A note on irreducible representations of symmetric groups and Sergeev superalgebras

A NOTE ON IRREDUCIBLE REPRESENT A TIONS OF SYMMETRIC GR OUPS AND SER GEEV SUPERALGEBRAS MINJIA CHEN, JINKUI W AN AND HONGBO ZHAO Abstract. W e pro vide an explicit construction and a closed dimension formula in terms of ho ok lengths for the irreducible representations for the symmetric groups S p and the Sergeev sup eralgebras Y p o ver an algebraically closed field F of c haracteristic p > 0. Contents 1. In tro duction 1 2. Irreducible represen tations of symmetric group S p 2 3. Irreducible represen tations of Sergeev sup eralgebra Y p 6 References 18 1. Introduction Let F b e an algebraically closed field of characteristic p > 0. Let S n denote the symmetric group on n letters, generated by the adjacen t transp ositions s 1 , s 2 , . . . , s n − 1 . W e denote by Y n = C n ⋊ F S n the asso ciated Sergeev sup eralgebra, where C n is the Clifford algebra ov er F generated by c 1 , . . . , c n sub ject to the relations c 2 k = 1 and c k c l = − c l c k for 1 ≤ k  = l ≤ n . It is a classical result that b oth F S n and Y n are semisimple if and only if n < p . In this semisimple situation, the irreducible representations are well-c haracterized, with established seminormal forms and explicit dimension form ulas (cf. [ K1 , WW ]). Ho w ever, in the mo dular case where n ≥ p , the representation theory b ecomes sub- stan tially more complex. F or ov er a century , determining a general dimension form ula for the irreducible represen tations of F S n has remained a ma jor op en problem; a similar c hallenge p ersists for the Sergeev sup eralgebra Y n . Significan t progress has b een made by fo cusing on sp ecific classes of mo dules. In [ Ma ], Mathieu determined the dimensions of irreducible F S n -mo dules associated with partitions λ = ( λ 1 , . . . , λ ℓ ) of n with ℓ ( λ ) = ℓ satisfying the condition λ 1 − λ ℓ − ℓ ≤ p by emplo ying classical Sch ur-W eyl duality . Subsequently , Kleshchev [ K2 ] demonstrated that these rep- resen tations are precisely the completely splittable mo dules—those whose restrictions to the subgroup S k remain semisimple for all k < n , or equiv alently , those up on which the Jucys-Murph y elements act semisimply . These results hav e b een generalized to degener- ate affine Heck e algebras b y Ruff [ Ru ], to affine Heck e algebras of t yp e A [ Ch , Ra ], and 1 2 MINJIA CHEN, JINKUI W AN AND HONGBO ZHA O to Khov anov-Lauda-Rouquier algebras [ KR ]. F urthermore, the second author extended these results to degenerate affine Heck e-Clifford algebras; as a byproduct, an explicit con- struction and dimension formula for the completely splittable representations of Y n (for p  = 2) were obtained using sp ecial standard Y oung tableaux in [ W a ]. The present note pro vides a closed dimension formula and an explicit construction for the irreducible represen tations of F S n and Y n sp ecifically in the case n = p . Our approac h is based on the observ ation that when n ≤ p , all irreducible representations of F S n and Y n are completely splittable. More sp ecifically , by comparing the parametrization set of irreducible modules with that of completely splittable representations, w e observ e that these t w o sets coincide if and only if n ≤ p . Under the assumption n ≤ p , we prov e that the combinatorial dimension form ulas for these irreducible mo dules can b e expressed explicitly in terms of the ho ok lengths asso ciated to the corresp onding Y oung diagrams. Notably , in the non-semsimple case where n = p , we obtain a closed dimension formula for all irreducible represen tations of S p (resp. Y p ) and further give the explicit action of the generators s 1 , s 2 , . . . , s p − 1 of S p (resp. generators c 1 , . . . , c p , s 1 , . . . , s p − 1 of Y p ) on the underlying v ector spaces. Inspired b y this work, in the forthcoming article [ CW ], w e develop an analogous construction for Hec k e-Clifford algebra H n ( q ) which is the q -analog of Y n and as an application we then further establish a semi-simplicit y criterion for H n ( q ) at ro ots of unit y . The pap er is organized as follo ws. In Section 2 , w e recall preliminary results regarding the symmetric group and derive a closed dimension formula for irreducible mo dules in terms of ho ok lengths, alongside a construction of these mo dules via the detailed action of the generators s 1 , . . . , s p − 1 in the case n = p . In Section 3 , we provide an analogous dimension formula and explicit construction for the irreducible representations of the Sergeev sup eralgebra. 2. Irreducible represent a tions of symmetric group S p In this section, we will derive a closed dimension formula and an explicit construction for irreducible representations of F S p o v er an algebraically closed field F of c haracteristic p > 0. 2.1. Basics on partitions and Spech t modules. Let P ( n ) denote the set of partitions of n . F or a partition λ = ( λ 1 , λ 2 , . . . ) ∈ P ( n ), we alwa ys assume λ i ≥ λ i +1 ≥ 0 and P i λ i = n . Set ℓ ( λ ) = ♯ { i | λ i > 0 } . W e can also write a partition as λ = (1 a 1 2 a 2 · · · ) with a i b eing the num b er of parts of λ equal to i for i ≥ 1. It is known that any partition λ ∈ P ( n ) can b e identified with its Y oung diagram, that is λ = { ( i, j ) ∈ Z 2 | 1 ≤ i ≤ ℓ ( λ ) , 1 ≤ j ≤ λ i } . F or λ ∈ P ( n ), the conjugate partition λ ′ is defined b y λ ′ j = max { i : λ i ≥ j } . The ( i, j )-hook of λ is the set of no des in the Y oung diagram that are either in the same ro w as ( i, j ) and to the righ t of ( i, j ), or in the same column as ( i, j ) and b elow ( i, j ), including ( i, j ) itself. The ho ok length of the ( i, j )-hook is defined as h λ ( i, j ) = λ i + λ ′ j − i − j + 1, which equiv alen tly counts the num ber of no des in the ( i, j )-ho ok. Denote by T ( λ ) the set of tableaux of shap e λ ; that is, a tableau is a lab elling of the no des in the Y oung diagram λ with the entries 1 , 2 , . . . , n . A tableau T is called SYMMETRIC GROUPS AND SERGEEV SUPERALGEBRAS 3 standar d if its en tries strictly increase from left to righ t along each ro w and down eac h column. W e denote by Std( λ ) the subset of T ( λ ) consisting of standard tableaux of shap e λ . W e then hav e the following remark able ho ok length formula (cf. [ Mac , Chapter I, Section 5, Example 2]): (2.1) f λ := ♯ Std( λ ) = n ! Q ( i,j ) ∈ λ h λ ( i, j ) where the pro duct in the denominator is ov er all no des in the Y oung diagram λ . 2.2. Irreducible F S n -mo dules in the case n = p . Let P p ( n ) b e the set of p -regular partition of n , that is, the subset of P ( n ) consisting of λ such that no part of λ is rep eated more than or equal to p times. The element in P p ( n ) can also b e written in the form P p ( n ) = { λ = (1 a 1 2 a 2 · · · ) ∈ P ( n ) | 0 ≤ a k < p for all k } . It is known that for eac h λ ∈ P ( n ), there exists a Sp e cht mo dule S λ of dimension f λ , whic h further admits a symmetric S n -in v ariant inner pro duct ( , ). The radical of this inner pro duct, denoted by Rad λ , is a S n -submo dule of S λ . W e refer the reader to [ Ja ] for details regarding the construction of S λ and the inner pro duct ( , ), which we omit here as they are not required for our curren t purp oses. Set D λ = S λ / Rad λ . Theorem 2.1. [ Ja ] F or λ ∈ P ( n ) , D λ  = 0 if and only if λ ∈ P p ( n ) . Mor e over, the set { D λ | λ ∈ P p ( n ) } is a c omplete set of non -isomorphic irr e ducible F S n -mo dules. Define the Jucys-Murph y elemen ts in F S n as follo ws: L 1 = 0 , L k = X 1 ≤ m i ′ , j < j ′ and i + j ′ + 1 − i ′ − j = p satisfy T ( i ′ ,j ′ ) > T ( i,j ) . Denote b y Std p ( λ ) the set of p -standard tableaux of shap e λ for eac h λ ∈ C P p ( n ), that is Std p ( λ ) = n T ∈ Std( λ ) | T ( i ′ ,j ′ ) > T ( i,j ) for an y ( i, j ) , ( i ′ , j ′ ) ∈ λ with i > i ′ , j < j ′ , i + j ′ + 1 − i ′ − j = p o . (2.2) Example 2.3. T ak e p = 5 , λ = (4 , 3 , 1). Then the tableau T = 1 2 5 6 3 7 8 4 is p -standard. But the tableau s 4 T = 1 2 4 6 3 7 8 5 is standard but not p -standard. Theorem 2.4. [ K2 , Theorem 2.1 and Corollary 2.4] L et λ ∈ P p ( n ) . Then D λ is c om- pletely splittable if and only if λ ∈ C P p ( n ) . Mor e over, if λ ∈ C P p ( n ) , then dim D λ = ♯ Std p ( λ ) . Supp ose λ ∈ C P p ( n ). F or a no de A = ( i, j ) in a Y oung diagram λ , define the residue of A to b e res( A ) = j − i mo d p . F or 1 ≤ k ≤ n and T ∈ Std p ( λ ), set T ( k ) b e the no de in T o ccupied by k . It is straightforw ard to chec k that res( T ( a + 1))  = res( T ( a )) if T ∈ Std p ( λ ) and hence ρ a ( T ) := 1 res( T ( a + 1)) − res( T ( a )) is w ell defined for each 1 ≤ a ≤ n − 1. Theorem 2.5. [ Ru , Theorem 4.9] Supp ose λ ∈ C P p ( n ) . Ther e exists an F S n -mo dule isomorphism D λ ∼ = ⊕ T ∈ Std p ( λ ) F v T as a ve ctor sp ac e , wher e the action of simple tr ansp osition s i ∈ S n for 1 ≤ i ≤ n − 1 is given as fol lows: (2.3) s i v T = ( ρ i ( T ) v T + q 1 − ( ρ i ( T )) 2 v s i T , if s i T ∈ Std p ( λ ) , ρ i ( T ) v T , otherwise SYMMETRIC GROUPS AND SERGEEV SUPERALGEBRAS 5 for e ach T ∈ Std p ( λ ) . Lemma 2.6. C P p ( n ) = P p ( n ) if and only if n ≤ p . Henc e, al l irr e ducible F S n -mo dules ar e c ompletely splittable if and only if n ≤ p . Pr o of. Supp ose n ≤ p . F or any p -regular partition λ = ( λ 1 , λ 2 , · · · , λ ℓ ) of n with ℓ ( λ ) = ℓ , w e hav e χ ( λ ) = λ 1 − λ ℓ + ℓ = n − ( λ 2 + λ 3 + · · · + λ ℓ ) − λ ℓ + ℓ ≤ n ≤ p . Hence P p ( n ) ⊆ C P p ( n ) whic h leads to C P p ( n ) = P p ( n ). Conv ersely , if n > p , then obviously χ (( n − 1 , 1)) = n > p while ( n − 1 , 1) ∈ P n ( n ). This means C P p ( n ) ⊊ P p ( n ) and th us C P p ( n )  = P p ( n ) if n > p . This pro v es the lemma. □ Prop osition 2.7. Supp ose n ≤ p and λ ∈ C P p ( n ) , then we have ♯ Std p ( λ ) = ( f ( k − 1 , 1 n − k ) =  n − 2 k − 2  , if λ = ( k , 1 n − k ) for some 2 ≤ k ≤ n − 1 , f λ = n ! Q ( i,j ) ∈ λ h λ ( i,j ) , otherwise . Pr o of. Clearly in the case n < p w e ha ve C P p ( n ) = P p ( n ) = P ( n ) and moreo v er Std p ( λ ) = Std( λ ) by ( 2.2 ). Th us the prop osition holds by ( 2.1 ). No w assume n = p . It is direct to v erify P p ( n ) = P ( n ) \ { (1 n ) } . and hence C P p ( n ) = { λ ∈ P ( n ) | λ  = (1 n ) } b y Lemma 2.6 . Assume λ ∈ P ( n ) with λ  = (1 n ). Notice that for an y t w o no des ( i, j ) and ( i ′ , j ′ ) in the Y oung diagram λ with i > i ′ , j < j ′ , w e ha ve (2.4) i + j ′ + 1 − i ′ − j ≤ λ ′ j + λ i ′ + 1 − i ′ − j = h λ ( i ′ , j ) and moreov er the equality in ( 2.4 ) holds if and only if i = λ ′ j , j ′ = λ i ′ . Meanwhile if λ 2 ≥ 2 then (2.5) h λ ( i, j ) < n = p for any no de ( i, j ) in the Y oung diagram λ . Then by ( 2.4 ) and ( 2.5 ) we obtain that every T ∈ Std( λ ) is p -standard if λ 2 ≥ 2. Th us (2.6) Std p ( λ ) = Std( λ ) if λ  = ( k , 1 , 1 , . . . , 1 | {z } n − k ) = ( k , 1 n − k ) No w supp ose λ 2 ≤ 1. Then λ must b e of the form λ = ( k , 1 n − k ) with k ≥ 2. Supp ose ( i, j ) and ( i ′ , j ′ ) are t w o no des in the Y oung diagram λ satisfying i > i ′ , j < j ′ and i + j ′ + 1 − i ′ − j = p . Then clearly w e hav e ( i ′ , j ′ ) = (1 , k ) and ( i, j ) = ( n − k + 1 , 1). Hence a tableaux T ∈ Std( λ ) b elongs to Std p ( λ ) if and only if T (1 ,k ) > T ( n − k +1 , 1) whic h means T (1 ,k ) = n . This implies that Std p ( λ ) = { T ∈ Std( λ ) | T (1 ,k ) = n } and th us (2.7) ♯ Std p ( λ ) = ♯ Std( λ − ) , where λ − = ( k − 1 , 1 n − k ) . Then the prop osition follows from ( 2.6 ), ( 2.7 ) and ( 2.1 ). □ Then w e are ready to introduce one of our main results. 6 MINJIA CHEN, JINKUI W AN AND HONGBO ZHAO Theorem 2.8. Supp ose n = p and λ ∈ C P p ( n ) . (1) If λ  = ( k , 1 n − k ) , then ther e exists an isomorphism of F S n -mo dules D λ ∼ = ⊕ T ∈ Std( λ ) F v T as a ve ctor sp ac e over F , wher e the action s i ∈ S n for 1 ≤ i ≤ n − 1 satisfies (2.8) s i v T = ( ρ i ( T ) v T + q 1 − ( ρ i ( T )) 2 v s i T , if s i T ∈ Std( λ ) , ρ i ( T ) v T , otherwise. In addition, dim D λ = n ! Q ( i,j ) ∈ λ h λ ( i,j ) . (2) If λ = ( k , 1 n − k ) with k ≥ 2 , ther e exists an isomorphism F S n -mo dules D λ ∼ = ⊕ T ∈ Std( λ − ) F v T as a ve ctor sp ac e over F , with λ − = ( k − 1 , 1 n − k ) and the action s i ∈ S n for 1 ≤ i ≤ n − 1 satisfies (2.9) s i v T = ( ρ i ( T ) v T + q 1 − ( ρ i ( T )) 2 v s i T if 1 ≤ i ≤ n − 2 and s i T ∈ Std( λ − ) , ρ i ( T ) v T , if 1 ≤ i ≤ n − 2 and s i T / ∈ Std( λ − ) s n − 1 v T =  v T , if n − 1 = T (1 ,k − 1) , − v T , if n − 1 = T ( n − k +1 , 1) . In addition, dim D λ =  n − 2 k − 2  . (3) The set { D λ | λ ∈ P ( n ) , λ  = (1 n ) } is a c omplete set of non-isomorphic irr e ducible F S n -mo dules. Pr o of. The first part (1) clearly is due to ( 2.3 ), Theorem 2.4 and Prop osition 2.7 . It remains to prov e the second part (2). No w assume λ = ( k , 1 n − k ) with k ≥ 2. Set λ − = ( k − 1 , 1 n − k ). Then by the pro of of Prop osition 2.7 , we obtain that there exists a bijection (2.10) ϕ : Std p ( λ ) − → Std( λ − ) , T 7→ ϕ ( T ) = T \ (1 , k ) Clearly ϕ ( s i T ) = s i ϕ ( T ) for 1 ≤ i ≤ n − 2. Moreo ver s n − 1 T / ∈ Std p ( λ ). Then one can iden tify v T with v ϕ ( T ) . Again (2) follows from ( 2.3 ), Theorem 2.4 and Prop osition 2.7 . This pro v es the theorem. □ Corollary 2.9. Supp ose n = p and λ  = (1 n ) . (1) If λ  = ( k , 1 n − k ) , then D λ ∼ = S λ and henc e Rad λ = 0 . (2) If λ = ( k , 1 n − k ) with k ≥ 2 , then dim Rad λ =  n − 2 k − 1  . 3. Irreducible represent a tions of Sergeev superalgebra Y p In this section w e assume that F is an algebraically closed field of characteristic p with p  = 2 or equiv alently p ≥ 3. W e will derive a closed dimension formula and an explicit construction for irreducible represen tations of the Sergeev sup eralgebra Y p o v er F . SYMMETRIC GROUPS AND SERGEEV SUPERALGEBRAS 7 3.1. Basics on affine Sergeev superalgebras. W e first recall some basic on sup eralge- bras, referring the reader to [ K1 , Chapter 12]. A v ector sup erspace V means a Z 2 -graded space V = V ¯ 0 ⊕ V ¯ 1 o v er F . Denote by ¯ v ∈ Z 2 the parit y of a homogeneous v ector v of a v ector sup erspace. By a sup eralgebra, we mean a Z 2 -graded asso ciativ e algebra. Let A b e a sup eralgebra. An A -mo dule means a Z 2 -graded left A -mo dule and a homomorphism f : V → W of A -mo dules V and W means a linear map such that f ( av ) = ( − 1) ¯ f ¯ a af ( v ) . Note that this and other such expressions only make sense for homogeneous a, f and the meaning for arbitrary elemen ts is to b e obtained b y extending linearly from the homo- geneous case. Let V b e a finite dimensional A -mo dule. Let Π V b e the same underlying v ector space but with the opp osite Z 2 -grading. The new action of a ∈ A on v ∈ Π V is defined in terms of the old action by a · v := ( − 1) ¯ a av . Note that the identit y map on V defines an isomorphism from V to Π V . By a sup eralgebra analog of Sch ur’s Lemma, the endomorphism algebra of a finite dimensional irreducible mo dule ov er a sup eralgebra is either one dimensional or tw o di- mensional. In the former case, we call the mo dule of typ e M while in the latter case the mo dule is called of typ e Q . Giv en tw o sup eralgebras A and B , the tensor pro duct of sup erspaces A ⊗ B can b e view ed as a sup eralgebra with multiplication defined by ( a ⊗ b )( a ′ ⊗ b ′ ) = ( − 1) ¯ b ¯ a ′ ( aa ′ ) ⊗ ( bb ′ ) ( a, a ′ ∈ A , b, b ′ ∈ B ) . Supp ose V is an A -mo dule and W is a B -module. Then V ⊗ W affords an A ⊗ B -mo dule denoted b y V ⊠ W via ( a ⊗ b )( v ⊗ w ) = ( − 1) ¯ b ¯ v av ⊗ bw, a ∈ A, b ∈ B , v ∈ V , w ∈ W. If V is an irreducible A -mo dule and W is an irreducible B -mo dule, V ⊠ W may not b e irreducible. Indeed, we ha ve the follo wing standard lemma (cf. [ K1 , Lemma 12.2.13]). Lemma 3.1. L et V b e an irr e ducible A -mo dule and W b e an irr e ducible B -mo dule. (1) If b oth V and W ar e of typ e M , then V ⊠ W is an irr e ducible A ⊗ B -mo dule of typ e M . (2) If one of V or W is of typ e M and the other is of typ e Q , then V ⊠ W is an irr e ducible A ⊗ B -mo dule of typ e Q . (3) If b oth V and W ar e of typ e Q , then V ⊠ W ∼ = X ⊕ Π X for a typ e M irr e ducible A ⊗ B -mo dule X . Mor e over, al l irr e ducible A ⊗ B -mo dules arise as c onstituents of V ⊠ W for some choic e of irr e ducibles V , W . If V is an irreducible A -mo dule and W is an irreducible B -mo dule, denote by V ⊛ W an irreducible comp onen t of V ⊠ W . Thus, V ⊠ W =  V ⊛ W ⊕ Π( V ⊛ W ) , if b oth V and W are of type Q , V ⊛ W , otherwise . Definition 3.2. F or n ≥ 1, the affine Sergeev sup eralgebra H c n is the sup eralgebra gener- ated b y even generators s 1 , . . . , s n − 1 , x 1 , . . . , x n and o dd generators c 1 , . . . , c n sub ject to 8 MINJIA CHEN, JINKUI W AN AND HONGBO ZHAO the follo wing relations s 2 i = 1 , s i s j = s j s i , s i s i +1 s i = s i +1 s i s i +1 , | i − j | > 1 , x i x j = x j x i , 1 ≤ i, j ≤ n, (3.1) c 2 i = 1 , c i c j = − c j c i , 1 ≤ i  = j ≤ n, (3.2) s i x i = x i +1 s i − (1 + c i c i +1 ) , s i x j = x j s i , j  = i, i + 1 , s i c i = c i +1 s i , s i c i +1 = c i s i , s i c j = c j s i , j  = i, i + 1 , x i c i = − c i x i , x i c j = c j x i , 1 ≤ i  = j ≤ n. (3.3) R emark 3.3 . The affine Sergeev sup eralgebra H c n w as introduced b y Nazarov [ N ](called affine Sergeev algebra) to study the represen tations of C S − n . The quan tized v ersion of the H c n in tro duced later by Jones-Nazarov [ JN ] to study the q -analogues of Y oung symmetrizers for pro jectiv e represen tations of the symmetric group S n is often also called affine Hec k e-Clifford algebras. F or α = ( α 1 , . . . , α n ) ∈ Z n + and β = ( β 1 , . . . , β n ) ∈ Z n 2 , set x α = x α 1 1 · · · x α n n and c β = c β 1 1 · · · c β n n . Then we ha ve the follo wing. Lemma 3.4. [ BK , Theorem 2.2] The set { x α c β w | α ∈ Z n + , β ∈ Z n 2 , w ∈ S n } forms a b asis of H c n . F or each i ∈ Z , set q ( i ) = i ( i + 1) . (3.4) Denote b y Z + the set of nonnegativ e integers and let I =  Z + , if p = 0 , { 0 , 1 , . . . , p − 1 2 } , if p ≥ 3 . (3.5) Then it is easy to v erify (3.6) if i, j ∈ I , then q ( i ) = q ( j ) if and only if i = j. and moreo ver { q ( i ) | i ∈ Z } = { q ( i ) | i ∈ I } . This justifies the in tro duction of I . Denote b y P c n the sup eralgebra generated b y even generators x 1 , . . . , x n and o dd generators c 1 , . . . , c n sub ject to the relations ( 3.1 ), ( 3.2 ) and ( 3.3 ). By Lemma 3.4 , P c n can b e identified with the subalgebra of H c n generated b y x 1 , . . . , x n and c 1 , . . . , c n . F or a comp osition µ = ( µ 1 , µ 2 , . . . , µ r ) of n , we define H c µ to b e the subalgebra of P c n generated b y P c n and s j ∈ S µ = S µ 1 × · · · × S µ r . Note that P c n = H c (1 n ) . Let us denote b y Rep I H c µ the category of so-called inte gr al finite dimensional H c µ -mo dules on which the x 2 1 , . . . , x 2 n ha v e eigen v alues of the form q ( i ) for i ∈ I . F or each i ∈ I , denote by L ( i ) the 2-dimensional P c 1 -mo dule with L ( i ) ¯ 0 = F v 0 and L ( i ) ¯ 1 = F v 1 and (3.7) x 1 v 0 = p q ( i ) v 0 , x 1 v 1 = − p q ( i ) v 1 , c 1 v 0 = v 1 , c 1 v 1 = v 0 . SYMMETRIC GROUPS AND SERGEEV SUPERALGEBRAS 9 Note that L ( i ) is irreducible of t ype M if i  = 0, and irreducible of type Q if i = 0. Moreov er { L ( i ) | i ∈ I } forms a complete set of pairwise non-isomorphic irreducible P c 1 -mo dule in the category Rep I P c 1 . Observ e that P c n ∼ = P c 1 ⊗ · · · ⊗ P c 1 , and hence we hav e the following result b y Lemma 3.1 . Lemma 3.5. [ BK , Lemma 4.8] The set of P c n -mo dules { L ( i ) = L ( i 1 ) ⊛ L ( i 2 ) ⊛ · · · ⊛ L ( i n ) | i = ( i 1 , . . . , i n ) ∈ I n } forms a c omplete set of p airwise non-isomorphic irr e ducible P c n -mo dule in the c ate gory Rep I P c n . Mor e over, denote by γ 0 the numb er of 1 ≤ j ≤ n with i j = 0 . Then L ( i ) is of typ e M if γ 0 is even and typ e Q if γ 0 is o dd. F urthermor e, dim L ( i ) = 2 n −⌊ γ 0 2 ⌋ , wher e ⌊ γ 0 2 ⌋ denotes the gr e atest inte ger less than or e qual to γ 0 2 . R emark 3.6 . Note that each p erm utation τ ∈ S n defines a sup eralgebra isomorphism τ : P c n → P c n b y mapping x k to x τ ( k ) and c k to c τ ( k ) , for 1 ≤ k ≤ n . F or i ∈ I n , the twist of the action of P c n on L ( i ) with τ − 1 leads to a new P c n -mo dule denoted by L ( i ) τ with L ( i ) τ = { z τ | z ∈ L ( i ) } , f z τ = ( τ − 1 ( f ) z ) τ , for an y f ∈ P c n , z ∈ L ( i ) . So in particular w e hav e ( x k z ) τ = x τ ( k ) z τ and ( c k z ) τ = c τ ( k ) z τ . It is easy to see that (3.8) L ( i ) τ ∼ = L ( τ · i ) , where τ · i := ( i τ − 1 (1) , . . . , i τ − 1 ( n ) ) for i = ( i 1 , . . . , i n ) ∈ I n and τ ∈ S n . Definition 3.7. [ W a , Definition 3.2] A finite dimensional H c n -mo dules is said to b e com- pletely splittable if the elemen ts x 1 , x 2 , . . . , x n act semisimply . It is known that a classification of irreducible completely splittable H c n -mo dule has been obtained in [ W a , Theorem 4.5, Theorem 5.7, Theorem 5.15 ]. W e will give a brief review in the follo wing. Definition 3.8. Let P ( H c n ) ⊂ I n b e the subset consisting of i = ( i 1 , i 2 , . . . , i n ) ∈ I n satisfying the follo wing conditions: (1) i k  = i k +1 for all 1 ≤ k ≤ n − 1. (2) The element p − 1 2 ∈ I app ears at most once in i . (3) If i k = i l = 0 for some 1 ≤ k < l ≤ n , then 1 ∈ { i k +1 , . . . , i l − 1 } . (4) If i k = i l ≥ 1 for some 1 ≤ k < l ≤ n , then either of the following holds: (a) { i k − 1 , i k + 1 } ⊆ { i k +1 , . . . , i l − 1 } , (b) there exists a sequence of in tegers k ≤ r 0 < r 1 < · · · < r p − 3 2 − i k < q < t p − 3 2 − i k < · · · < t 1 < t 0 ≤ l such that i q = p − 1 2 , i r j = i t j = i k + j and i k + j do es not app ear b et ween i r j and i t j in i for eac h 0 ≤ j ≤ p − 3 2 − i k . F or i ∈ I n and 1 ≤ k ≤ n − 1, the simple transp osition s k is said to b e admissible with resp ect to i if i k  = i k +1 ± 1. Define an equiv alence relation ∼ on I n b y declaring that i ∼ j if there exist s k 1 , . . . , s k t for some t ∈ Z + suc h that j = ( s k t · · · s k 1 ) · i and s k l is admissible with resp ect to ( s k l − 1 · · · s k 1 ) · i for 1 ≤ l ≤ t . Observe that if i ∈ P ( H c n ) and 10 MINJIA CHEN, JINKUI W AN AND HONGBO ZHAO s k is admissible with resp ect to i , then the conditions in Definition 3.8 hold for s k · i and hence s k · i ∈ P ( H c n ). This means there is an equiv alence relation denoted by ∼ on P ( H c n ) inherited from the equiv alence relation ∼ on I n . F or each i ∈ P ( H c n ), set Λ i = { j ∈ P ( H c n ) | j ∼ i } , P i = { τ = s k t · · · s k 1 | s k a is admissible with resp ect to s k a − 1 · · · s k 1 · i , 1 ≤ a ≤ t, t ∈ Z + } . Then b y [ W a ] we hav e the following. Lemma 3.9. [ W a , Lemma 4.1] F or e ach i ∈ P ( H c n ) , the map ϕ : P i → Λ i , τ 7→ τ · i = ( i τ − 1 (1) , i τ − 1 (2) , . . . , i τ − 1 ( n ) ) (3.9) is bije ctive. F or eac h i ∈ P ( H c n ), b y Definition 3.8 w e hav e i k  = i k +1 for each 1 ≤ k ≤ n − 1 and then by ( 3.6 ) we ha ve q ( i k )  = q ( i k +1 ). Th us we can define tw o linear op erators Ξ k and Ω k on the P c n -mo dule L ( i ) such that for any z ∈ L ( i ), Ξ k z := −  x k + x k +1 x 2 k − x 2 k +1 + c k c k +1 x k − x k +1 x 2 k − x 2 k +1  z , (3.10) Ω k z := s 1 − 2( x 2 k + x 2 k +1 ) ( x 2 k − x 2 k +1 ) 2 ! z = s 1 − 2( q ( i k ) + q ( i k +1 )) ( q ( i k ) − q ( i k +1 )) 2 ! z . (3.11) Supp ose i ∈ P ( H c n ). Recall the definition of L ( i ) τ from Remark 3.6 for τ ∈ P i . Denote b y V i the P c n -mo dule defined via V i = ⊕ τ ∈ P i L ( i ) τ . (3.12) Then w e ha ve the following due to [ W a ]. Theorem 3.10. [ W a , Theorem 4.5] Supp ose i, j ∈ P ( H c n ) . Then, (1) V i affor ds an irr e ducible H c n -mo dule via s k z τ =  Ξ k z τ + Ω k z s k τ , if s k is admissible with r esp e ct to τ · i, Ξ k z τ , otherwise , (3.13) for 1 ≤ k ≤ n − 1 , z ∈ L ( i ) and τ ∈ P i . It has the same typ e as the irr e ducible P c n -sup ermo dule L ( i ) . (2) V i ∼ = V j if and only if i ∼ j . (3) Every irr e ducible c ompletely splittable H c n -mo dule in Rep I H c n is isomorphic to V i for some i ∈ P ( H c n ) . Henc e the e quivalenc e classes P ( H c n ) / ∼ p ar ametrize irr e- ducible c ompletely splittable H c n -sup ermo dules in the c ate gory Rep I H c n . SYMMETRIC GROUPS AND SERGEEV SUPERALGEBRAS 11 3.2. Irreducible completely splittable represen tations of Y n . Denote by C n the subalgebra of H c n generated b y c 1 , . . . , c n , which is known as the Clifford algebra. The Sergeev sup er algebra Y n = C n ⋊ F S n is isomorphic to the subalgebra of H c n generated by c 1 , . . . , c n , s 1 , . . . , s n − 1 . W e first recall the classification of irreducible Y n -mo dules obtained in [ BK ] (cf. [ K1 ]). A partition λ is called p -strict if p divides λ r whenev er λ r = λ r +1 for r ≥ 1. W e sa y that a p -strict partition is p -r estricte d if in addition ( λ r − λ r +1 < p if p | λ r , λ r − λ r +1 ≤ p if p ∤ λ r , Denote by RP p ( n ) b e the set of p -restricted p -stricted partition of n . It is known that for eac h λ ∈ RP p ( n ), there exists an irreducible Y n -sup ermodule M λ . W e refer the reader to [ K1 ] for details of the construction. Let b ( λ ) := ♯ { r ≥ 1 | p ∤ λ r > 0 } b e the num ber of (non-zero) parts of λ that are not divisible by p . Theorem 3.11. [ K1 , Theorem 22.2.1] The set { M λ | λ ∈ RP p ( n ) } forms a c omplete set of p airwise non-isomorphic irr e ducible Y n -mo dule. Mor e over, for λ ∈ RP p ( n ) , M λ is of typ e M if b ( λ ) is even, typ e Q if b ( λ ) is o dd. The Jucys-Murph y elemen ts L k (1 ≤ k ≤ n ) in Y n are defined as L k = X 1 ≤ j λ 2 > · · · > λ ℓ > 0. Denote by S P ( n ) the set of strict partition of n . Similar to the case of partitions, a strict partition λ ∈ S P ( n ) can b e identified with the shifted Y oung diagram whic h is obtained from the ordinary Y oung diagram by shifting the k -th row to the righ t b y k − 1 squares for all k > 1, that is, (3.16) λ s = { ( i, j ) | 1 ≤ i ≤ ℓ ( λ ) , i ≤ j ≤ i + λ i − 1 } . The ( i, j )-ho ok of a shifted Y oung diagram contains all no des that are either in the same ro w as ( i, j ) and to the right of ( i, j ), or in the same column as ( i, j ) and b elo w ( i, j ) including ( i, j ). Additionally if ( j, j ) is included in the ho ok then no des in the ( j + 1)-row are also included. Denote by h s λ ( i, j ) the num ber of no des in the ( i, j )-ho ok. Example 3.14. Supp ose λ = (7 , 5 , 3 , 2), then the (1 , 2)-ho ok and (1 , 4)-ho ok of shifted Y oung diagram λ s are as b elo w • • • • • • • • • • , • • • • • • • , and accordingly h s λ (1 , 2) = 10 and h s λ (1 , 4) = 7. Denote by T s ( λ ) the set of shifted tableaux of shap e λ s ; that is, a shifted tableau is a lab elling of the no des in the s hifted Y oung diagram λ s with the entries 1 , 2 , . . . , n . Let T ( i,j ) denote the en try in the no de ( i, j ) and let T ( k ) be the no de whic h is o ccupied b y the n um b er k for each 1 ≤ k ≤ n . So if T ( i,j ) = k then T ( k ) = ( i, j ). A shifted tableau T is called standar d if its en tries strictly increase from left to right along each ro w and do wn eac h column. W e denote b y Std s ( λ ) the subset of T s ( λ ) consisting of standard tableaux of shap e λ s . W e then ha v e the follo wing remark able ho ok length formula (cf. [ Mac , Chapter I II, Section 8, Example 12]) (3.17) ♯ Std s ( λ ) = n ! Q ( i,j ) ∈ λ s h s λ ( i, j ) where the pro duct in the denominator is ov er all no des in the shifted Y oung diagram λ s . F ollowing [ W a , Lemma 6.6-6.7], we set C P s p ( n ) = n ξ = ( ξ 1 , ξ 2 , . . . ) ∈ S P ( n ) | ξ 1 = p − u, ξ 2 ≤ u for some 1 ≤ u ≤ p − 3 2 or 1 ≤ ξ 1 ≤ p + 1 2 o (3.18) SYMMETRIC GROUPS AND SERGEEV SUPERALGEBRAS 13 and in addition define Std s p ( ξ ) =     T ∈ Std s ( ξ ) | T (2 ,ξ 2 +1) > T (1 ,ξ 1 )  , if ξ 1 = p − u, ξ 2 = u for some 1 ≤ u ≤ p − 3 2 , Std s ( ξ ) , otherwise . (3.19) for eac h ξ ∈ C P s p ( n ). A tableau T is called p -standar d if T ∈ Std s p ( ξ ) for some ξ ∈ C P s p ( n ). W e lab el the residue of no des in the shifted Y oung diagram of λ ∈ C P s p ( n ) using the set I in ( 3.5 ) via the wa y that the first no de in each row has residue 0 and then follow the rep eating pattern (3.20) 0 , 1 , . . . , p − 3 2 , p − 1 2 , p − 3 2 , . . . , 1 , 0 . The residue in ( 3.20 ) is actually to compute q -v alues of the usual residue j − i of the no des ( i, j ) and the reason for this pattern is due to the observ ation q ( a ) = q ( b ) if a = b mo d p or a + b + 1 = 0 mo d p for any a, b ∈ Z . Let ξ ∈ C P s p ( n ) and supp ose T ∈ Std s p ( ξ ). Let i T = (res( T (1)) , res( T (2)) , · · · , res( T ( n ))) ∈ I n b e the residue sequence corresp onding to T . Example 3.15. Let p = 7, so p − 1 2 = 3. Then ξ = (5 , 2 , 1) b elongs to C P s p (8). The residues of no des in the shifted Y oung diagram ξ s are as follo ws: 2 3 2 1 1 0 0 0 . In addition for T = 8 7 5 6 4 3 2 1 ∈ Std s p ( ξ ) , w e ha ve i T = (0 , 1 , 2 , 3 , 0 , 2 , 1 , 0). Lemma 3.16. [ W a , Lemma 6.6-6.7] L et ξ ∈ C P s p ( n ) and supp ose T ∈ Std s p ( ξ ) . Then (1) i T ∈ P ( H c n ) and mor e over if j ∈ P ( H c n ) , then j ∼ i T if and only if j = i S for some S ∈ Std s p ( ξ ) . Henc e Λ i T = { i S | S ∈ Std s p ( ξ ) } . (2) If ξ  = γ ∈ C P s p ( n ) , then i T ≁ i S for any T ∈ Std s p ( ξ ) and S ∈ Std s p ( γ ) . Assume ξ ∈ C P s p ( n ) and T ∈ Std s p ( ξ ). Recall the mo dule V i for each i ∈ P ( H c n ) defined in ( 3.12 ) and set (3.21) V ξ := V i T . Then V ξ admits a H c n -mo dule by Theorem 3.10 . Clearly b y Lemma 3.16 and Theorem 3.10 the H c n -mo dule V ξ up to isomorphism is indep enden t of the choice of T ∈ Std s p ( ξ ). 14 MINJIA CHEN, JINKUI W AN AND HONGBO ZHAO Theorem 3.17. (cf. [ W a , Theorem 6.8] ) The set { V ξ | ξ ∈ C P s p ( n ) } is a c omplete set of non-isomorphic irr e ducible c ompletely splittable Y n -mo dules. Mor e over dim V ξ = 2 n −⌊ ℓ ( ξ ) 2 ⌋ ♯ Std s p ( ξ ) . Pr o of. Fix T ∈ C P s p ( n ). Observ e that for eac h τ ∈ P i T , there exists a unique S ∈ Std s p ( ξ ) suc h that τ · i T = i S ∈ Λ i T b y Lemma 3.9 and Lemma 3.16 . This together with ( 3.8 ) giv es rise to an isomorphism ψ τ T : L ( i T ) τ ∼ = − → L ( i S ) (3.22) Then b y ( 3.7 ) and Lemma 3.5 w e obtain x 1 = 0 on L ( i T ) τ and hence x 1 = 0 on V ξ . This means V ξ is actually a Y n -mo dule. Then b y Theorem 3.10 and Lemma 3.16 , one can obtain that V ξ ∼ = V γ if and only if ξ = γ ∈ C P s p ( n ). Then the theorem follo ws from [ W a , Theorem 6.8] b y comparing the parametrizing set. □ Clearly for eac h n ≥ 1, C P s p ( n ) ⊆ RP p ( n ) and b y [ W a , Remark 6.9] we hav e (3.23) V ξ ∼ = M ξ for eac h ξ ∈ C P s p ( n ). 3.3. Irreducible represen tations of Y n in the case n = p . Lemma 3.18. F or n ≥ 1 , the fol lowing holds: RP p ( n ) = ( S P ( n ) , if p > n ; S P ( n ) \ { ( n ) } , if p = n. (3.24) Pr o of. Supp ose p > n . F or any λ ∈ RP p ( n ), if λ r = λ r +1 for some r , then p | λ r . Since λ r ≤ n < p , it follows that λ r = 0, which implies λ ∈ S P ( n ). Th us, RP p ( n ) ⊆ S P ( n ). Con v ersely , an y µ ∈ S P ( n ) is clearly p -strict since no non-zero parts are equal. Moreov er, µ is p -restricted b ecause µ r − µ r +1 ≤ µ 1 ≤ n < p for all r . Th us, µ ∈ RP p ( n ), showing that S P ( n ) ⊆ RP p ( n ). Therefore, RP p ( n ) = S P ( n ) when n < p . No w assume n = p . Note that the partition ( n ) is not p -restricted since λ 1 − λ 2 = n − 0 = p , and p | p . Consequen tly , any λ ∈ RP p ( n ) m ust satisfy λ 1 ≤ n − 1. If λ r = λ r +1 , then p | λ r ; since λ r ≤ λ 1 < p , we must hav e λ r = 0, implying λ ∈ S P ( n ). Con v ersely , for an y µ ∈ S P ( n ) such that µ  = ( n ), w e ha ve µ 1 ≤ n − 1. It follows that µ is p -strict and µ r − µ r +1 ≤ µ 1 ≤ n − 1 < p for all r . Th us, µ ∈ RP p ( n ), which yields RP p ( n ) = S P ( n ) \ { ( n ) } . □ Lemma 3.19. F or n ≥ 1 , the fol lowing holds: C P s p ( n ) = ( S P ( n ) , if p > n ; S P ( n ) \ { ( n ) } , if p = n. (3.25) Pr o of. Supp ose p > n . By ( 3.18 ), it suffices to show that every λ ∈ S P ( n ) is contained in C P s p ( n ). This clearly holds if λ 1 ≤ p +1 2 . Otherwise, supp ose p +3 2 ≤ λ 1 ≤ n ≤ p − 1 and SYMMETRIC GROUPS AND SERGEEV SUPERALGEBRAS 15 let u = p − λ 1 . Then 1 ≤ u ≤ p − 3 2 . Since λ ∈ S P ( n ), we ha v e λ 2 ≤ n − λ 1 ≤ p − 1 − λ 1 < p − λ 1 = u . Thus, λ satisfies the conditions for C P s p ( n ). No w assume n = p . Clearly ( n ) / ∈ C P s p ( n ) b ecause n > p − 1. F or an y λ ∈ S P ( n ) with λ  = ( n ), w e hav e λ 1 ≤ n − 1 = p − 1. As in the previous case, either λ 1 ≤ p +1 2 or the condition 1 ≤ u = p − λ 1 ≤ p − 3 2 is satisfied. In the latter case, λ 2 ≤ n − λ 1 = p − λ 1 = u , whic h implies λ ∈ C P s p ( n ). Therefore, C P s p ( n ) = S P ( n ) \ { ( n ) } . □ Prop osition 3.20. RP p ( n ) = C P s p ( n ) if and only if n ≤ p . Henc e, al l irr e ducible Y n - mo dules ar e c ompletely splittable if and only if n ≤ p . Pr o of. By Lemma 3.18 and Lemma 3.19 , we hav e RP p ( n ) = C P s p ( n ) if n ≤ p . No w supp ose p < n, then n = ap + b for some a ∈ Z + , 0 ≤ b ≤ p − 1. Obviously either λ = ( p a , b )(the case b  = 0) or µ = ( p a − 1 , p − 1 , 1)(the case b = 0 and then a > 1) b elongs to RP p ( n ). But neither of them b elongs to C P s p ( n ) since λ 1 = µ 1 = p > p − 1. Hence RP p ( n )  = C P s p ( n ). This prov es the prop osition. □ Prop osition 3.21. Supp ose n ≤ p and ξ ∈ C P s p ( n ) . Then ♯ Std s p ( ξ ) = ( n − 2 u +1 n − u  n − 2 u − 1  , if ξ = ( p − u, u ) for some 1 ≤ u ≤ p − 3 2 , n ! Q h s ξ ( i,j ) , otherwise . Pr o of. Clearly in the case n < p w e hav e ξ / ∈ { ( p − u, u, . . . ) ∈ C P s p ( n ) | 1 ≤ u ≤ p − 3 2 } and then the prop osition follows from ( 3.19 ) and ( 3.17 ). It remains to consider the case n = p . F or any 1 ≤ u ≤ p − 3 2 , the only partition satisfying ξ 1 = p − u and ξ 2 = u is ξ = ( n − u, u ) = ( p − u, u ). Then by ( 3.19 ) we hav e (3.26) Std s p ( ξ ) = Std s ( ξ ) if ξ / ∈ { ( p − u, u, . . . ) ∈ C P s p ( n ) | 1 ≤ u ≤ p − 3 2 } and then the prop osition follo ws from ( 3.17 ). Now supp ose ξ = ( n − u, u ) = ( p − u, u ) with 1 ≤ u ≤ p − 3 2 . By ( 3.19 ), a tableau T ∈ Std s ( ξ ) b elongs to Std s p ( ξ ) if and only if T (2 ,u +1) > T (1 ,n − u ) , which implies T (2 ,u +1) = n . This leads to Std s p ( ξ ) = { T ∈ Std s ( ξ ) | T (2 ,u +1) = n } and hence (3.27) ♯ Std s p ( ξ ) = ♯ Std s ( ξ − ) , where ξ − = ( n − u, u − 1) ∈ S P ( n − 1) due to n − u > u − 1 as 1 ≤ u ≤ p − 3 2 . Now for ξ − = ( n − u, u − 1), it is straightforw ard to verify that the ho ok lengths h s ξ − ( i, j ) of ( i, j ) ∈ ( ξ − ) s are n − 1 , n − u, n − u − 1 , . . . , n − 2 u + 2 , n − 2 u, n − 2 u − 1 , . . . , 1 , u − 1 , . . . , u and hence b y ( 3.17 ) we obtain ♯ Std s ( ξ − ) = ( n − 2 u + 1)( n − 1)! ( n − 1) · ( n − u )!( u − 1)! = n − 2 u + 1 n − u  n − 2 u − 1  . Then the prop osition follows from ( 3.26 ), ( 3.27 ) and ( 3.17 ). □ If ξ ∈ C P s p ( n ) and T ∈ Std s p ( ξ ), w e write κ T ( k ) = q (res( T ( k ))) = res( T ( k ))(res( T ( k )) + 1) . 16 MINJIA CHEN, JINKUI W AN AND HONGBO ZHAO By ( 3.16 ) and Definition 3.8 , w e obtain res( T ( k ))  = res( T ( k + 1)) for 1 ≤ k ≤ n − 1 and hence κ T ( k )  = κ T ( k + 1). Thus w e can introduce the following tw o w ell-defined elements in P c n : (3.28) Ξ T k = − 1 p κ T ( k ) − p κ T ( k + 1) + c k c k +1 p κ T ( k ) + p κ T ( k + 1) ! and (3.29) Ω T k = s 1 − 2( κ T ( k ) + κ T ( k + 1)) ( κ T ( k ) − κ T ( k + 1)) 2 for 1 ≤ k ≤ n − 1. Then we are ready to in tro duce our main result. Theorem 3.22. Supp ose n = p . L et ξ ∈ RP p ( n ) = C P s p ( n ) . Then (1) If ξ / ∈ { ( p − u, u ) | 1 ≤ u ≤ p − 3 2 } , then ther e exists a Y n -mo dule isomorphism M ξ ∼ = V ξ ∼ = ⊕ T ∈ Std s ( ξ ) L ( i T ) with the action of C n ⊂ P c n on L ( i T ) given via L emma 3.5 and the action of s k ∈ S n for 1 ≤ k ≤ n − 1 is given by (3.30) s k z =    Ξ T k z + Ω T k z s k , if s k T ∈ Std s ( ξ ) , Ξ T k z , otherwise , for e ach z ∈ L ( i T ) and T ∈ Std s ( ξ ) . Mor e over (3.31) dim M ξ = 2 n −⌊ ℓ ( ξ ) 2 ⌋ n ! Q ( i,j ) ∈ ξ s h s ξ ( i, j ) . (2) If ξ = ( p − u, u ) for some 1 ≤ u ≤ p − 3 2 , then ther e exists a Y n -mo dule isomorphism satisfies M ξ ∼ = V ξ ∼ = ⊕ T ∈ Std s p ( ξ ) L ( i T ) with the action of C n ⊂ P c n on L ( i T ) given via L emma 3.5 and the action of s k ∈ S n for 1 ≤ k ≤ n − 1 is given by (3.32) s k z =            Ξ T k z + Ω T k z s k , if 1 ≤ k ≤ n − 2 , s k T ∈ Std s p ( ξ ) , Ξ T k z , if 1 ≤ k ≤ n − 2 , s k T / ∈ Std s p ( ξ ) , Ξ T n − 1 z , if k = n − 1 . for e ach z ∈ L ( i T ) and T ∈ Std s p ( ξ ) , wher e Ξ T n − 1 satisifies (3.33) Ξ T n − 1 =      ( 1 √ u ( u +1) − √ u ( u − 1) + c n − 1 c n √ u ( u +1)+ √ u ( u − 1) ) , if T (1 ,n − u ) = n − 1 , ( 1 √ ( u − 1)( u − 2) − √ u ( u − 1) + c n − 1 c n √ ( u − 1)( u − 2)+ √ u ( u − 1) ) , if T (2 ,u ) = n − 1 . SYMMETRIC GROUPS AND SERGEEV SUPERALGEBRAS 17 Mor o ever (3.34) dim M ξ = 2 n −⌊ ℓ ( ξ ) 2 ⌋ ( n − 2 u + 1) n − u  n − 2 u − 1  . In addition, the set { V ξ | ξ ∈ C P s p ( n ) = S P ( n ) \ ( n ) } is a c omplete set of non-isomorphic irr e ducible Y n -mo dules. Pr o of. Firstly , b y Prop osition 3.20 and ( 3.23 ) we hav e RP p ( n ) = C P s p ( n ) = S P ( n ) \ ( n ) and M ξ ∼ = V ξ for each ξ ∈ RP p ( n ). Observe that eac h p erm utation τ naturally acts on an arbitrary tableau of shifted Y oung diagram ξ s to a get a new tableau τ · S by p erm uting the en tries in S and moreov er (3.35) τ · i S = i τ · S . In particular, fixing T ξ ∈ Std s p ( ξ ), one can obtain i τ · T ξ ∈ Λ i T ξ for each τ ∈ P i T ξ b y Lemma 3.9 and then b y Lemma 3.16 we ha ve τ · T ξ ∈ Std s p ( ξ ). This leads to Std s p ( ξ ) = { τ · T ξ | τ ∈ P i T ξ } . and moreo v er (3.36) P i T ξ = { τ = s k t · · · s k 2 s k 1 | s k a · · · s k 2 s k 1 · T ξ ∈ Std s p ( ξ ) for 1 ≤ a ≤ t } This together with ( 3.35 ) and Remark 3.6 leads to V i T ξ = ⊕ τ ∈ P i T ξ L ( i T ξ ) τ ∼ = ⊕ T ∈ Std s p ( ξ ) L ( i T ) Moreo v er, for each z ∈ L ( i T ξ ) and τ ∈ P i T ξ , we hav e z τ ∈ L ( i T ξ ) τ ∼ = L ( i T ) with T = τ · T ξ b y ( 3.35 ) and moreov er if s k is admissible with resp ect to τ · i T ξ then (3.37) z s k τ = ( z τ ) s k b y using the t wist action in Remark 3.6 . Then by the isomorphism ψ τ T ξ : L ( i T ξ ) τ → L ( i T ) in ( 3.22 ) with T = τ · T ξ , we can identify z τ with its image in L ( i T ) and then the action of s k in ( 3.30 ) follows from ( 3.13 ) in Theorem 3.10 . In addition, the dimensional formula ( 3.31 ) is due to Lemma 3.5 and Prop osition 3.21 . This together ( 3.21 ) and ( 3.23 ) with pro v es part (1). One can prov e the action form ula ( 3.32 ) and the dimension formula ( 3.34 ) in part (2) applying the same approach with some extra computation need to v erify ( 3.33 ) for the action of s n − 1 . Actually , if ξ = ( p − u, u ) for some 1 ≤ u ≤ p − 3 2 , then T (2 ,u +1) = n since an y T ∈ Std s p ( ξ ) satisfies T (2 ,u +1) > T (1 ,p − u ) b y ( 3.19 ). This means s n − 1 T / ∈ Std s p ( ξ ). Then b y ( 3.13 ) in Theorem 3.10 we obtain s n − 1 z = Ξ T n − 1 z for eac h z ∈ L ( i T ) and T ∈ Std s p ( ξ ). Since T (2 ,u +1) = n , one can deduce that either T (2 ,u ) = n − 1 with u ≥ 2 or T (1 ,p − u ) = n − 1. This means res( T ( n )) = u − 1 ∈ I and res( T ( n − 1)) = u − 2 ∈ I with u ≥ 2 or res( T ( n − 1)) = u ∈ I according to the residue lab eling pattern defined in ( 3.20 ). Then by ( 3.28 ) the equation ( 3.33 ) is verified. This pro v es the theorem. □ 18 MINJIA CHEN, JINKUI W AN AND HONGBO ZHAO References [BK] J. Brundan and A. 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W an and W. W ang, L e ctur es on spin r epresentation the ory of symmetric gr oups , Bull. Inst. Math. Acad. Sinica (N.S.) 7 (2012), 91–164. Minjia Chen, School of Ma thema tics and St a tistics, Beijing Institute of Technology, Beijing 100081, China Email address : scarleturanus@163.com Jinkui W an, School of Ma thema tical Sciences, Shenzhen University, Shenzhen, 518060, P.R. China Email address : wjk302@hotmail.com Hongbo Zhao, School of Ma thema tics and St a tistics, Beijing Institute of Technology, Beijing 100081, China Email address : 2767451726@qq.com