters χ sup ( ∇ ( µ )), where µ runs over polyno-mial dominant weights with | µ | = r .The assumption M > r , Theorem 5.4 and Proposition 5.6 of [2]imply that for the highest weight µ = ( µ + | µ − ) we have µ − = pµ and ∇ ( µ ) ≃ ∇ ( µ + | ⊗ F ( ∇ ( µ )), where F is the Frobenius map and ∇ ( µ )is the costandard GL ( n )-module with the highest weight µ . Therefore χ sup ( ∇ ( µ )) = χ sup ( ∇ ( µ + | χ ( ∇ ( µ )) p . ENERATORS OF SUPERSYMMETRIC POLYNOMIALS 5 Since χ ( ∇ ( µ )) p is a polynomial in σ j ( y, n ) p , all that remains to showis that χ sup ( ∇ ( µ + | A ns ( M | n ) and use Lemma 1.1.By Theorem 6.6 of [2] the character χ sup ( ∇ ( π | π does not depend on the characteristic of the ground field.Therefore we can temporarily assume that charK = 0. In this casethe category S ( M | n, r ) − mod is semisimple and its simple modules are ∇ ( λ ), where λ runs over ( M | n )-hook weights.An exterior power Λ t ( E ( M | n )) for t ≤ M has a unique maximalweight (1 t | S ( M | n, r )-supermodule V =Λ M ( E ( M | n )) ⊗ π M ⊗ Λ M − ( E ( M | n )) ⊗ ( π M − − π M ) ⊗ . . . ⊗ Λ ( E ( M | n )) ⊗ ( π − π ) has the unique maximal weight ( π | 0) and supercharacter χ sup ( V ) = c π − π . . . c π M − − π M M − c π M M . The module V is a direct sum of L ( π | 0) = ∇ ( π | 0) and L ( κ ) = ∇ ( κ )with κ < ( π | κ is a polynomial weight, it implies that κ =( κ + | 0) and κ + < π . Using induction on π we derive that χ sup ( ∇ ( π | c , . . . , c M hence it belongs to A ns ( M | n ). (cid:3) Corollary 1.1. The morphism p e maps A ns ( M | N, t ) onto A ns ( m | n, t ) . Proof of Theorem 1 We will need the following crucial observation. Lemma 2.1. If f ∈ A s ( m | n ) is divided by x m , then f is divided by anonconstant element of A ns ( m | n ) .Proof. We can assume f = 0 and use the symmetricity of f in vari-ables x , . . . , x m and y , . . . , y n to write f = x a . . . x am y b . . . y bn g , whereexponents a > , b ≥ g such that g | x m = y n =0 = 0 areunique. Then f | x m = y n = T = T a + b x a . . . x am − y b . . . y bn − g | x m = y n = T = T a + b x a . . . x am − y b . . . y bn − g + T a + b +1 x a . . . x am − y b . . . y bn − g , where we write g | x m = y n = T = g + T g . The requirement g | x m = y n =0 = 0implies g = 0. Since ddT f | x m = y n = T = 0, this is only possible if a + b ≡ p ). Since a > 0, the polynomial x a . . . x am y b . . . y bn is not constant,and is a product of σ m ( x, m ) p , σ n ( y, n ) p and u k ( m | n ) which belongs to A ns ( m | n ). In fact, since a > 0, we have that f is divisible either by σ m ( x, m ) p or by some u k ( m | n ). (cid:3) Proof of Theorem 1. Using Proposition 5.1 of [3] the statement ofthe theorem is equivalent to the equality A s ( m | n ) = A ns ( m | n ). A.N. GRISHKOV, F. MARKO, AND A.N. ZUBKOV Fix n and assume that m is minimal such that there exists a polyno-mial f ∈ A s ( m | n ) \ A ns ( m | n ) and choose f such that it is homogeneousand of the minimal degree. Then its reduction f | x m =0 ∈ A ns ( m − | n ) isa nonzero polynomial h ( c t ( m − | n ) , σ i ( x, m − p , σ j ( y, n ) p , u k ( m − | n ))in elements c t ( m − | n ), σ i ( x, m − p , σ j ( y, n ) p and u k ( m − , n ) where t ≥ 0, 1 ≤ i ≤ m − 1, 1 ≤ j ≤ n and 0 < k < p . By Corol-lary 1.1 there are elements v k ∈ A ns ( m | n ) of degree mk + ( p − k ) n such that v k | x m =0 = u k ( m − | n ). Since c t ( m | n ) | x m =0 = c t ( m − | n ), σ i ( x, m ) p | x m =0 = σ i ( x, m − p and σ j ( y, n ) p | x m =0 = σ j ( y, n ) p , thepolynomial l = f − h ( c t ( m | n ) , σ i ( x, m ) p , σ j ( y, n ) p , v k ( m | n )) satisfies l | x m =0 = 0. Since the degree of l does not exceed the degree of f , l ∈ A s ( m | n ) and x m divides l , Lemma 2.1 implies that l = l l , where l ∈ A ns ( m | n ) and the degree of l is strictly less than the degree of f .But l ∈ A s ( m | n ) \ A ns ( m | n ) which is a contradiction with our choiceof f . (cid:3) Elementary proof of Theorem 1 A closer look at the proof of Theorem 1 reveals that Corollary 1.1 isthe only result from Section 1 that was used in it. Actually, only thefollowing weaker statement was required in the proof of Theorem 1. Proposition 3.1. For each < k < p there is a polynomial v k ∈ A ns ( m | n ) of degree ( m − k +( p − k ) n such that v k | x m =0 = u k ( m − | n ) . In this section we give a constructive elementary proof of Proposi-tion 3.1 that bypasses a use of the Schur functor and results aboutcostandard modules derived in [2].Fix 0 < k < p and denote s = ⌈ kp − k ⌉ . Then for i = 0 , . . . s − k i = ( i + 1) k − ip > k p = sp − ( s + 1) k ≥ . The relations k i + ( p − k ) = k i − , k p + k = s ( p − k ) , k i + k p = ( s − i )( p − k )will be used repeatedly.A symbol ∆ will denote a nondecreasing sequence ( i ≤ . . . ≤ i t ) ofnatural numbers, where 0 ≤ t < s . We denote || ∆ || = t and | ∆ | = P tj =1 i j . In particular, we allow ∆ = ( ∅ ) and set || ( ∅ ) || = | ( ∅ ) | = 0. De-note by Supp (∆) a maximal increasing subsequence of ∆. Further, de-note by Sym M ( x a . . . x a r r ) and Sym N ( y b . . . y b r r ) respectively a homo-geneous symmetric polynomial in variables x , . . . , x M and y , . . . , y N respectively with a general monomial term x a . . . x a r r and y b . . . y b r r respectively. Denote(∆ , j ) M,N = Sym M ( x k . . . x kM − t x k i M − t +1 . . . x k it M ) Sym N ( y p − k . . . y p − kN − j − y k p N − j ) ENERATORS OF SUPERSYMMETRIC POLYNOMIALS 7 for 0 ≤ || ∆ || = t ≤ M and 0 ≤ j < N , and (∆ , l ) M,N = 0 otherwise;[∆ , j ] M,N = Sym M ( x k . . . x kM − t x k i M − t +1 . . . x k it M ) Sym N ( y p − k . . . y p − kN − j )for 0 ≤ || ∆ || = t ≤ M and 0 ≤ j ≤ N , and [∆ , j ] M,N = 0 otherwise; { ∆ , l, j } M,N = Sym M ( x k . . . x kM − t − x l ( p − k ) M − t x k i M − t +1 . . . x k it M ) Sym N ( y p − k . . . y p − kN − j )for 0 ≤ || ∆ || = t < M and 0 ≤ j ≤ N and any l , and { ∆ , l, j } M,N = 0otherwise.For f ∈ K [ x , . . . , x m , y , . . . , y n ] define ψ ( f ) = f | x m = y n = T and for g, h ∈ K [ x , . . . , x m − , y , . . . , y n − , T ] write g ≡ h if and only if ddT ( g − h ) = 0 . For simplicity write (∆ , j ), [∆ , j ] and { ∆ , l, j } short for (∆ , j ) m − ,n − ,[∆ , j ] m − ,n − and { ∆ , l, j } m − ,n − . Lemma 3.1. We have ψ { ∆ , l, j } m,n ≡ T k { ∆ , l, j − } + T ( l +1)( p − k ) [∆ , j ] + T l ( p − k ) [∆ , j − P i ∈ Supp (∆) ( T k i { ∆ \ i, l, j − } + T k i − { ∆ \ i, l, j } ) . and ψ (∆ , j ) m,n ≡ T k (∆ , j − 1) + T s ( p − k ) [∆ , j ]+ P i ∈ Supp (∆) ( T k i (∆ \ i, j − 1) + T k i − (∆ \ i, j ) + T ( s − i )( p − k ) [∆ \ i, j ]) . Proof. The first relation follows from ψ ( Sym m ( x k . . . x km − t − x l ( p − k ) m − t x k i m − t +1 . . . x k it m )) = δ t,m − T k Sym m − ( x k . . . x km − − t − x l ( p − k ) m − − t x k i m − − t +1 . . . x k it m − )+ T l ( p − k ) Sym m − ( x k . . . x km − t − x k i m − t . . . x k it m − )+ P i ∈ Supp (∆) T k i Sym m − ( x k . . . x km − t − x l ( p − k ) m − t x k i m − t +1 . . . \ x k i m − t + i . . . x k it m − ) ,ψ ( Sym n ( y p − k . . . y p − kn − j )) = δ j,n T p − k Sym n − ( y p − k . . . y p − kn − − j ) + δ j, Sym n − ( y p − k . . . y p − kn − j )and definitions of (∆ , j ), [∆ , j ] and { ∆ , l, j } .Second relations follows from ψ ( Sym m ( x k . . . x km − t x k i m − t +1 . . . x k it m )) = δ t,m T k Sym m − ( x k . . . x km − t − x k i m − t . . . x k it m − )+ P i ∈ Supp (∆) T k i Sym m − ( x k . . . x km − t x k i m − t +1 . . . \ x k i m − t + i . . . x k it m − ) ,ψ ( Sym n ( y p − k . . . y p − kn − j y k p n − − j )) = δ j,n − T p − k Sym n − ( y p − k . . . y p − kn − − j − y k p n − − j )+ T k p Sym n − ( y p − k . . . y p − kn − − j ) + δ j, Sym n − ( y p − k . . . y p − kn − − j y k p n − j ) A.N. GRISHKOV, F. MARKO, AND A.N. ZUBKOV and definitions of (∆ , j ), [∆ , j ] and { ∆ , l, j } . (cid:3) Let us define w = P s − l =1 P l − n ≤| ∆ |≤ l ( − | ∆ | + s + l ( s − l ) { ∆ , l, l − | ∆ |} m,n + P s − − n< | ∆ | 2) + T s ( p − k ) [∆ , s − − | ∆ | ]+ P i ∈ Supp (∆) ( T k i (∆ \ i, s − | ∆ | − 2) + T k i − (∆ \ i, s − | ∆ | − T ( s − i )( p − k ) [∆ \ i, s − | ∆ | − (cid:17) . To analyze this expression, denote by C ( a ) the coefficient corre-sponding to a term a of ψ ( w ).Then C ( T k i (∆ , s − − i − | ∆ | ) = ( − | ∆ | + i + ( − | ∆ | + i +1 = 0 foreach i = 0 , . . . , s − C ( T k i { ∆ , l, l − − | ∆ | − i } ) = ( − s + l + | ∆ | + i + ( − s + l + | ∆ | + i +1 = 0 for each i = 0 , . . . , s − C ( T s ( p − k ) [∆ , s − | ∆ | − − | ∆ | + s + s − ( s − ( s − − | ∆ | = 0 and if l − − | ∆ | 6 = 0 or ∆ = ∅ , then C ( T l ( p − k ) [∆ , l −| ∆ | − − | ∆ | + s + l ( s − l ) + ( − | ∆ | + s + l − ( s − l + 1) + ( − | ∆ l | , where ∆ l = ∆ ∪ ( s − l ) . Since ( − | ∆ l | = ( − | ∆ | + s − l we conclude that C ( T l ( p − k ) [∆ , l − | ∆ | − C ( T p − k [ ∅ , − s +1 ( s − 1) + ( − | ∆ | = ( − s +1 s. Therefore ψ ( w ) = ( − s +1 sT p − k [ ∅ , − s +1 sT p − k Sym m − ( x k . . . x km − ) Sym n − ( y p − k . . . y p − kn − ) . We can now easily prove Proposition 3.1. Proof of Proposition 3.1. Since s < p we can take v k = ( − s s w + Sym m ( x k . . . x km − )( y p − k . . . y p − kn ) . ENERATORS OF SUPERSYMMETRIC POLYNOMIALS 9 Then ψ ( v k ) = 0, hence v k ∈ A s ( m | n ) and v k | x m =0 = u k ( m − | n ). It iseasy to check that v k is homogeneous of degree ( m − k +( p − k ) n . (cid:3) .4. Concluding remarks The proof of Theorem 1 was influenced by Theorem 1 in [4]. Al-though our proof uses different arguments we would like to remarkthat an analogue of Theorem 1 of [4] remains valid over arbitrary com-mutative ring A of any characteristic. Proposition 4.1. The algebra of polynomials f ( x , . . . , x m , y , . . . , y n ) over a commutative ring A , symmetric in variables x , . . . , x m and y , . . . , y n separately and such that f | x m = y n = T does not depend on T is generated by polynomials c r ( x | y ) .Proof. Proof is a complete analogue of Theorem 1 of [4]. Just replaceevery appearance of σ ( r ) m,n by c r ( x | y ). (cid:3) Let us comment that if characteristic of A is positive, then conditionthat f | x m = y n = T does not depend on T is stronger than ddT f | x m = y n = T = 0.Proposition 3.1 of [1] states that, in the case of characteristic zero, thealgebra A s is infinitely generated. In the case of positive characteristicwe have the following. Proposition 4.2. Algebra A s is finitely generated.Proof. The algebra A s is contained in B = K [ σ i ( x | m ) , σ j ( y | n ) | ≤ i ≤ m, ≤ j ≤ n ]. Algebra B is finitely generated over its subalgebra B ′ = K [ σ i ( x | m ) p , σ j ( y | n ) p | ≤ i ≤ m, ≤ j ≤ n ] hence is a Noetherian B ′ -module. However, A s contains B ′ and is therefore finitely generated B ′ -module. Since B ′ is finitely generated, so is A s . (cid:3) References [1] I.Kantor and I.Trishin, The algebra of polynomial invariants of the adjointrepresentation of the Lie superalgebra gl ( m | n ), Comm. Algebra, 25(7) 1997,2039-2070.[2] R. La Scala and A.N.Zubkov, Costandard modules over Schur superalgebras incharacteristic p , J.Algebra and its Appl., 7(2008), no 2, 147-166.[3] R. La Scala and A.N.Zubkov, Donkin-Koppinen filtration for general linearsupergroup , submitted to Algebra and Representation Theory.[4] J.R.Stembridge, A characterization of supersymmetric polynomials , J. Algebra95 (1985), N.2, 439-444. Departamento de Matematica, Universidade de Sao Paulo, CaixaPostal 66281, 05315-970 - So Paulo, Brazil E-mail address : [email protected] Penn State Hazleton, 76 University Drive, Hazleton PA 18202, USA E-mail address : [email protected] Omsk State Pedagogical University, Chair of Geometry, 644099Omsk-99, Tuhachevskogo Embankment 14, Russia E-mail address ::