DESCRIPTION OF SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(2j2)

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1 DESCRIPTION OF SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(22) A.N. GRISHKOV AND F. MARKO Abstract. The goal of this paper is to describe explicitly simple modules for Schur superalgebra S(22) over an algebraically closed eld K of characteristic zero or of positive characteristic p > 2. Notation and outline of the paper Notation. Throughout the paper we shall work over an algebraically closed eld K of characteristic p = 0 or p > 2 and use the basic terminology of bialgebras A(mn), general linear supergroups GL(mn), Schur superagebras S(mn) and superderivations i D from papers [8, 6]. All modules considered in this paper will be left modules and all superderivations will be right superderivations. In this manuscript we shall only work with G = GL(22) and S(22) and we can therefore describe them in a down-to-earth fashion as follows. Start by de ning the parity i of symbols i = 1; : : : ; 4 by 1 = 2 = 0 and 3 = 4 = 1, and the parity c i of an element c i by c i = i+ (mod 2). Elements of parity 0 will be called even and of parity 1 will be called odd. Let A = A(22) be a commutative superalgebra freely generated over K by elements c i for 1 i; 4, where c 11 ; c 12 ; c 21 ; c 22 ; c 33 ; c 34 ; c 43 and c 44 are even and c 13 ; c 14 ; c 23 ; c 24 ; c 31 ; c 32 ; c 41 and c 42 are odd. The superalgebra A has a structure of a bialgebra given by comultiplication : A! AA de ned as (c i ) = P k c ik c k. The superalgebra A has a natural grading given by the total degree, and is a direct sum A = r0 A(22; r) of its homogeneous components A(22; r). Each component A(22; r) is a coalgebra and its dual A(22; r) is the component S(22; r) of degree r of the Schur superalgebra S(22) = r0 S(22; r). The localization of A(22) by elements d 12 = c 11 c 22 c 12 c 21 and d 34 = c 33 c 44 c 34 c 43 is the coordinate superalgebra K[G] of the general linear supergroup G. The general linear supergroup G is a group functor from the category SAlg K of commutative superalgebras over K to the category of groups, represented by its coordinate ring K[G], that is G(A) = Hom SAlgK (K[G]; A) for A 2 SAlg K. Here, for g 2 G(A) and f 2 K[G] we de ne f(g) = g(f). Modules over Schur superalgebra S(22) correspond to polynomial representations of G. In order to study the structure of G-modules, we will use the superalgebra of distributions Dist(G) of G described in Section 3 of [3]. Denote Dist 1 (G) = (K[G]=m 2 ), where is the duality Hom K ( ; K) and m is the kernel of the augmentation map of the Hopf algebra K[G], and by e i the elements of Dist 1 (G) determined by e i (c hk ) = ih k and e i (1) = 0. Denote the parity of e i to be sum of parities i of i and of. Then e i belong to the Lie superalgebra 2000 Mathematics Subect Classi cation. 17A70;20G05;15A72;13A50;05E15. Key words and phrases. general linear group, Schur superalgebra, character of a simple module. Research was partially supported by CAPES, FAPESP and CNPq(Brazil). 1

2 2 A.N. GRISHKOV AND F. MARKO Lie(G) = (m=m 2 ) which is identi ed with the general linear Lie superalgebra gl(mn). Under this identi cation e i corresponds to the matrix unit which has all entries zeroes except the entry at the position (i,) which equals one. The commutation relations for the matrix units e i are given as [e ab ; e cd ] = e ad bc + ( 1) (a+b)(c+d) e cb ad : Let U C be the universal enveloping algebra of gl(mn) over the eld of complex numbers. Then the Kostant Z-form U Z is generated by elements e i for odd e i, e (r) i = er i r! for even e i, and eii r = e ii(e ii 1):::(e ii r+1) r! for all r > 0. We will consider all G-modules as left modules and use the terminology of right superderivations i D of A(22) de ned on generators c kl as (c kl ) l D = c k and (c kl ) i D = 0 for l 6= i. There is a surective map Dist(G)! S(22; r) explicitly described in Lemma 4.2 of [6]. Composition of this map with the representation S(22; r)! End K (A(22; r)) given by left action of S(22; r) on A(22; r) gives a left action of Dist(G) on A(22; r). Under this action, the generators e i, e (t) i and eii t, respectively, of Dist(G) correspond to i D, i D (t) and iid t, respectively, where id (t) = idt t! and iid t = iid( iid 1):::( iid t+1) t! - for more details see section 4 of [6]. Therefore the action of S(22; r) on A(22; r) is completely determined by right superderivations i D. While doing computations, we extend the superderivations id of A(22) to superderivations of A(22) d = K[G]. Since the simple module L S(22) () is included in the costandard module r S(22) (), which in turn is included in A(22; r) (by Proposition 3.1 of [6]), we conclude that the action of S(22; r) on L S(22) () is completely determined using the action of superderivations i D. Let G ev ' GL(2) GL(2) be an even supersubgroup of G, Lie(G ev ) ' gl(2) gl(2) be the corresponding Lie algebra of G ev and S = sl(2) sl(2). Let B be the lower triangular Borel subsupergroup of G. Fix a dominant weight = ( 1 ; 2 3 ; 4 ) of G, that is 1 2 and 3 4. Following [8], we denote by HG 0 () the induced G-module H 0 (G=B; K ), where K is the one-dimensional (even) B- supermodule corresponding to the weight. Finally, using the restriction of B to G ev de ne by HG 0 ev () the induced G ev -module corresponding to the weight. The induced G ev -module HG 0 ev (), denoted by V, can identi ed with the subspace of superalgebra K[c 11 ; c 12 ; c 21 ; c 22 ; c 33 ; c 34 ; c 43 ; c 44 ], generated by polynomials d 2 12 ca 1 2 a 11c12 d 4 34 cb 3 4 b 33c34 ; where d = d 12 = c 11 c 22 c 12 c 21, d 34 = c 33 c 44 c 34 c 43 and 0 a 1 2, 0 b 3 4. The induced G-supermodule HG 0 () can be described explicitly using the isomorphism : HG 0 ev ()K[c 13 ; c 14 ; c 23 ; c 24 ]! HG 0 () de ned in Lemma 5.2 and on p.163 of [8]. This isomorphism is given by (d 12 ) = d 12 ; (c 11 ) = c 11 ; (c 12 ) = c 12 ; and (c 13 ) = c 22c 13 c 12 c 23 d = y 13 ; (c 14 ) = c 22c 14 c 12 c 24 d = y 14 ; (c 23 ) = c 21c 13 + c 11 c 23 = y 23 ; (c 24 ) = c 21c 14 + c 11 c 24 = y 24 ; d d (c 33 ) = c 33 c 31 y 13 c 32 y 23 = z 1 ; (c 34 ) = c 34 c 31 y 14 c 32 y 24 = z 2 (d 34 ) = (c 33 c 31 y 13 c 32 y 23 )(c 44 c 41 y 14 c 42 y 24 ) (c 34 c 31 y 14 c 32 y 24 )(c 43 c 41 y 13 c 42 y 23 ) = x:

3 SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(22) 3 Then the supermodule HG 0 () has a basis w(a; b; 13 ; 14 ; 23 ; 24 ) = d 2 c a 1 2 a 11c12 x 4 z1z b 3 4 b 2 y y14 y23 y24 with a; b as before and 13 ; 14 ; 23 ; 24 2 f0; 1g: The weight of w(a; b; 13 ; 14 ; 23 ; 24 ) is ( 2 + a ; 1 a b ; 3 b ): We shall write v a;b for w(a; b; 0; 0; 0; 0) and v = v 1 2; 3 4. Then v is the highest vector of the simple G-module L(). Outline of the paper. We will now explain the basic approach of the paper. For analogous results for S(21) and S(31) see [4] and [7]. The basis of our investigation of the description of the G ev -module structure of HG 0 () and its simple submodule L(). Although it would be natural to describe the G ev -module structure using the Lie algebra Lie(G ev ) ' gl(2)gl(2), it is easier to work with modules over the Lie algebra S = sl(2) sl(2) since the S-weights are described by only two parameters (instead of four for Lie(G ev )). The structure of G ev -modules can be easily retrieved once their S-module structure is known. One advantage of this approach is exhibited in section 4.1, where we use a certain isomorphism V of S-modules. If the characteristic p of the ground eld K is bigger than two, then using the Steinberg Theorem (see Theorem 4.4 of [5]), it is enough to determine the structure of the simple S(22)-module L S(22) () for restricted, that is when 1 2 ; 3 4 < p. If is restricted, then the action of even elements e (pr ) i 2 Dist(G) for r > 0 on HG 0 () is trivial. Since the G ev-structure of Hev() 0 is known, the G-structure of HG 0 () is then determined completely by the action of superderivations id. In Section 1 we compute the action of superderivations i D on elements in HG 0 ev () and on elements (X 12 ). Furthermore, we de ne the concept of atypicality of the weight (extending the clasical de nition of Kac from characteristic zero case). The module HG 0 () decomposes into a direct sum of S-submodules F 0() F 1 ()F 2 ()F 3 ()F 4 (), where the submodule F k (), that will be called the k- oor, is given as a K-span of all vectors w(a; b; 13 ; 14 ; 23 ; 24 ), where P 4 i=0 i4 = k. Equivalently, F k () is spanned by vectors of weights = ( 1 ; 2 3 ; 4 ) such that k = = Clearly each F i () is a S-module. Denote by Y a four-dimensional S-module spanned by elements y 13 ; y 14 ; y 23 and y 24. Then F 0 () = V,F 1 () = V Y, F 2 () = V (Y ^ Y ), F 3 () = V (Y ^ Y ^ Y ) and F 4 () = V (Y ^ Y ^ Y ^ Y ). The complete description of the S-module structure of each oor F i will be carried out in sections 2 through 5. In order to describe L S(22) (), we will use PBW theorem, and corresponding to our choice of the Borel subsupergroup B, we order the generators of Dist(G) as follows: e (r) i for i < rst, followed by eii (r) r and e i for i >, and then by odd e i where i >. The simple module L S(22) () is generated by the vector v. The elements e (r) e i 2 Dist(G) for i < act trivially on v and elements ii (r) r and e i for i > applied to v generate the module V. Using the previously discussed action of Dist(G) on A(22; r) we conclude that L S(22) () is generated by V and its images under compositions of superderivations i D for i <. We can identify these images with elements of various oors F k. Since the superderivations i D supercommute,

4 4 A.N. GRISHKOV AND F. MARKO the maps 1 : F 1 ()! F 1 () given by 2 : F 2 ()! F 2 () given by 3 : F 3 ()! F 3 () given by v a;b y i 7! (v a;b ) i D; v a;b (y i1 1 ^ y i2 2 ) 7! (v a;b ) i1 1 D i2 2 D; v a;b (y i1 1 ^ y i2 2 ^ y i3 3 ) 7! (v a;b ) i1 1 D i2 2 D i33 D; and 4 : F 4 ()! F 4 () given by v a;b (y 13 ^ y 23 ^ y 14 ^ y 24 ) 7! (v a;b ) 13 D 23 D 14 D 24 D are well de ned. It is easy to check that they are S-morphisms. We will compute images 1 (F 1 ), 2 (F 2 ), 3 (F 3 ) and 4 (F 4 ) in sections 2 through 5. These images together with V constitute the whole module L S(22) (). In each section 2 through 5 we follow this procedure: We determine primitive vectors in characteristics zero rst, then we establish the S-module structure of each oor. Special care is taken in the cases when either 1 2 or 3 4 is equal to p 2 or p 1, since in those cases HG 0 () is not semisimple as an S-module. Afterwards we compute the S-module structure of the image k (F k ()). Finally, in section 6, we combine the results of preceeding sections and determine the character and dimension of the simple module L S(22) (). 1. Basic formulas 1.1. Basic formulas for S(22). It is clear that V = H 0 G ev () = L() is an irreducible S-module if is restricted. Lemma 1.1. The actions of superderivations 12 D; 21 D; 13 D; 14 D; 23 D; 24 D; 34 D and 43 D on elements d; x; c 11 ; c 12 ; y 13 ; y 23 ; y 14 ; y 24 ; z 1 and z 2 is given in the following table. 12D 21 D 13D 14D 23D 24D 34D 43 D d 0 0 dy 13 dy 14 dy 23 dy x 0 0 xy 13 xy 14 xy 23 xy c 11 c 12 0; c 11 y 13 + c 12 y 23 c 11 y 14 + c 12 y c 12 0 c c 11 y 13 + c 12 y 23 c 11 y 14 + c 12 y y 13 0 y 23 0 y 13 y 14 y 13 y 23 y 23 y 14 y 14 0 y 23 y 13 0 y 23 y 13 y 13 y 24 0 y 23 y 24 y 24 0 y 14 0 y 24 y 14 y 13 0 y 13 y 24 y 14 y 24 0 y 13 y 24 y 14 0 y 23 y 14 y 24 y 14 y 24 y y 23 z z 1 y 13 z 2 y 13 z 1 y 23 z 2 y 23 z 2 0 z z 1 y 14 z 2 y 14 z 1 y 24 z 2 y 24 0 z 1 Proof. It is straightforward computation using the properties (c kl ) i D = li c k, where li is the Kronecker delta, and (ab) i D = ( 1) idb (a) i Db+a(b) i D, where the symbol denotes the parity. As a consequence, we obtain the following fundamental formulas.

5 SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(22) 5 Lemma 1.2. (v a;b ) 12D = av a 1;b ; (v a;b ) 21D = ( 1 2 a)v a+1;b ; (v a;b ) 13D = ( b + a)v a;b y 13 + av a 1;b y 23 + ( 3 4 b)v a;b+1 y 14; (v a;b ) 14D = ( b + a)v a;b y 14 + av a 1;b y 24 + bv a;b 1 y 13; (v a;b ) 23D = ( b a)v a;b y 23 + ( 1 2 a)v a+1;b y 13 + ( 3 4 b)v a;b+1 y 24; (v a;b ) 24D = ( b a)v a;b y 24 + ( 1 2 a)v a+1;b y 14 + bv a;b 1 y 23; (v a;b ) 34D = bv a;b 1 (v a;b ) 43D = ( 3 4 b)v a;b+1 : Proof. It follows by repeated application of Lemma 1.1. Other identities of interest are dy 13 y 23 = c 13 c 23 ; dy 14 y 24 = c 14 c 24 ; d(y 13 y 24 + y 14 y 23 ) = c 13 c 24 + c 14 c 23 : 1.2. Further notation. The simple S-module of the highest weight and of the highest vector w shall be denoted either by L() or by L(w) depending on the circumstances. Denote 1 2 = A and 3 4 = B. Further denote! 12 = 1 2,! 34 = 3 4,! 13 = ,! 14 = 1 + 4,! 23 = and! 24 = If p = 0, then we shall write i = 0 if! i = 0 and i = 1 otherwise, and 1 i = 1 if! i = 1 and 1 i = 1 otherwise. If p > 2, then we denote i = 0 if! i 0 (mod p) and i = 1 otherwise, and 1 i = 0 if! i 1 (mod p) and 1 i = 1 otherwise. De nition 1.3. A weight is called typical if = 1, is called 13-atypical if 13 = 0 but = 1, is called 14-atypical if 14 = 0 but = 1, is called 23-atypical if 23 = 0 but = 1, is called 24-atypical if 24 = 0 but = 1, is called (13,14)-atypical if 13 = 14 = 0 but = 1, is called (13,23)-atypical if 13 = 23 = 0 but = 1, is called (13,24)-atypical if 13 = 24 = 0 but = 1, is called (14,23)-atypical if 14 = 23 = 0 but = 1, is called (14,24)-atypical if 14 = 24 = 0 but = 1, is called (23,24)-atypical if 23 = 24 = 0 but = 1, is called (13,14,23,24)-atypical if 13 = 14 = 23 = 24 = 0. It is easy to see that if p = 0, then every dominant weight is either typical, 14-atypical, 23-atypical, 24-atypical or (14,23)-atypical. If p > 2, then weights of all above atypical types are possible. In this case observe the following. If is 13-atypical, then B 6 p 1 (mod p). If is 23-atypical, then B 6 p 1 (mod p) and A 6 p 1 (mod p). If is 24-atypical, then A 6 p 1 (mod p). If is (13,14)-atypical, then B p 1 (mod p) and A 6 p 1 (mod p). If is (13,23)-atypical, then A p 1 (mod p) and B 6 p 1 (mod p). If is (14,24)-atypical, then A p 1 (mod p) and B 6 p 1 (mod p). If is (23,24)- atypical, then B p 1 (mod p) and A 6 p 1 (mod p). If is (13,24)-atypical, then A + B p 2 (mod p) and A; B 6 p 1 (mod p). If is (14,23)-atypical,

6 6 A.N. GRISHKOV AND F. MARKO then A B (mod p) and A; B 6 p 1 (mod p). If is (13,14,23,24)-atypical, then A; B p 1 (mod p). Furthermore, denote v w if and only if both v; w 6= 0 and one of them is a constant multiple of the other one. 2. First floor 2.1. Characteristic zero. In order to describe V Y as an S-module consider the following elements: l 1 = v AB y 23, l 2 = v AB y 24 v A;B 1 y 23, l 3 = v AB y 13 + v A 1;B y 23, l 4 = v AB y 14 + v A 1;B y 24 v A;B 1 y 13 v A 1;B 1 y 23. Lemma 2.1. The module V Y is isomorphic to the direct sum L(l 1 ) 34 L(l 2 ) 12 L(l 3 ) L(l 4 ). Proof. The vectors l 1 ; l 2 ; l 3 and l 4 are primitive vectors. A dimension count completes the argument. The image of the rst oor under the action of superderivations is given in the following Proposition. Proposition 2.2. Let 1 : V Y! V Y be a morphism of S-modules given by vy i 7! (v) i D. Then the image 1 (V Y ) = 23 L(l 1 ) L(l 2 ) L(l 3 ) L(l 4 ). Proof. We compute 1 (l 1 ) =! 23 l 1, 1 (l 2 ) =! 24 l 2, 1 (l 3 ) =! 13 l 3, 1 (l 4 ) =! 14 l 4, and the claim follows Characteristic p. Assume that the weight is restricted. The question of describing the S-module structure of V Y is related to a classical problem of decomposing the tensor product of a simple module and the natural or dual of the natural module over the general linear group GL(n). These questions were studied in [2] in relation to a complete reducibility criterion, primitive vectors and socles of the tensor products of the above type. In particular, costandard ltrations of these modules were described on p. 88 of [2]. We will only need a description of the tensor product of the simple module with the natural module for G + = GL(2). Actually, we will only determine its explicit structure over S + = sl(2). Assign to a G + -weight + = ( 1 ; 2 ) its corresponding restricted S + -weight A = 1 2 < p. Let V + be a simple S + -module generated by an element v + A of highest S+ - weight A. Then the dimension of V + is A + 1 and V + is a span of vectors v + A i for 0 i A such that (v + A i ) 12D = (A i)v + A i 1, (v+ A i ) 21D = iv + A i+1, and 12D (pk) and 21 D (pk) for k 1 act trivially. Denote by W + the two-dimensional S + -module which is a K-span of elements w + 1 and w + 1 for which (w + 1 ) 21D = 0; (w + 1 ) 21D = w + 1, (w+ 1 ) 12D = w + 1, (w + 1 ) 12D = 0 and 12 D (pk) and 21 D (pk) for k 1 act trivially. The following lemma describes the S + -module structure of V + W +. Although it is a classical result, we include it here for the convenience of the reader.

7 SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(22) 7 Lemma 2.3. The S + -module structure of the module V + W + is given as follows. If A = 0, then V + W + = U 1 +, where U 1 + = hu + 1 = v+ A w+ 1 i, is a simple S + -module. If 0 < A < p 1, then V + W + = U 1 + U 2 +, where U 1 + is as above and U 2 + = hu+ 2 = v+ A 1 w+ 1 v + A w+ 1 i, is a simple S+ -module. If A = p 1, then V + W + has a composition series U + 3 V + W + = U + 1 ; U + 2 where U + 1, U + 2 as above, and U + 3 = hu+ 3 = v+ A w+ 1 i is a simple S+ -module. Proof. Since (u + 1 ) 21D = 0, u + 1 is a primitive vector of the highest weight A + 1. If A = 0, then dimensions of V + W + and U 1 + are both equal to 2. If A > 0, then (u + 2 ) 21D = 0 shows that u + 2 is a primitive vector of the highest weight A 1 and dimension A. Assume 0 < A < p 1. Then the dimension of U 1 + is A+2, and the dimensions of U 1 + and U 2 + add up to the dimension of V + W +. Since (u + 1 ) 12D 6 u + 2, Ext1 S (U ; U 2 + ) = 0, and the S+ -module structure of V + W + follows. Assume now A = p 1. Then (u + 3 ) 21D = u + 1 shows that u+ 3 is a primitive vector of weight A 1 and dimension p 2. The vector u + 1 is primitive of weight p and L(u + 1 ) has dimension 2 (and is spanned by u+ 1 and (u+ 1 ) 12D (p) ). Since u + 2 is a primitive vector of weight A 1 and dimension p 2, dimensions of U 1 +, U 2 + and U 3 + add up to the dimension of V + W +. Finally, (u + 1 ) 12D = u + 2 implies that the S + -module structure of V + W + is as stated. The even supergroup G ev of G is a product of two copies of GL(2), the rst copy (based on letter 1 and 2) can be be identi ed with G + and we can denote the second copy (based on letters 3 and 4) by G. Assign to a G -weight = ( 3 ; 4 ) its corresponding restricted S = sl(2)-weight B = 3 4 < p. Then we can de ne V and W analogously to V + and W + and obtain the following analogous result for the S -module structure of V W. Lemma 2.4. The S -module structure of the module V W is given as follows. If B = 0, then V W = U 1, where U 1 = hu 1 = v B w 1 i, is a simple S -module. If 0 < B < p 1, then V W = U 1 U 2, where U 1 as above and U 2 = hu 2 = v B 1 w 1 v B w 1 i, is a simple S -module. If B = p 1, then V W has a composition series U 3 V W = U 1 ; where U 1, U 2 as above, and U 3 = hu 3 = v B w 1 i is a simple S -module. Now we are ready to describe the S-module structure of the rst oor. U 2

8 8 A.N. GRISHKOV AND F. MARKO Proposition 2.5. The S-module V Y is described as follows. 1) If A; B < p 1, then V Y = L(l 1 ) 34 L(l 2 ) 12 L(l 3 ) L(l 4 ). 2) If A = p 1 and B < p 1, then V Y = L 5 34 L 6. Here the indecomposable module L 5 is given as L(l 5 ) L 5 = L(l 1 ) ; L(l 3 ) where l 5 = v A;B y 13. The indecomposable module L 6 given as L(l 6 ) L 6 = L(l 2 ) L(l 4 ) where l 6 = v A;B 1 y 13 + v A;B y 14. 3) If A < p 1 and B = p 1, then V Y = L 7 12 L 8. Here the indecomposable module L 7 is given as L(l 7 ) L 7 = L(l 1 ) ; L(l 2 ) where l 7 = v A;B y 24. The indecomposable module module L 8 given as L(l 8 ) L 8 = L(l 3 ) L(l 4 ) where l 8 = v A 1;B y 24 + v A;B y 14. 4) If A = B = p 1, then V Y = L 9 has the composition series L(l 9 ) L(l 5 ) L(l 7 ) L(l 6 ) L(l 1 ) L(l 8 ) L(l 2 ) L(l 3 ) L(l 4 ) of simple S-modules, where l 9 = v A;B y 14. Proof. There is an isomorphism of S-modules W + W and Y which sends w 1 + w 1 7! y 23, w 1 + w 1 7! y 24, w + 1 w 1 7! y 13 and w + 1 w 1 7! y 14. For an S-module V of highest weight (A; B) there is an isomorphism V = V + V, where V + and V are de ned as above. ; ;

9 SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(22) 9 The claim follows from Lemmas 2.3 and 2.4 using the standard properties of tensor products Image under 1. The structure of the S-module 1 (V Y ) is given as follows. Proposition 2.6. The following statements describe S-modules isomorphic to V 1 = 1 (V Y ). If A; B < p 1, then V 1 = 23 L(l 1 ) L(l 2 ) L(l 3 ) L(l 4 ). Assume A = p 1 and B < p 1. If is typical, then V 1 = V Y. If is (13; 23)- atypical, then V 1 = L(l3 ) 34 L 6. If is (14; 24)-atypical, then V 1 = L5 34 L(l 4 ). Assume A < p 1 and B = p 1. If is typical, then V 1 = V Y. If is (13; 14)- atypical, then V 1 = L7 12 L(l 4 ). If is (23; 24)-atypical, then V 1 = L(l2 ) 12 L 8. Assume A = B = p 1. If typical, then V 1 = V Y. If is (13; 14; 23; 24)- atypical, then V 1 = L( l 6 + l 8 ) L 10 = L(l 2 ) L(l 3 ) : L(l 4 ) Proof. Assume rst that A; B < p 1. The images of generators l 1 ; l 2 ; l 3 and l 4 of V Y were determined earlier. The image V 1 = 23 L(l 1 ) L(l 2 ) L(l 3 ) L(l 4 ) as in the characteristic zero case. Next, assume that A = p 1 and B < p 1. The images of additional primitive vectors under 1 equal 1 (l 5 ) =! 13 l 5 l 3 and 1 (l 6 ) =! 14 l 6 l 4. The structure of V 1 follows. Now assume that A < p 1 and B = p 1. Then the images of additional primitive vectors under 1 equal 1 (l 7 ) =! 24 l 7 + l 2 and 1 (l 8 ) =! 14 l 8 + l 4. The structure of V 1 follows. Finally, assume that A = B = p 1. We compute rst the image of the generator l 9 under 1 as 1 (l 9 ) =! 14 l 9 l 8 + l 6. If is typical, then 1 (V Y ) = V Y. If is (13; 14; 23; 24)-atypical, then 1 (l 5 ) = l 3, 1 (l 7 ) = l 2, 1 (l 6 ) = l 4, 1 (l 8 ) = l 4, and the images of the remaining elements l i vanish. In this case V 1 is an indecomposable module generated by l 6 + l 8 = v A;B 1 y 13 + v A 1;B y 24 that has the structure described in the statement of the proposition. 3. Second floor 3.1. Characteristic zero. The S-module Y ^ Y is a direct sum of 2 irreducible modules Y 1 = hy 23 ^ y 24 i and Y 2 = hy 23 ^ y 13 i. Consider the following elements: m 1 = v AB y 23 ^ y 24, m 2 = 2v A 1;B y 23 ^ y 24 v AB y 13 ^ y 24 + v AB y 14 ^ y 23, m 3 = v A 2;B y 23 ^y 24 +v A 1;B y 13 ^y 24 v A 1;B y 14 ^y 23 +v AB y 13 ^y 14, n 1 = v AB y 13 ^ y 23, n 2 = 2v A;B 1 y 13 ^ y 23 v AB y 13 ^ y 24 v AB y 14 ^ y 23, and n 3 = v A;B 2 y 13 ^y 23 v A;B 1 y 13 ^y 24 v A;B 1 y 14 ^y 23 +v AB y 14 ^y 24. Lemma 3.1. The module V (Y ^ Y ) is isomorphic to the direct sum L(m 1 ) L(n 1 ) 12 L(m 2 ) 34 L(n 2 ) L(m 3 ) L(n 3 ).

10 10 A.N. GRISHKOV AND F. MARKO Proof. The action of 34 D; 43 D on V Y 1 is given by (v a;b y 1 ) 34 D = bv a;b 1 y 1 and (v a;b y 1 ) 43 D = ( 3 4 b)v a;b+1 y 1 for any y 1 2 Y 1. The action of 12D; 21 D on V Y 2 is given by (v a;b y 2 ) 12 D = av a 1;b y 2 and (v a;b y 2 ) 21 D = ( 1 2 a)v a+1;b y 2 for any y 2 2 Y 2. Since the vectors m 1, m 2 and m 3 are primitive, the dimension count gives that the module V Y 1 is a direct sum of simple modules L(m 1 ) 12 L(m 2 ) L(m 3 ). Analogously, since the vectors n 1, n 2 and n 3 are primitive, the dimension count gives that the module V Y 2 is a direct sum of simple modules L(n 1 ) 34 L(n 2 ) L(n 3 ). Therefore V (Y ^ Y ) = L(m 1 ) L(n 1 ) 12 L(m 2 ) 34 L(n 2 ) L(m 3 ) L(n 3 ). If t; s > 0, then L(m 2 ) = L(n 2 ) = L( 13;24 ). We shall denote the simple module L( 13;24 ) by L. Lemma 3.2. The S-morphism 2 is described completely by images of generating vectors 2 (m 1 ) =! 23! 24 m 1 ; 2 (n 1 ) =! 13! 23 n 1 ; 2 (m 2 ) = ( )( ) + (1 2 (n 2 ) = 2 2 (m 3 ) =! 13! 14 m 3 ; m 2 + ( )( ) 2 (n 3 ) =! 14! 24 n 3. (1 2)(2+3 4) Proof. It is a straightforward computation. 2)(22+3+4) ( 3 4)(2+ 1 2) 2 m2 ( 3 4)( ) 2 2 n 2 ; n2 ; Lemma 3.3. The image 2 (m 2 ) vanishes if and only if! 23! 13 = 0 and s = 0. The image 2 (n 2 ) vanishes if and only if! 23! 24 = 0 and t = 0. If s; t > 0 and! 13! 14! 23! 24 = 0, then 2 (m 2 ) 2 (n 2 ). Proof. Lemma 3.2 implies that 2 (m 2 ) = 0 implies s = 0, and 2 (n 2 ) = 0 implies t = 0. It is easy to verify that 2 (m 2 ) =! 23! 13 m 2 for s = 0 and 2 (n 2 ) =! 23! 24 m 2 for t = 0. If s; t > 0, then the restriction ~ 2 of the map 2 to the 2-dimensional space of primitive vectors hm 2 ; n 2 i of weight, is represented by a matrix! ( )( ) + ( 1 2 )( ) ( 1 2 )( ) 2 2 ( 3 4 )( ) ( )( ) The determinant of this matrix is! 13! 14! 23! 24 and our claim follows. ( 3 4 )( ) 2 Proposition 3.4. The image 2 (V Y ^ Y ) = L(m 1 ) L(n 1 ) L(m 3 ) L(n 3 ) Z, where Z = L L if! 13! 14! 23! 24 6= 0; Z = L if! 13! 14! 23! 24 = 0 and one of the following conditions is satis ed: - s; t > 0, - t = 0; s > 0 and! 23! 24 6= 0, and - s = 0; t > 0 and! 23! 13 6= 0; and Z = 0 if one of the following conditions is satis ed: - t = 0; s > 0 and! 23! 24 = 0, - s = 0; t > 0 and! 23! 13 = 0, and - s = t = 0 Proof. It follows from Lemma 3.2 and 3.3. :

11 SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(22) Characteristic p. Assume that the weight is restricted. As in the case of characteristic zero, we also have that the S-module Y ^ Y is a direct sum of 2 irreducible S-modules Y 1 = hy 23 ^ y 24 i and Y 2 = hy 23 ^ y 13 i. We shall determine the S-module structures of V Y 1 and V Y 2 rst and then combine them. Since S acts trivially on V Y 1 and S + acts trivially on V Y 2, instead of S-modules we shall deal with appropriate sl(2)-modules and the computations shall become easier Module V Y 1. De ne the elements m 4 = v A;B (y 13 ^ y 24 y 14 ^ y 13 ) and m 5 = 1 2 v A 2;B (y 23 ^ y 24 ) + v A;B (y 13 ^ y 14 ). Lemma 3.5. The S-module of the module V Y 1 is given as follows. 1) If A < p 2, then V Y 1 = L(m1 ) 12 L(m 2 ) L(m 3 ). 2) If A = p 2, then V Y 1 = M4 1 12L(m 3 ), where the S-module M 4 has the composition series L(m 4 ) M 4 = L(m 1 ) : L(m 2 ) 3) If A = p 1, then V Y 1 = M5 L(m 2 ), where the S-module M 5 has the composition series L(m 5 ) M 5 = L(m 1 ) : L(m 3 ) Proof. Since (m 1 ) 21 D = (m 2 ) 21 D = (m 3 ) 21 D = 0, we obtain that the vectors m 1, m 2 and m 3 are primitive, of highest S + -weights A + 2,A and A 2 respectively. To nd possible extension between L(m 1 ), L(m 2 ) and L(m 3 ), we compute that (m 2 ) 12 D 6 m 3 ; (m 1 ) 12 D m 2 if and only if A = p 2; and (m 1 ) 12 D 2 m 3 if and only if A = p 1. If A < p 2, the dimensions of modules L(m 1 ), L(m 2 ) and L(m 3 ) are (A + 3)(B + 1), (A + 1)(B + 1) and (A 1)(B + 1), respectively. Since (A + 3) + 12 (A + 1) (A 1) = 3(A + 1), the dimension of the module L(m 1 ) 12 L(m 2 ) L(m 3 ) is the same as the dimension of V Y 1. Since there are no extensions between L(m 1 ), L(m 2 ) and L(m 3 ), part 1) follows. If A = p 2, then (m 4 ) 21 D = 2m 1 shows that the vector m 4 is a primitive vector of the highest S + -weight A. Since L(m 1 ) has dimension 2(B + 1), L(m 2 ) and L(m 4 ) have dimensions (p 1)(B +1), and if A 6= 1, then L(m 3 ) has dimension (p 3)(B + 1); the dimensions of these modules add up to the dimension of V Y 1. Since (m 4 ) 21 D = 2m 1 and (m 1 ) 12 D = m 2, and (m 4 ) 12 D 6 m 3 and (m 2 ) 12 D 6 m 3, we infer the S-module structure of the module V Y 1. If A = p 1, then (m 5 ) 21 D 2 = m 1 shows that the vector m 5 is a primitive vector of the highest S + -weight A 2. Since the dimensions of the modules L(m 1 ), L(m 2 ), L(m 3 ) and L(m 5 ) are 4(B + 1), p(b + 1), (p 2)(B + 1) and (p 2)(B + 1), respectively, they add up to the dimension of V Y 1. Since (m 5 ) 21 D 2 = m 1,

12 12 A.N. GRISHKOV AND F. MARKO (m 1 ) 12 D 2 = 2m 3, and (m 1 ) 12 D 6 m 2, we infer the S-submodule structure of the module V Y Module V Y 2. De ne the elements n 4 = v A;B (y 13 ^ y 24 + y 14 ^ y 23 ) and n 5 = 1 2 v A;B 2 y 13 ^ y 23 + v A;B y 14 ^ y 24. Lemma 3.6. The S-module of the module V Y 2 is given as follows. 1) If B < p 2, then V Y 2 = L(n1 ) 34 L(n 2 ) L(n 3 ). 2) If B = p 2, then V Y 2 = N4 1 34L(n 3 ), where the S-module N 4 has the composition series L(n 4 ) N 4 = L(n 1 ) : L(n 2 ) 3) If B = p 1, then V Y 2 = N5 L(n 2 ), where the S-module N 5 has the composition series L(n 5 ) N 5 = L(n 1 ) L(n 3 ) Proof. It is symmetric to the proof of Lemma 3.5. : V (Y ^ Y ). Combining the descriptions of V Y 1 and V Y 2 we get Proposition 3.7. The S-module V (Y ^ Y ) is isomorphic to 1) Case 0 < A; B < p 2 L(m 1 ) L(n 1 ) L(m 2 ) L(n 2 ) 1 12L(m 3 ) 1 34L(n 3 ) 2) Case A = 0; 0 < B < p 2 L(m 1 ) L(n 1 ) L(n 2 ) 1 34L(n 3 ) 3) Case 0 < A < p 2; B = 0 L(m 1 ) L(n 1 ) L(m 2 ) 1 12L(m 3 ) 4) Case A = B = 0 L(m 1 ) L(n 1 ) 5) Case A = p 2; 0 < B < p 2 M 4 L(n 1 ) L(n 2 ) 1 12L(m 3 ) 1 34L(n 3 ) 6) Case A = p 2; B = 0 M 4 L(n 1 ) 1 12L(m 3 ) 7) Case 0 < A < p 2; B = p 2 L(m 1 ) N 4 L(m 2 ) 1 12L(m 3 ) 1 34L(n 3 ) 8) Case A = 0; B = p 2 L(m 1 ) N L(n 3 ) 9) Case A = B = p 2 M 4 N L(m 3 ) 1 34L(n 3 ) 10) Case A = p 1; 0 < B < p 2 M 5 L(n 1 ) L(m 2 ) L(n 2 ) 1 34L(n 3 )

13 SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(22) 13 11) Case A = p 1; B = 0 M 5 L(n 1 ) L(m 2 ) 12) Case 0 < A < p 2; B = p 1 L(m 1 ) N 5 L(m 2 ) L(n 2 ) 1 12L(m 3 ) 13) Case A = 0; B = p 1 L(m 1 ) N 5 L(n 2 ) 14) Case A = p 1; B = p 2 M 5 N 4 L(m 2 ) 1 34L(n 3 ) 15) Case A = p 2; B = p 1 M 4 N 5 L(n 2 ) 1 12L(m 3 ) 16) Case A = B = p 1 M 5 N 5 L(m 2 ) L(n 2 ) Proof. Combine Lemmas 3.5 and 3.6. The reason why we split the above Proposition into 16 cases instead of 9 is to prepare for its application in the next subsection Image under 2. We shall analyze the modules 2 (V Y 1 ) and 2 (V Y 2 ) rst, and then determine 2 (V (Y ^ Y )). The starting point is Lemma 3.2 which holds in positive characteristic as well. Lemma 3.3 is modi ed as follows. Lemma 3.8. If! 23! 13 = 0 and B = 0, then 2 (m 2 ) = 0. If! 23! 13 = 0 and B = p 2, then 2 (n 2 ) = 0. If! 23! 24 = 0 and A = 0, then 2 (n 2 ) = 0. If! 23! 24 = 0 and A = p 2, then 2 (m 2 ) = 0. In all other cases, when de ned, the images 2 (m 2 ) and 2 (n 2 ) are nonzero, and if A; B > 0 and! 13! 14! 23! 24 = 0, then 2 (m 2 ) 2 (n 2 ). Proof. Lemma 3.2 shows that 2 (m 2 ) = 0 implies A = p 2 or B = 0, and 2 (n 2 ) = 0 implies A = 0 or B = p 2. It is easy to verify that 2 (m 2 ) =! 23! 13 m 2 for B = 0, 2 (n 2 ) =! 23! 13 n 2 for B = p 2, 2 (n 2 ) =! 23! 24 m 2 for A = 0, and 2 (m 2 ) =! 23! 24 m 2 for A = p 2. The remaining arguments are as in Lemma 3.3. For computation of 2 (V Y 1 ) and 2 (V Y 2 ) we shall use Lemmas 3.2 and 3.8 repeatedly. To combine both 2 (V Y 1 ) and 2 (V Y 2 ), we shall need the following lemma. Denote a K-span of 2 (m 2 ) and 2 (n 2 ) by X. Lemma 3.9. The dimension of the space X is described as follows: dimx = 2 if and only if! 13! 14! 23! 24 6= 0; dimx = 1 if and only if! 13! 14! 23! 24 = 0 and one of the following conditions is satis ed: - A; B > 0, - A = 0; B > 0 and! 23! 24 6= 0, and - A > 0; B = 0 and! 23! 13 6= 0; and dimx = 0 if and only if one of the following conditions is satis ed: - A = 0; B > 0 and! 23! 24 = 0, - A > 0; B = 0 and! 23! 13 = 0, and - A = B = 0 Proof. Use Lemmas 3.2 and 3.8.

14 14 A.N. GRISHKOV AND F. MARKO If 2 (M) = M and L( 2 (w)) = L(w), respectively, then we shall denote M = 2 (M) and w = 2 (w), respectively Module 2 (V Y 1 ). We shall need the following lemma, a part of which shall be useful for the determination of 2 (V (Y ^ Y )). Lemma If A = p 2 and! 24 = 0, then 2 (m 4 ) = 2 (n 2 ) 6= 0. Assume A = p 2 and! 23 = 0. If B 6= 0; p 2, then 2 (m 4 ) 2 (n 2 ). If B = 0, then 2 (m 4 ) = 0 and 2 (n 2 ) 6= 0. If B = p 2, then 2 (m 4 ) 6= 0 and 2 (n 2 ) = 0. Proof. If! 24 = 0, then 2 (m 4 ) = (2 + B)m 2 Bn 2 = 2 (n 2 ). If! 23 = 0, then 2 (m 4 ) = B(m 2 + n 2 ) and 2 (n 2 ) = (B + 2)(m 2 + n 2 ). The structure of the S-module 2 (V Y 1 ) is given as follows. Proposition The following statements describe S-modules isomorphic to V 21 = 2 (V Y 1 ). Assume A < p 2. If is typical, then V 21 = L(m1 ) 12 L(m 2 ) L(m 3 ). If is 13-atypical, then V 21 = L(m1 ) L(m 2 ). If is 14- or (13; 14)-atypical, then V 21 = L(m1 ) 12 L(m 2 ). If is 23-atypical, then V 21 = L(m 2 ) L(m 3 ). If is 24- or (23; 24)-atypical, then V 21 = 12 L(m 2 ) L(m 3 ). If is (13; 24)-atypical or (14; 23)-atypical, then V 21 = L(m 2 ). Assume now A = p 2. If is typical, then V 21 = M L(m 3 ). If is 13-, 14- or (13; 14)-atypical, then V 21 = M 4. If is 24- or (23; 24)-atypical, then V 21 = L((B + 2)m2 Bn 2 ) 1 12L(m 3 ). If is (13; 24)-atypical, then V 21 = L((B + 2)m2 Bn 2 ). If is 23-atypical, then V 21 = 34 L(m 2 + n 2 ) 1 12L(m 3 ). If is (14; 23)-atypical, then V 21 = L(m2 + n 2 ). Finally, assume A = p 1. If is typical, then V 21 = M 5 L(m 2 ). If is (13; 23)-atypical and B 6= 0, then V 21 = L(m5 ) L(m 2 ). If is (13; 23)-atypical and B = 0, then V 21 = L(m5 ). If is (14; 24)-atypical, then V 21 = L(m5 ) L(m 2 ). If is (13; 14; 23; 24)-atypical, then V 21 = L(m2 ). Proof. If A < p 2, then V Y 1 is semisimple by the rst part of Proposition 3.5 and the highest weights of primitive vectors are pairwise di erent. Therefore it is enough to determine whether the images of these primitive vectors under 2 vanish or not, and this follows from Lemma 3.2 and 3.8. For A = p 2, we use the second part of Proposition 3.5. Since L(m 2 ) is the S-socle of M 4, Lemma 3.8 shows that 2 (M 4 ) = M 4 provided! 23! 24 6= 0. If! 23! 24 = 0, then using Lemma 3.2 we infer 2 (m 1 ) = 0 and therefore 2 (M 4 ) = 2 (L(m 4 )). Lemma 3.10 determines 2 (m 4 ) and Lemma 3.2 concludes the proof in the case A = p 2. Assume now that A = p 1. If is typical, then Lemma 3.2 shows that 2 (V 21 ) = V 21. Assume now is not typical. Then! 13 =! 23 and! 14 =! 24 imply that! 23! 24 = 0. Since 2 (m 1 ) =! 23! 24 m 1, using the third part of Proposition 3.5 we

15 SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(22) 15 conclude 2 (M 5 ) = 2 (L(m 5 )). We verify that! 23 = 0 implies 2 (m 5 ) = (B+1)m 3 and! 24 = 0 implies 2 (m 5 ) = (B + 1)m 3. Therefore 2 (m 5 ) = 0 if and only if B = p 1. Lemma 3.2 concludes the proof Module 2 (V Y 2 ). We shall need the following lemma, a part of which shall be useful for the determination of 2 (V (Y ^ Y )). Lemma If B = p 2 and! 13 = 0, then 2 (n 4 ) = 2 (m 2 ) 6= 0. Assume B = p 2 and! 23 = 0. If A 6= 0; p 2, then 2 (n 4 ) 2 (m 2 ). If A = 0, then 2 (n 4 ) = 0 and 2 (m 2 ) 6= 0. If A = p 2, then 2 (n 4 ) 6= 0 and 2 (m 2 ) = 0. Proof. If! 13 = 0, then 2 (n 4 ) = Am 2 + (2 + A)n 2 = 2 (m 2 ). If! 23 = 0, then 2 (n 4 ) = A(m 2 + n 2 ) and 2 (m 2 ) = (2 + A)(m 2 + n 2 ). The structure of the S-module 2 (V Y 2 ) is given as follows. Proposition The following statements describe S-modules isomorphic to V 22 = 2 (V Y 2 ). Assume B < p 2. If is typical, then V 22 = L(n1 ) 34 L(n 2 ) L(n 3 ). If is 14- or (14; 24)-atypical, then V 22 = L(n1 ) 34 L(n 2 ). If is is 24-atypical, then V 22 = L(n1 ) L(n 2 ). If is 13- or (13; 23)-atypical, then V 22 = 34 L(n 2 ) L(n 3 ). If is 23-atypical, then V 22 = L(n 2 ) L(n 3 ). If is (13; 24)-atypical or (14; 23)-atypical, then V 22 = L(n 2 ). Assume now B = p 2. If is typical, then V 22 = N L(n 3 ). If is 14- or 24- or (14; 24)-atypical, then V 22 = N 4. If is 13- or (13; 23)-atypical, then V 22 = L( Am2 + (2 + A)n 2 ) 1 34L(n 3 ). If is (13; 24)-atypical, then V 22 = L( Am2 + (2 + A)n 2 ). If is 23-atypical, then V 22 = 12 L(m 2 + n 2 ) 1 34L(n 3 ). If is (14; 23)-atypical, then V 22 = L(m2 + n 2 ). Finally, assume B = p 1. If is typical, then V 22 = N 5 L(n 2 ). If is (23; 24)-atypical and A 6= 0, then V 22 = L(n5 ) L(n 2 ). If is (23; 24)-atypical and A = 0, then V 22 = L(n5 ). If is (13; 14)-atypical, then V 22 = L(n5 ) L(n 2 ). If is (13; 14; 23; 24)-atypical, then V 22 = L(n2 ). Proof. The proof is analogous to the proof of Proposition 3.11 and uses the following identities. If B = p 1 and! 13 = 0, then 2 (n 5 ) = (A + 1)n 3. If B = p 1 and! 23 = 0, then 2 (n 5 ) = (A + 1)n Module 2 (V (Y ^ Y )). Now we shall determine 2 (V (Y ^ Y )). Proposition The S-module V 2 = 2 (V (Y ^ Y )) is isomorphic to the following modules. 1) Case 0 < A; B < p 2 - L(m 1 ) L(n 1 ) L(m 2 ) L(n 2 ) 1 12L(m 3 ) 1 34L(n 3 ), if is typical - L(m 1 ) L(m 2 ) 1 34L(n 3 ), if is 13-atypical

16 16 A.N. GRISHKOV AND F. MARKO - L(m 1 ) L(n 1 ) L(m 2 ), if is 14-atypical - L(m 2 ) 1 12L(m 3 ) 1 34L(n 3 ), if is 23-atypical - L(n 1 ) L(m 2 ) 1 12L(m 3 ), if is 24-atypical - L(m 2 ), if is (13; 24)-atypical or (14; 23)-atypical 2) Case A = 0; 0 < B < p 2 - L(m 1 ) L(n 1 ) L(n 2 ) 1 34L(n 3 ), if is typical - L(m 1 ) L(n 2 ) 1 34L(n 3 ), if is 13-atypical - L(m 1 ) L(n 1 ) L(n 2 ), if is 14-atypical L(n 3 ), if is 23-atypical - L(n 1 ), if is 24-atypical 3) Case 0 < A < p 2; B = 0 - L(m 1 ) L(n 1 ) L(m 2 ) 1 12L(m 3 ), if is typical - L(m 1 ), if is 13-atypical - L(m 1 ) L(n 1 ) L(m 2 ), if is 14-atypical L(m 3 ), if is 23-atypical - L(m 2 ) L(n 1 ) 1 12L(m 3 ), if is 24-atypical 4) Case A = B = 0 - L(m 1 ) L(n 1 ), if is typical - L(m 1 ), if is 13-atypical - L(n 1 ), if is 24-atypical - 0, if is (14; 23)-atypical 5) Case A = p 2; 0 < B < p 2 - M 4 L(n 1 ) L(n 2 ) 1 12L(m 3 ) 1 34L(n 3 ), if is typical - M L(n 3 ), if is 13-atypical - L(n 1 ) M 4, if is 14-atypical - L(m 2 + n 2 ) 1 12L(m 3 ) 1 34L(n 3 ), if is 23-atypical - L(n 1 ) L((B + 2)m 2 Bn 2 ) 1 12L(m 3 ), if is 24-atypical 6) Case A = p 2; B = 0 - M 4 L(n 1 ) 1 12L(m 3 ), if is typical - L(n 1 ) M 4, if is 14-atypical L(m 3 ), if is 23-atypical - L(m 2 ), if is (13; 24)-atypical 7) Case 0 < A < p 2; B = p 2 - L(m 1 ) N 4 L(m 2 ) 1 12L(m 3 ) 1 34L(n 3 ), if is typical - L(m 1 ) N 4, if is 14-atypical - N L(m 3 ), if is 24-atypical - L(m 1 ) L( Am 2 + (A + 2)n 2 ) 1 34L(n 3 ), if is 13-atypical - L(m 2 + n 2 ) 1 12L(m 3 ) 1 34L(n 3 ), if is 23-atypical 8) Case A = 0; B = p 2 - L(m 1 ) N L(n 3 ), if is typical - L(m 1 ) N 4, if is 14-atypical L(n 3 ), if is 23-atypical - L(n 2 ), if is (13; 24)-atypical 9) Case A = B = p 2 - M 4 N L(m 3 ) 1 34L(n 3 ), if is typical - M L(n 3 ), if is 13-atypical - N L(m 3 ), if is 24-atypical

17 SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(22) 17 - L(m 2 + n 2 ), if is (14; 23)-atypical 10) Case A = p 1; 0 < B < p 2 - M 5 L(n 1 ) L(m 2 ) L(n 2 ) L(m 3 ) 1 34L(n 3 ), if is typical - L(m 3 ) L(m 2 ) 1 34L(n 3 ), if is (13; 23)-atypical - L(m 3 ) L(n 1 ) L(m 2 ), if is (14; 24)-atypical 11) Case A = p 1; B = 0 - M 5 L(n 1 ) L(m 2 ) L(m 3 ), if is typical - L(m 3 ), if is (13; 23)-atypical - L(m 3 ) L(m 2 ) L(n 1 ), if is (14; 24)-atypical 12) Case 0 < A < p 2; B = p 1 - L(m 1 ) N 5 L(m 2 ) L(n 2 ) 1 12L(m 3 ) L(n 3 ), if is typical - L(m 1 ) L(n 3 ) L(m 2 ), if is (13; 14)-atypical - L(n 3 ) L(m 2 ) 1 12L(m 3 ), if is (23; 24)-atypical 13) Case A = 0; B = p 1 - L(m 1 ) N 5 L(n 2 ) L(n 3 ), if is typical - L(m 1 ) L(n 3 ) L(n 2 ), if is (13; 14)-atypical - L(n 3 ), if is (23; 24)-atypical 14) Case A = p 1; B = p 2 - M 5 N 4 L(m 2 ) L(m 3 ) 1 34L(n 3 ), if is typical - L(m 3 ) L(m 2 + n 2 ) 1 34L(n 3 ), if is (13; 23)-atypical - L(m 3 ) N 4, if is (14; 24)-atypical 15) Case A = p 2; B = p 1 - M 4 N 5 L(n 2 ) 1 12L(m 3 ) L(n 3 ), if is typical - L(n 3 ) L(m 2 + n 2 ) 1 12L(m 3 ), if is (23; 24)-atypical - L(n 3 ) M 4, if is (13; 14)-atypical 16) Case t = s = p 1 - M 5 N 5 L(m 2 ) L(n 2 ) L(m 3 ) L(n 3 ), if is typical - L(m 2 ), if is (13; 14; 23; 24)-atypical. Proof. It follows from Propositions 3.11, 3.13, and Lemmas 3.2, 3.8, 3.9, 3.10 and For the convenience of the reader we shall point out cases when the nonzero images under 2 of di erent primitive vectors of the highest weight (A; B) are colinear. If A = p 2; 0 < B < p 2 and is 23-atypical or 24-atypical, then 2 (m 4 ) 2 (n 2 ). If 0 < A < p 2; B = p 2 and is 13-atypical or 23-atypical, then 2 (n 4 ) 2 (m 2 ). If A = B = p 2 and is 13-atypical, then 2 (n 4 ) 2 (m 2 ). If A = B = p 2 and is 24-atypical, then 2 (m 4 ) 2 (n 2 ). If A = B = p 2 and is (14; 23)-atypical, then 2 (m 4 ) 2 (n 4 ) while 2 (m 2 ) = 2 (n 2 ) = 0. If t = A 1; 0 < B < p 2 and is (13; 23)- or (14; 24)-atypical, then 2 (n 2 ) 2 (m 2 ). If 0 < A < p 2; B = p 1 is (13; 14)- or (23; 24)-atypical, then 2 (m 2 ) 2 (n 2 ). If A = p 1; B = p 2 and is (13; 23)-atypical, then 2 (n 4 ) 2 (m 2 ). If A = p 1; B = p 2 and is (14; 24)-atypical, then 2 (m 2 ) 2 (n 2 ). If A = p 2; B = p 1 and is (23; 24)-atypical, then 2 (m 4 ) 2 (n 2 ). If A = p 2; B = p 1 and is (13; 14)-atypical, then 2 (m 2 ) 2 (n 2 ).

18 18 A.N. GRISHKOV AND F. MARKO If A = B = p 1 and is (13; 14; 23; 24)-atypical, then 2 (m 2 ) 2 (n 2 ). 4. Third floor 4.1. Characteristic zero. Denote z 23 = y 13 ^ y 23 ^ y 24, z 24 = y 14 ^ y 23 ^ y 24, z 13 = y 13 ^ y 14 ^ y 23, z 14 = y 13 ^ y 14 ^ y 24. Furthermore, set k 1 = v AB z 23, k 2 = v AB z 24 v A;B 1 z 23, k 3 = v AB z 13 + v A 1;B z 23, and k 4 = v AB z 14 + v A 1;B z 24 v A;B 1 z 13 v A 1;B 1 z 23. Let : Y! Y ^Y ^Y be a map that sends y 23 7! z 23, y 24 7! z 24, y 13 7! z 13 and y 14 7! z 14. Then is an isomorphism of S-modules and it induces an isomorphism of S-modules V : V Y! V (Y ^ Y ^ Y ) via V (v y i ) = v (y i ) = v z i for appropriate i;. Lemma 4.1. The module V (Y ^ Y ^ Y ) is isomorphic to the direct sum L(k 1 ) 34 L(k 2 ) 12 L(k 3 ) L(k 4 ). Proof. Since the map : Y! Y ^ Y ^ Y is an isomorphism of S-modules, the module Y ^ Y ^ Y is irreducible of the highest vector z 23. Since the map V is an isomorphism of S-modules and the S-module structure of V Y was already determined, we can use V to describe the S-module structure of V (Y ^ Y ^ Y ). Using Lemma 2.1 we establish V (Y ^ Y ^ Y ) = L(k 1 ) 34 L(k 2 ) 12 L(k 3 ) L(k 4 ). Proposition 4.2. The image 3 (V (Y ^Y ^Y )) = K K K K 4. Proof. The S-morphism 3 is described completely by images of generating vectors 3 (k 1 ) =! 13! 23! 24 k 1, 3 (k 2 ) =! 14! 23! 24 k 2, 3 (k 3 ) =! 13! 14! 23 k 3, and 3 (k 4 ) =! 13! 14! 24 k 4 : 4.2. Characteristic p. Assume that the weight is restricted. Recall that the map V : V Y! V (Y ^ Y ^ Y ) is an isomorphism of S-modules, and for each 5 i 9, denote V (l i ) = k i and K i = V (L i ). Proposition 4.3. The S-module V (Y ^ Y ^ Y ) is isomorphic to 1) Case A; B < p 1 L(k 1 ) 34 L(k 2 ) 12 L(k 3 ) L(k 4 ). 2) Case A = p 1, B < p 1 K 5 34 K 6. 3) Case A < p 1, B = p 1 K 7 12 K 8. 4) Case A = B = p 1 K 9. Here, the composition series of the S-module K i, for every 5 i 9, is analogous to that of corresponding S-module L i from Proposition 2.5. Proof. The map V is an isomorphism of S-modules. Since the S-module structure of V Y was already determined, we can use V to describe the S-module structure of V (Y ^ Y ^ Y ). All that is necessary to do is to replace every appearance of l and L in Proposition 2.5 with k and K, respectively.

19 SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(22) Image under 3. The structure of the S-module 3 (V (Y ^Y ^Y )) is given as follows. Proposition 4.4. The following statements describe S-modules isomorphic to V 3 = 3 (V (Y ^ Y ^ Y )). If A; B < p 1, then V 3 = K K K K 4. Assume A = p 1 and B < p 1. If is typical, then V 3 = V (Y ^Y ^Y ). If is (13; 23)-atypical, then V 3 = 34 L(k 4 ). If is (14; 24)-atypical, then V 3 = L(k3 ). Assume A < p 1 and B = p 1. If is typical, then V 3 = V (Y ^Y ^Y ). If is (13; 14)-atypical, then V 3 = L(k2 ). If is (23; 24)-atypical, then V 3 = 12 L(k 4 ). Assume A = B = p 1. If is typical, then V 3 = V (Y ^ Y ^ Y ). If is (13; 14; 23; 24)-atypical, then V 3 = 0. Proof. If A; B < p 1, then V 3 = K K K K 4 as in the characteristic zero case. If A = p 1 and B < p 1, then the images of generators of K 5 and K 6 are 3 (k 5 ) =! 13! 14! 23 k 5! 13! 23 k 3 and 3 (k 6 ) =! 13! 14! 24 k 6! 14! 24 k 4. The structure of V 3 follows. If A < p 1 and B = p 1, then the images of generators of K 7 and K 8 are 3 (k 7 ) =! 14! 23! 24 k 7 +! 23! 24 k 2 and 3 (k 8 ) =! 13! 14! 24 k 8! 13! 14 k 4. The structure of V 3 follows. If A = B = p 1, then! 13 =! 14 =! 23 =! 24 =!. The image of the generator k 9 of V (Y ^ Y ^ Y ) is 3 (k 9 ) =! 3 k 9! 2 k 6! 2 k 8 2!k 4 and the claim follows. 5. Fourth floor 5.1. Characteristic zero. The S-module Y ^ Y ^ Y ^ Y is the trivial module of the highest vector y 13 ^ y 14 ^ y 23 ^ y 24. The S-module V (Y ^ Y ^ Y ^ Y ) is irreducible of the highest vector l = v AB y 13 ^ y 14 ^ y 23 ^ y 24 and is isomorphic to V as S-module. Proposition 5.1. The image 4 (V (Y ^ Y ^ Y ^ Y )) = V (Y ^ Y ^ Y ^ Y ). Proof. The morphism 4 is given by 4 (l) =! 13! 14! 23! 24 l Characteristic p. Assume that the weight is restricted. The S-module V (Y ^Y ^Y ^Y ) = L(v A;B (y 13 ^y 14 ^y 23 ^y 24 )) is irreducible and isomorphic to V as S-module Image under 4. The S-module structure of 4 (V (Y ^Y ^Y ^Y )) is given as follows. Proposition 5.2. The S-module 4 (V (Y ^ Y ^ Y ^ Y )) is isomorphic to V (Y ^ Y ^ Y ^ Y ) ' V if is typical, and is isomorphic to 0 if is atypical. 6. Character and dimension of simple module L S(22) () Combining the previous results we obtain the following theorem. Theorem 6.1. The S-module H 0 G () is isomorphic to the direct sum V (V Y ) (V (Y ^ Y )) (V (Y ^ Y ^ Y )) (V (Y ^ Y ^ Y ^ Y )), where the middle summands are described in Propositions 2.5, 3.7 and 4.3.

20 20 A.N. GRISHKOV AND F. MARKO Theorem 6.2. The S-module L S(22) () is isomorphic to V 1 (F 1 ()) 2 (F 2 ()) 3 (F 3 ()) 4 (F 4 ()); where the images V 1 = 1 (F 1 ()), V 2 = 2 (F 2 ()), V 3 = 3 (F 3 ()) and V 4 = 4 (F 4 ()) respectively are described in Propositions 2.6, 3.14, 4.4 and 5.2, respectively Characteristic zero. Combining previous results we obtain the following theorem. Theorem 6.3. The simple module L S(22) (), viewed as an S-module, is isomorphic to the direct sum V 1 (F 1 ()) 2 (F 2 ()) 3 (F 3 ()) 4 (F 4 ()), where the images 1 (F 1 ()), 2 (F 2 ()), 3 (F 3 ()) and 4 (F 4 ()), respectively, are described in Propositions 2.2, 3.4, 4.2 and 5.1, respectively. Corollary 6.4. The induced module H 0 G () is isomorphic to L S(22)() if and only if is typical which happens if and only if 2 2. In order to relate the last results to Hook Schur functions, we need to explain how a simple module L S(22) corresponds to a (2,2)-hook partition = ( 1 ; : : : ; k ). The correspondence is such that 1 = 1, 2 = 2 and the partition ( 3 ; : : : ; k ) is the transpose of ( 3 ; 4 ). The character of the induced module HG 0 () is given by the formula (H0 G ()) = (1 + y1 x 1 )(1 + y2 x 1 )(1 + y1 x 2 )(1 + y2 x 2 )s (1; 2)(x 1 ; x 2 )s (3; 4)(y 1 ; y 2 ), where s (1; 2)(x 1 ; x 2 ) denotes the Schur function corresponding to the partition ( 1 ; 2 ) and s (3; 4)(y 1 ; y 2 ) denotes the Schur function corresponding to the transpose of the partition ( 3 ; 4 ). The character of L S(22) () is given by the Hook Schur function HS (x 1 ; x 2 ; y 1 ; y 2 ). Therefore, we have the following equivalence which strengthens Theorem 6.20 of [1] in the case of (2; 2)-hook partitions. Proposition 6.5. For a (2; 2)-hook partition, the following are equivalent: 1) 2 2 2) HS (x 1 ; x 2 ; y 1 ; y 2 ) = (H 0 G ()) 3) H 0 G () is isomorphic to L S(22)(). Proof. If 2 2, then in the notation of Theorem 6.20 of [1] we have (HG 0 ()) = (x 1 + y 1 )(x 1 + y 2 )(x 2 + y 1 )(x 2 + y 2 )s (x 1 ; x 2 )s (y 1 ; y 2 ) and HS (x 1 ; x 2 ; y 1 ; y 2 ) = (HG 0 ()). The remaining statements follow from Corollary Characteristic p. We give a compact formula for the character and dimension of a simple S(22) module of restricted weight. Using the Steinberg Tensor Product theorem we can then determine the same for an arbitrary highest weight. If ( 1 2 ) is a dominant weight for the algebra sl(2), then the character of a simple sl(2)-module with highest weight ( 1 ; 2 ) is given by the Schur function S (1; 2)(x 1 ; x 2 ). For a dominant weight = ( ) for the algebra S, the character S() of the simple S-module L() of the highest weight is given by S (1 ; 2 )(x 1 ; x 2 ) S (3 ; 4 )(x 3 ; x 4 ). Denote by S( 1 ; 2 3 ; 4 ) the product S (1; 2)(x 1 ; x 2 )S (3; 4)(x 3 ; x 4 ) of two Schur functions if 1 2 and 3 4 and S( 1 ; 2 3 ; 4 ) = 0 otherwise. For short, write it as S() and call it the Schur function corresponding to. Then S(k; 0l; 0) = p(k; l).

21 SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(22) 21 Denote 13 = ( 1; 01; 0), 14 = ( 1; 00; 1), 23 = (0; 11; 0) and 24 = (0; 10; 1) and write the "decorated" weights derived from as follows: ~ 13 = + 13, ~ 14 = + 14, ~ 23 = + 23, ~ 24 = + 24, 13;14 = , 13;23 = , 13;24 = = 14;23 = , 14;24 = , 23;24 = , 14 = , 13 = , 23 = , 24 = , ^ = Then the S-weights of these decorated weights are given in the following table. 13 ~ 14 ~ 23 ~ 24 ~ (A; B) (A 1; B + 1) (A 1; B 1) (A + 1; B + 1) (A + 1; B 1) 13;14 13;23 13;24 14;23 14;24 23;24 (A 2; B) (A; B + 2) (A; B) (A; B) (A; B 2) (A + 2; B) ^ (A + 1; B + 1) (A + 1; B 1) (A 1; B 1) (A 1; B + 1) (A; B) We have already seen that weight 13;24 = 14;23 is of special signi cance and we shall denote it by. The space spanned by all elements of the simple module L S(22) () that lie on the i-th oor shall be called the sector of that oor corresponding to and shall be denoted by L i (). Each L i () is a S- module and to each L i () we assign a "partial" character i () that records the multiplicities of weight spaces of L i (). Then the character () of the simple module L S(22) () equals () = 0 () + 1 () + 2 () + 3 () + 4 (). The character and dimension of a simple S(22)-module L S(22) () of restricted weight are given below. Theorem 6.6. Let L S(22) () be a simple S(22)-module of restricted highest weight. If is typical, then (L S(22) ()) = S() + S( ~ 23 ) + S( ~ 13 ) + S( ~ 24 ) + S( ~ 14 ) + S( 13;23 ) + S( 23;24 ) + 12 S() + 34 S() + S( 13;14 ) + S( 14;24 ) + S( 14 ) + S( 24 ) + S( 13 ) + S( 23 ) + S(^) and diml() = 16(A + 1)(B + 1). If is 13-atypical or (13,14)-atypical, then (L S(22) ()) = S() + S( ~ 23 ) + S( ~ 24 ) + S( ~ 14 ) + S( 23;24 ) + 34 S() + S( 14;24 ) + S( 13 ) and diml() = 8 + 4A + 12B + 8AB. If is 14-atypical, then (L S(22) ()) = S() + S( ~ 23 ) + S( ~ 13 ) + S( ~ 24 ) + S( 13;23 ) + S( 23;24 ) + S() + S( 14 ) and diml() = A + 12B + 8AB. If is 23-atypical,(13,23)-atypical or (23,24)-atypical, then (L S(22) ()) = S() + S( ~ 13 ) + S( ~ 24 ) + S( ~ 14 ) S() + S( 13;14 ) + S( 14;24 ) + S( 23 ) and diml() = 4A + 4B + 8AB. If is 24-atypical,(14,24)-atypical, then (L S(22) ()) = S() + S( ~ 23 ) + S( ~ 13 ) + S( ~ 14 ) + S( 13;23 ) + 12 S() + S( 13;14 ) + S( 24 ) and diml() = A + 4B + 8AB. If is (13,24)-atypical, then (L S(22) ()) = S() + S( ~ 23 ) + S( ~ 14 ) + S()