Jacobi-Trudi identity and Drinfeld functor for super Yangian
aa r X i v : . [ m a t h . QA ] J u l JACOBI-TRUDI IDENTITY AND DRINFELD FUNCTOR FOR SUPER YANGIAN
KANG LU AND EVGENY MUKHINA
BSTRACT . We show that the quantum Berezinian which gives a generating function of the integrals of motionsof XXX spin chains associated to super Yangian Y( gl m | n ) can be written as a ratio of two difference operatorsof orders m and n whose coefficients are ratios of transfer matrices corresponding to explicit skew Youngdiagrams.In the process, we develop several missing parts of the representation theory of Y( gl m | n ) such as q -charactertheory, Jacobi-Trudi identity, Drinfeld functor, extended T-systems, Harish-Chandra map. Keywords: super Yangian, Jacobi-Trudi identity, degenerate affine Hecke algebra, Drinfeld functor, T-systems,transfer matrices.
1. I
NTRODUCTION
This paper deals with the representation theory of Y( gl m | n ) , the Yangian associated with the generalLie superalgebra gl m | n . However, the original motivation comes from the theory of integrable systems.The superalgebra Y( gl m | n ) contains a commutative subalgebra (called the Bethe subalgebra ) given by theexpansion of a certain quantum Berezinian, see [MR14] and (5.1). The coefficients of the expansion aretransfer matrices related to representations associated to single-column Young diagrams, see (5.2). On theother hand, the method of Bethe ansatz suggests that it is natural to expect that the quantum Bereziniancan be compactly written as in the form D D − where D and D are difference operators of orders m and n respectively, see [HMVY19, HLM19]. We show that this is indeed the case and, moreover, the i -thcoefficients of D and D are given by transfer matrices corresponding to skew Young diagrams ,n mi ,n mi respectively, divided by the transfer matrix corresponding to the m × n rectangle, see Theorem 5.12. Thatexplains the formulas for the rational difference operators (5.9) corresponding to solutions of Bethe ansatzequations (5.8) and makes a step towards understanding the so-called gl m | n spaces of rational functions[HMVY19], see the discussion in Section 5.5.The proof of the above result is obtained by a direct computation with the use of the Jacobi-Trudi identityfor the Y( gl m | n ) -modules related to the participating skew Young diagrams.More generally, we study irreducible Y( gl m | n ) -modules L ( λ/µ ) corresponding to arbitrary skew Youngdiagrams. These are supersymmetric analogs of the tame modules [NT98], also known as snake modules KANG LU AND EVGENY MUKHIN [MY12a]. We establish the formulas for the q -characters (that is the joint generalized eigenvalues of theGelfand-Tsetlin subalgebra) of such modules in terms of the semi-standard Young tableaux, see Theorem3.4, and prove the Jacobi-Trudi identity for the q -characters. The ways to attack the Jacobi-Trudi identityare well-known. Here we use the Lindstr¨om-Gessel-Viennot method with the appropriate modifications, seeTheorem 3.16.We take the opportunity to define the Drinfeld functor which constructs a Y( gl m | n ) -module from a rep-resentation of the degenerate affine Hecke algebra H l . The key fact is that the Jacobi-Trudi formula doesnot depend on m and n . It implies that the Y( gl m | n ) -module L ( λ/µ ) comes from the same H l -modulefor all m and n . Since the Drinfeld functor is exact and an equivalence of categories in the even case, see[Dri86, CP96] or Theorem 4.3, we obtain a tool to translate the information about representations of evenYangian Y( gl N ) to super Yangian Y( gl m | n ) .We give a couple of such examples, describing sufficient conditions for tensor products of evaluation Y( gl m | n ) -modules to be irreducible, see Theorem 4.18 and establishing the extended T-systems, see Corol-lary 4.24.The paper is constructed as follows. We start by organizing known facts about the general Lie superalgebra gl m | n and the super Yangian Y( gl m | n ) in Section 2. In addition we compute some information about thecoproduct, see Proposition 2.7, which allows us to introduce the q -character ring homomorphism in Section2.7. Section 3 is devoted to the study of Y( gl m | n ) -modules related to skew Young diagrams. Drinfeldfunctor is defined and studied in Section 4. The applications in the form of irreducibility conditions fortensor products and the extended T-systems are given in Sections 4.5 and 4.6, respectively. Section 5 dealswith transfer matrices and, in particular, with quantum Berezinians. We study the Harish-Chandra map whichconnects the transfer matrices to q -characters, see Section 5.3. The main result of this section is Theorem5.12. Acknowledgments.
We thank V. Tarasov for stimulating discussions. This work was partially supportedby a grant from the Simons Foundation
UPER Y ANGIAN Y( gl m | n ) Lie superalgebra gl m | n . Through out the paper, we work over C . In this section, we recall the basicsof the Lie superalgebra gl m | n , see e.g. [CW12] for more detail. We simply write gl m for gl m | .A vector superspace W = W ¯0 ⊕ W ¯1 is a Z -graded vector space. We call elements of W ¯0 even andelements of W ¯1 odd . We write | w | ∈ { ¯0 , ¯1 } for the parity of a homogeneous element w ∈ W . Set ( − ¯0 = 1 and ( − ¯1 = − .Fix m, n ∈ Z > . Set I := { , , . . . , m + n − } and ¯ I := { , , . . . , m + n } . We also set | i | = ¯0 for i m and | i | = ¯1 for m < i m + n . Define s i = ( − | i | for i ∈ ¯ I .The Lie superalgebra gl m | n is generated by elements e ij , i, j ∈ ¯ I , with the supercommutator relations [ e ij , e kl ] = δ jk e il − ( − ( | i | + | j | )( | k | + | l | ) δ il e kj , where the parity of e ij is | i | + | j | . Denote by U( gl m | n ) the universal enveloping superalgebra of gl m | n . Thesuperalgebra U( gl m | n ) is a Hopf superalgebra with the coproduct given by ∆( x ) = 1 ⊗ x + x ⊗ for all x ∈ gl m | n .The Cartan subalgebra h of gl m | n is spanned by e ii , i ∈ ¯ I . Let ǫ i , i ∈ ¯ I , be a basis of h ∗ (the dual spaceof h ) such that ǫ i ( e jj ) = δ ij . There is a bilinear form ( , ) on h ∗ given by ( ǫ i , ǫ j ) = s i δ ij . The root system Φ is a subset of h ∗ given by Φ := { ǫ i − ǫ j | i, j ∈ ¯ I and i = j } . We call a root ǫ i − ǫ j even (resp. odd ) if | i | = | j | (resp. | i | 6 = | j | ). ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 3
Set α i := ǫ i − ǫ i +1 for i ∈ I . Denote by P := ⊕ i ∈ ¯ I Z ǫ i , Q := ⊕ i ∈ I Z α i , and Q > := ⊕ i ∈ I Z > α i the weight lattice , the root lattice , and the cone of positive roots , respectively. Define a partial ordering > on h ∗ : α > β if α − β ∈ Q > . There is a natural Z -grading on P such that the parity p α of α ∈ P is given by p α := X i ∈ ¯ I ( α, ǫ i ) | i | . (2.1)A module M over a superalgebra A is a vector superspace M with the homomorphism of superalgebras A →
End( M ) . A gl m | n -module is a module over U( gl m | n ) .Let α ∈ P . We call a nonzero vector v in a gl m | n -module M a singular vector of weight α if v satisfies e ii v = α ( e ii ) v, e jk v = 0 , for i ∈ ¯ I and j < k m + n . Denote by L p ( α ) the irreducible gl m | n -module generated by a singularvector of weight α and of parity p . We simply write L ( α ) for L p α ( α ) .For a gl m | n -module M , define the weight subspace of weight α by ( M ) α := { v ∈ M | e ii v = α ( e ii ) v, i ∈ ¯ I } . If ( M ) α = 0 , we call α a weight of M . Denote by wt( M ) the set of all weights of M . If v ∈ ( M ) α and v isnon-zero, then we write wt( v ) = α . We focus on the gl m | n -modules M such that ( M ) α = 0 unless α ∈ P .We say that M is P -graded . We call a vector v ∈ M singular if e ij v = 0 for i < j m + n .Let V := C m | n be the vector superspace with a basis v i , i ∈ ¯ I , such that | v i | = | i | . Let E ij ∈ End( V ) be the linear operators such that E ij v k = δ jk v i . The map ρ V : gl m | n → End( V ) , e ij E ij defines a gl m | n -module structure on V . As a gl m | n -module, V is isomorphic to L ( ǫ ) . The vector v i has weight ǫ i .The highest weight vector is v and the lowest weight vector is v m + n .We call it the vector representation of gl m | n .Let gl n | m be the Lie superalgebra defined in the same way as gl m | n with m and n interchanged. Thereexists a Lie superalgebra isomorphism between gl m | n and gl n | m given by the map ς m | n : e ij e m + n +1 − i,m + n +1 − j . (2.2)Fix m ′ , n ′ ∈ Z > and consider the Lie superalgebra gl m ′ | n ′ . For this algebra we also choose the standardparity, namely, we set | i | = ¯0 if and only if i m ′ .We also consider a larger Lie superalgebra gl m ′ + m | n ′ + n . For gl m ′ + m | n ′ + n , we fix the parity by | i | = ( ¯0 , if i m ′ or m ′ + n ′ + 1 i m ′ + n ′ + m, ¯1 , if m ′ + 1 i m ′ + n ′ or m ′ + n ′ + m + 1 i m ′ + n ′ + m + n. (2.3)Clearly, we have the embeddings of Lie superalgebras given by gl m ′ | n ′ → gl m ′ + m | n ′ + n , e ij e ij , gl m | n → gl m ′ + m | n ′ + n , e ij e m ′ + n ′ + i,m ′ + n ′ + j . Define the supertrace str : End( C m | n ) → C , str( E ij ) = s j δ ij . The supertrace is supercyclic, that is str([ E ij , E rs ]) = 0 . Here [ · , · ] is the supercommutator of linear operators.Denote by sl m | n the Lie subalgebra of gl m | n consisting of all elements acting on V as matrices with zerosupertrace. KANG LU AND EVGENY MUKHIN
Define the supertranspositions t and ⊤ , t : End( V ) → End( V ) , E tij = ( − | i || j | + | j | E ji , (2.4) ⊤ : End( V ) → End( V ) , E ⊤ ij = ( − | i || j | + | i | E ji . (2.5)Both supertranspositions are anti-homomorphisms and respect the supertrace, ( AB ) ∗ = ( − | A || B | B ∗ A ∗ , str( A ) = str( A ∗ ) , (2.6)for all ( m + n ) × ( m + n ) matrices A and B , where ∗ is either t or ⊤ . We also have t = ⊤ and t = ⊤ = 1 .2.2. Hook partitions, skew Young diagrams, and polynomial modules.
Let λ = ( λ > λ > · · · ) be apartition of ℓ : λ i ∈ Z > , λ i = 0 if i ≫ , and | λ | := P ∞ i =1 λ i = ℓ . We denote by λ ′ the conjugate of thepartition λ . The number λ ′ is the length of the partition λ , namely the number of nonzero parts of λ . Let µ = ( µ > µ > · · · ) be another partition such that µ i λ i for all i = 1 , , . . . . Consider the skew Youngdiagram λ/µ which is defined as the set of pairs { ( i, j ) ∈ Z | i > , λ i > j > µ i } . When µ is the zero partition, then λ/µ is the usual Young diagram corresponding to λ .We use the standard representation of skew Young diagrams on the coordinate plane R with coordinates ( x, y ) . Here we use the convention that x increases from north to south while y increases from west to east.Moreover, the pair ( i, j ) ∈ λ/µ is represented by the unit box whose south-eastern corner has coordinate ( i, j ) ∈ Z . We also define the content of the box corresponding to ( i, j ) ∈ λ/µ by c ( i, j ) = j − i .A semi-standard Young tableau of shape λ/µ is the skew Young diagram λ/µ with an element from { , , . . . , m + n } inserted in each box such that the following conditions are satisfied:(i) the numbers in boxes are weakly increasing along rows and columns;(ii) the numbers from { , , . . . , m } are strictly increasing along columns;(iii) the numbers from { m + 1 , m + 2 , . . . , m + n } are strictly increasing along rows.For a semi-standard Young tableau T of shape λ/µ , denote by T ( i, j ) the number in the box representingthe pair ( i, j ) ∈ λ/µ . Example 2.1.
Let λ = (5 , , , , , µ = (3 , , , , m = n = 2 , then the skew Young diagram λ/µ hasthe shape as one of the following. − − − − In the picture above, the left one is an example of a semi-standard Young tableau of shape λ/µ . We have T (1 ,
4) = 1 , T (1 ,
5) = T (5 ,
1) = 2 , T (3 ,
3) = T (4 ,
3) = T (5 ,
2) = 3 , and T (5 ,
3) = 4 . In the right, wewrote in each box its content. (cid:3) A standard Young tableau of shape λ/µ is the skew Young diagram λ/µ with an element from { , . . . , | λ |−| µ |} inserted in each box such that the numbers in boxes are strictly increasing along rows and columns. Example 2.2.
There is a distinguished standard Young tableau obtained by filling numbers along rows fromleft to right and top to bottom. We call it the row tableau . Similarly, one defines the column tableau . Here are
ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 5 the row (on the left) and the column (on the right) tableaux for the skew Young diagram λ/µ in the previousexample. (cid:3)
Recall that V = C m | n denotes the vector representation of gl m | n . A gl m | n -module is called a polynomialmodule if it is a submodule of V ⊗ l for some l ∈ Z > . We call a partition λ an ( m | n ) - hook partition if λ m +1 n . Let P l ( m | n ) be the set of all ( m | n ) -hook partitions of l and P ( m | n ) the set of all ( m | n ) -hookpartitions. In particular, P l ( m ) := P l ( m | is the set of all partitions of l with length m . It is well-knownthat irreducible polynomial gl m | n -modules are parameterized by P ( m | n ) .For λ ∈ P ( m | n ) , define the gl m | n -weight λ ♮ by λ ♮ = m X i =1 λ i ǫ i + n X j =1 max { λ ′ j − m, } ǫ m + j . (2.7)We sometimes use the notation λ ♯ [ m | n ] to stress the dependence of λ ♯ on m and n .Let P ∈
End( V ⊗ V ) be the super flip operator, P = X i,j ∈ ¯ I s j E ij ⊗ E ji . Let S l be the symmetric group permuting { , , . . . , l } . The symmetric group S l acts naturally on V ⊗ l ,where the simple transposition σ k = ( k, k + 1) acts as P ( k,k +1) = X i,j ∈ ¯ I s j E ( k ) ij E ( k +1) ji ∈ End( V ⊗ l ) , (2.8)where we use the standard notation E ( k ) ij = 1 ⊗ ( k − ⊗ E ij ⊗ ⊗ ( l − k ) ∈ End( V ⊗ l ) , k = 1 , . . . , l. Let S ( λ ) be the finite-dimensional irreducible representation of S l corresponding to the partition λ . Theorem 2.3 (Schur-Sergeev duality [Ser85]) . The S l -action and gl m | n -action on V ⊗ l commute. Moreover,as a U( gl m | n ) ⊗ C [ S l ] -module, we have V ⊗ l ∼ = M λ ∈ P l ( m | n ) L ( λ ♮ ) ⊗ S ( λ ) . (cid:3) For λ ∈ P ( m | n ) , we have λ ′ ∈ P ( n | m ) . The gl m | n -module obtained by pulling back the gl n | m -module L (( λ ′ ) ♯ [ n | m ] ) through the isomorphism ς : gl m | n → gl n | m , see (2.2), is isomorphic to L ( λ ♯ [ m | n ] ) . KANG LU AND EVGENY MUKHIN
For λ ∈ P ( m ′ + m | n ′ + n ) , define a gl m ′ + m | n ′ + n -weight λ ◦ by λ ◦ = m ′ X i =1 λ i ǫ i + n ′ X j =1 max { λ ′ j − m ′ , } ǫ m ′ + j + m ′ + m X i = m ′ +1 max { λ i − n ′ , } ǫ n ′ + i + n ′ + n X j = n ′ +1 max { λ ′ j − m ′ − m, } ǫ m ′ + m + j . (2.9)This definition is dictated by parity (2.3) we chose, see [BR83]. We will use λ ◦ in Section 3 to define skewrepresentations of super Yangian.2.3. Super Yangian Y( gl m | n ) . We recall the definition of super Yangian Y( gl m | n ) from [Naz91].The super Yangian Y( gl m | n ) is the Z -graded unital associative algebra over C with generators { t ( r ) ij | i, j ∈ ¯ I, r > } and defining relations [ t ( r ) ij , t ( s ) kl ] = ( − | i || j | + | j || k | + | j || k | min( r,s ) − X a =0 ( t ( a ) kj t ( r + s − − a ) il − t ( r + s − − a ) kj t ( a ) il ) , (2.10)where the generators t ( r ) ij have parities | i | + | j | .The super Yangian Y( gl m | n ) has the RTT presentation as follows. Define the rational R-matrix R ( u ) ∈ End( V ⊗ V ) by R ( u ) = 1 − P /u . The rational R-matrix satisfies the quantum Yang-Baxter equation R ( u − u ) R ( u − u ) R ( u − u ) = R ( u − u ) R ( u − u ) R ( u − u ) . (2.11)Define the generating series t ij ( u ) = δ ij + ∞ X k =1 t ( k ) ij u − k and the operator T ( u ) ∈ End( V ) ⊗ Y( gl m | n )[[ u − ]] , T ( u ) = X i,j ∈ ¯ I ( − | i || j | + | j | E ij ⊗ t ij ( u ) . Denote by T k ( u ) = X i,j ∈ ¯ I ( − | i || j | + | j | E ( k ) ij ⊗ t ij ( u ) ∈ End( V ⊗ l ) ⊗ Y( gl m | n )[[ u − ]] . (2.12)Then defining relations (2.10) can be written as R ( u − u ) T ( u ) T ( u ) = T ( u ) T ( u ) R ( u − u ) ∈ End( V ⊗ ) ⊗ Y( gl m | n )[[ u − ]] , In terms of generating series, defining relations (2.10) are equivalent to ( u − u )[ t ij ( u ) , t kl ( u )] = ( − | i || j | + | j || k | + | j || k | ( t kj ( u ) t il ( u ) − t kj ( u ) t il ( u )) . (2.13)The super Yangian Y( gl m | n ) is a Hopf superalgebra with coproduct, antipode, counit given by ∆ : t ij ( u ) X k ∈ ¯ I t ik ( u ) ⊗ t kj ( u ) , S : T ( u ) T ( u ) − , ε : T ( u ) . (2.14)Let ∆ op be the opposite coproduct of Y( gl m | n ) , ∆ op ( t ij ( u )) = X k ∈ ¯ I ( − ( | i | + | k | )( | j | + | k | ) t kj ( u ) ⊗ t ik ( u ) . (2.15) ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 7
For z ∈ C there exists an isomorphism of Hopf superalgebras, τ z : Y( gl m | n ) → Y( gl m | n ) , t ij ( u ) t ij ( u − z ) . (2.16)For any Y( gl m | n ) -module M , denote by M z the Y( gl m | n ) -module obtained from pulling back M throughthe isomorphism τ z .The super Yangian Y( gl m | n ) has a weight decomposition ( P -grading) with respect to Cartan subalgebraof U( gl m | n ) ⊂ Y( gl m | n ) . The generator t ( k ) ij has weight ǫ i − ǫ j .We have the standard PBW theorem. Theorem 2.4 ([Gow07]) . Fix some ordering on the generators t ( k ) ij , i, j ∈ ¯ I and k ∈ Z > , for the superYangian Y( gl m | n ) . Then the ordered monomials of these generators, with at most power 1 for odd generators,form a basis of Y( gl m | n ) . Set B := 1 + u − C [[ u − ]] . For any series ϑ ( u ) ∈ B , the map Γ ϑ : T ( u ) ϑ ( u ) T ( u ) (2.17)defines an automorphism of Y( gl m | n ) . Denote by Y( sl m | n ) the subalgebra of Y( gl m | n ) which consists of allelements that are fixed under automorphisms Γ ϑ for all ϑ ( u ) ∈ B .Let z m | n be the center of super Yangian Y( gl m | n ) . If m = n , then we have an isomorphism of algebras Y( gl m | n ) ∼ = z m | n ⊗ Y( sl m | n ) . see [Gow07, Proposition 8.1].Let Y( gl n | m ) be the super Yangian defined in the same way as Y( gl m | n ) by interchanging m and n .Let η m | n be the automorphism of Y( gl m | n ) given by η m | n : T ( u ) T ( − u ) − . (2.18)Define an isomorphism of superalgebras ̺ m | n : Y( gl m | n ) Y( gl n | m ) by ̺ m | n : t ij ( u ) t m + n +1 − i,m + n +1 − j ( − u ) . Denote by ˆ ς the composition of isomorphisms of superalgebras ˆ ς m | n = ̺ m | n ◦ η m | n , ˆ ς m | n : Y( gl m | n ) Y( gl n | m ) . (2.19)Finally, for fixed m ′ , n ′ ∈ Z > , one also defines a larger super Yangian Y( gl m ′ + m | n ′ + n ) following thechoice of parities as in (2.3).2.4. Gauss decomposition.
The Gauss decomposition of Y( gl m | n ) , see [Gow07, Pen16], gives generatingseries e ij ( u ) = X r > e ( r ) ij u − r , f ji ( u ) = X r > f ( r ) ji u − r , d k ( u ) = 1 + X r > d ( r ) k u − r , where i < j m + n and k ∈ ¯ I , such that t ii ( u ) = d i ( u ) + X k
For i ∈ I and k ∈ ¯ I , let e i ( u ) = e i,i +1 ( u ) = X r > e ( r ) i u − r , f i ( u ) = f i +1 ,i ( u ) = X r > f ( r ) i u − r ,d ′ k ( u ) = ( d k ( u )) − = 1 + X r > d ′ ( r ) k u − r . We use the convention d (0) k = d ′ (0) k = 1 .The parities of e ( r ) ij and f ( r ) ji are the same as that of t ( r ) ij while all d ( r ) k and d ′ ( r ) k are even. The super Yangian Y( gl m | n ) is generated by e ( r ) i , f ( r ) i , d ( r ) k , d ′ ( r ) k , where i ∈ I and k ∈ ¯ I , and r > . The full defining relationsare described in [Gow07, Lemma 4 or Theorem 3]. Here we only write down the following relations. Let φ i ( u ) = d ′ i ( u ) d i +1 ( u ) = 1 + P r > φ ( r ) i u − r . Then one has [ d ( r ) i , d ( s ) j ] = 0 , [ d ( r ) i , e ( s ) j ] = ( ǫ i , α j ) r − X t =0 d ( t ) i e ( r + s − − t ) j , [ d ( r ) i , f ( s ) j ] = − ( ǫ i , α j ) r − X t =0 f ( r + s − − t ) j d ( t ) i , (2.20) [ e ( r ) j , f ( s ) k ] = − s j δ jk r + s − X t =0 d ′ ( t ) j d ( r + s − − t ) j +1 = − s j δ jk φ ( r + s − j . (2.21)Moreover, the subalgebra Y( sl m | n ) is generated by the coefficients of the series φ i ( u ) , e i ( u ) , f i ( u ) for i ∈ I .Let Y + m | n , Y − m | n , and Y m | n be the subalgebras of Y( gl m | n ) generated by coefficients of the series e i ( u ) , f i ( u ) , and d j ( u ) , respectively. It is known from [Gow07] that Y( gl m | n ) ∼ = Y − m | n ⊗ Y m | n ⊗ Y + m | n as vector spaces and d ( r ) i are algebraically free generators of Y m | n .The Gauss decomposition for super Yangian associated to non-standard parity sequences is studied in[Pen16]. In particular, one obtains generating series e i ( u ) , f i ( u ) , d i ( u ) , d ′ i ( u ) and generators e ( r ) i , f ( r ) i , d ( r ) i , d ′ ( r ) i for Y( gl n | m ) with standard parities and for Y( gl m ′ + m | n ′ + n ) with parities in (2.3). We refer the readerto [Pen16, Tsy20] for the explicit relations of Y( gl m ′ + m | n ′ + n ) in these generators.We conclude this section with the following lemma used in Section 3.3. Lemma 2.5 ([Gow07, Proposition 4.2]) . For the isomorphism ˆ ς m | n : Y( gl m | n ) Y( gl n | m ) defined in (2.19) , we have ˆ ς m | n : d i ( u ) ( d m + n +1 − i ( u )) − , e j ( u )
7→ − f m + n − j ( u ) , f j ( u )
7→ − e m + n − j ( u ) , for i ∈ ¯ I and j ∈ I . (cid:3) Highest and lowest ℓ -weight representations. Recall B = 1 + u − C [[ u − ]] and set B := B ¯ I × Z .We call an element ζ ∈ B an ℓ -weight . We write ℓ -weights in the form ζ = ( ζ i ( u )) p ( ζ ) i ∈ ¯ I , where p ( ζ ) ∈ Z and ζ i ( u ) ∈ B for all i ∈ ¯ I .Clearly B is an abelian group with respect to the point-wise multiplication of the tuples and the additionof the parities. Let Z [ B ] be the group ring of B whose elements are finite Z -linear combinations of the form P a ζ [ ζ ] , where a ζ ∈ Z .Let M be a Y( gl m | n ) -module. We say that a nonzero Z -homogeneous vector v ∈ M is of ℓ -weight ζ if d i ( u ) v = ζ i ( u ) v for i ∈ ¯ I and the parity of v is given by p ( ζ ) . We say that a vector v ∈ M of ℓ -weight ζ is a highest (resp. lowest) ℓ -weight vector of ℓ -weight ζ if e ij ( u ) v = 0 (resp. f ji ( u ) v = 0 ) for all i < j m + n . The module M is called a highest (resp. lowest) ℓ -weight module of ℓ -weight ζ if M isgenerated by a highest (resp. lowest) ℓ -weight vector of ℓ -weight ζ . ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 9
In general, ℓ -weight vectors do not need to be eigenvectors of t ii ( u ) . However, from the Gauss decom-position one can deduce that v is a highest ℓ -weight vector of ℓ -weight ζ if and only if v is of parity p ( ζ ) and t ij ( u ) v = 0 , t kk ( u ) v = ζ k ( u ) v, i < j m + n, k ∈ ¯ I. (2.22)Note similar formulas do not hold for a lowest ℓ -weight vector v of ℓ -weight ζ .Let v and v ′ be highest ℓ -weight vectors of ℓ -weights ζ and ϑ , respectively. Then, by (2.22) and (2.14),we have t ij ( u )( v ⊗ v ′ ) = 0 , t kk ( u )( v ⊗ v ′ ) = ζ k ( u ) ϑ k ( u )( v ⊗ v ′ ) , i < j m + n, k ∈ ¯ I. Hence v ⊗ v ′ is a highest ℓ -weight vector of ℓ -weight ζϑ . In particular, we have e i ( u )( v ⊗ v ′ ) = 0 , d j ( u )( v ⊗ v ′ ) = ζ j ( u ) ϑ j ( u )( v ⊗ v ′ ) , i ∈ I, j ∈ ¯ I. (2.23)This formula will be used to obtain information about ∆( d j ( u )) .Every finite-dimensional irreducible Y( gl m | n ) -module is a highest ℓ -weight module. Let ζ ∈ B be an ℓ -weight. There exists a unique irreducible highest ℓ -weight Y( gl m | n ) -module of highest ℓ -weight ζ . Wedenote it by L ( ζ ) . The criterion for L ( ζ ) to be finite-dimensional is as follows. Theorem 2.6 ([Zh96]) . The irreducible Y( gl m | n ) -module L ( ζ ) is finite-dimensional if and only if there existmonic polynomials g i ( u ) , i ∈ ¯ I , such that ζ i ( u ) ζ i +1 ( u ) = g i ( u + s i ) g i ( u ) , ζ m ( u ) ζ m +1 ( u ) = g m ( u ) g m + n ( u ) , i ∈ I, i = m, and deg g m = deg g m + n . (cid:3) Finite-dimensional irreducible Y( gl m | n ) -modules stay irreducible under restriction to Y( sl m | n ) . Every ir-reducible finite-dimensional Y( sl m | n ) -module is a restriction of an irreducible finite-dimensional Y( gl m | n ) -module. The restrictions of two finite-dimensional irreducible Y( gl m | n ) -modules are isomorphic Y( sl m | n ) -modules if and only if one of these modules is obtained from the other by a twist by the automorphism Γ ϑ for ϑ ( u ) ∈ B .We finish this section by proving the following technical proposition. Define the length function ℓ : Q > → Z > by ℓ ( P i ∈ I n i α i ) = P i ∈ I n i . Proposition 2.7.
For i ∈ I , j ∈ ¯ I , k ∈ Z > , we have ∆( d ( k ) j ) − k X l =0 d ( l ) j ⊗ d ( k − l ) j ∈ X ℓ ( α ) > (Y( gl m | n )) α ⊗ (Y( gl m | n )) − α , (2.24) ∆( e ( k ) i ) − ⊗ e ( k ) i ∈ X ℓ ( α ) > (Y( gl m | n )) α ⊗ (Y( gl m | n )) α i − α , (2.25) ∆( f ( k ) i ) − f ( k ) i ⊗ ∈ X ℓ ( α ) > (Y( gl m | n )) α − α i ⊗ (Y( gl m | n )) − α . (2.26) Proof.
We simply write Y α for (Y( gl m | n )) α . Let N i be the subalgebra of Y( gl m | n ) generated by e ( r ) j for r ∈ Z > , j ∈ I \ { i } . Let A i be the unital subalgebra of Y( gl m | n ) generated by φ ( r ) i , r ∈ Z > . Let h (1) i = d (2) i −
12 ( d (1) i ) − d (1) i , then by (2.20), we have [ h (1) i , e ( s ) i ] = c i e ( s +1) i for some c i ∈ C × . Note that d (2) i = t (2) ii − P j (N i ) α ⊗ Y − α + X ℓ ( α − α i ) > (N i ) α ⊗ Y − α . Note that ∆( e (1) i ) = 1 ⊗ e (1) i + e (1) i ⊗ . Using [ h (1) i , e ( s ) i ] = c i e ( s +1) i and [ e (1) j ⊗ f (1) j , ⊗ e ( k ) i ] = 0 for j = i and k ∈ Z > , one shows inductively that ∆( e ( k ) i ) − ⊗ e ( k ) i ∈ k X s =1 e ( s ) i ⊗ A i + X ℓ ( α ) > (N i ) α ⊗ Y α i − α + X ℓ ( α − α i ) > (N i ) α ⊗ Y α i − α . In particular, we obtain (2.25). Similarly, one shows (2.26).We then show (2.24). Since φ ( k ) i = − ( ǫ i , ǫ i )[ e ( k ) i , f (1) i ] and f (1) i (super)commutes with N i , we have ∆( φ ( k ) i ) ∈ A i ⊗ A i + X ℓ ( α ) > Y α ⊗ Y − α . Note that d ( u ) = T ( u ) , we have ∆( d ( u )) = d ( u ) ⊗ d ( u ) + P ℓ ( α ) > Y α ⊗ Y − α [[ u − ]] . Hence itsuffices to show that ∆( φ i ( u )) ∈ φ i ( u ) ⊗ φ i ( u ) + X ℓ ( α ) > Y α ⊗ Y − α [[ u − ]] . Let ∆ i ( φ ( k ) i ) ∈ A i ⊗ A i be such that ∆( φ ( k ) i ) − ∆ i ( φ ( k ) i ) ∈ P ℓ ( α ) > Y α ⊗ Y − α . Clearly, A i = C [ φ ( k ) i ] k> is a polynomial algebra, so is A i ⊗ A i . Therefore an element x ∈ A i ⊗ A i is determined by the data ( χ ⊗ χ )( x ) where χ , χ are algebra homomorphisms A i → C . Note that P ℓ ( α ) > Y α ⊗ Y − α annihilatestensor products of highest ℓ -weight vectors. Therefore, we have ( χ ⊗ χ )(∆ i ( φ ( k ) i )) = k X s =0 χ ( φ ( s ) i ) χ ( φ ( k − s ) i ) = ( χ ⊗ χ ) (cid:16) k X s =0 φ ( s ) i ⊗ φ ( k − s ) i (cid:17) , where the first equality follows from (2.23). Therefore ∆ i ( φ ( k ) i ) = P ks =0 φ ( s ) i ⊗ φ ( k − s ) i , completing theproof of (2.24). (cid:3) Evaluation maps.
The universal enveloping superalgebra U( gl m | n ) is a Hopf subalgebra of Y( gl m | n ) via the embedding e ij s i t (1) ij . The left inverse of this embedding is the evaluation homomorphism π m | n :Y( gl m | n ) → U( gl m | n ) given by π m | n : t ij ( u ) δ ij + s i e ij u − . (2.27)The evaluation homomorphism is an algebra homomorphism but not a Hopf algebra homomorphism. Forany gl m | n -module M , it is naturally a Y( gl m | n ) -module obtained by pulling back M through the evaluationhomomorphism π m | n . We denote the corresponding Y( gl m | n ) -module by the same letter M and call it an evaluation module .Following [Naz04], define the modified evaluation map of Y( gl m | n ) by π ιm | n : t ij ( u ) δ ij + ( − ( | i | +1)( | j | +1) e ji u − . Given a gl m | n -module M , we call the Y( gl m | n ) -module obtained by pulling back through the modifiedevaluation map π ιm | n a modified evaluation module and denote it by M . ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 11
The evaluation map and modified one are related as follows. Let ι : Y( gl m | n ) → Y( gl m | n ) op be theisomorphism of Hopf superalgebras defined by ι : t ij ( u ) ( − | i || j | + | i | t ji ( − u ) , (2.28)where Y( gl m | n ) op is the Hopf superalgebra with opposite coproduct (2.15). For a Y( gl m | n ) -module M ,denote by M ι the pull back of M through ι .Clearly, one has π ιm | n = π m | n ◦ ι . Therefore, the modified evaluation module can be thought of as the pullback of an evaluation module through the isomorphism ι , namely for a gl m | n -module M , M = M ι . Notethat, more generally, for z ∈ C , we have M − z = ( M ι ) z .Define also the second modified evaluation map π ∨ m | n : Y( gl m | n ) → U( gl m | n ) by π ∨ m | n = ς n | m ◦ π n | m ◦ ˆ ς m | n , where ς n | m , π n | m , and ˆ ς m | n are defined in (2.2), (2.27), and (2.19), respectively. The second modifiedevaluation map will be used in Section 3.3.Given a gl m | n -module M , we call the Y( gl m | n ) -module obtained by pulling back through the secondmodified evaluation map π ∨ m | n a second modified evaluation module and denote it by M .Note that if v ∈ M is a gl m | n singular vector of a given weight, then in the evaluation Y( gl m | n ) -module M and second modified evaluation module M , v is a highest ℓ -weight vector, while in the modified evaluationmodule M , v is a lowest ℓ -weight vector.2.7. Category C and q -character map. Let C be the category of finite-dimensional Y( gl m | n ) -modules. Thecategory C is abelian and monoidal.Let M ∈ C be a finite-dimensional Y( gl m | n ) -module and ζ ∈ B an ℓ -weight. Let ζ i ( u ) = 1 + ∞ X j =1 ζ ( j ) i u − j , ζ ( j ) i ∈ C . Denote by M ζ the generalized ℓ -weight space corresponding to the ℓ -weight ζ , M ζ := { v ∈ M | ( d ( j ) i − ζ ( j ) i ) dim M v = 0 for all i ∈ ¯ I, j ∈ Z > , and | v | = p ( ζ ) } . For a finite-dimensional Y( gl m | n ) -module M , define the q - character (or Yangian character ) of M by theelement χ ( M ) := X ζ ∈ B dim( M ζ )[ ζ ] ∈ Z [ B ] . Let R ep ( C ) be the Grothendieck ring of C , then χ induces a Z -linear map from R ep ( C ) to Z [ B ] .Define the map ̟ : B → h ∗ , ζ ̟ ( ζ ) by ̟ ( ζ )( e ii ) = s i ζ (1) i . Lemma 2.8.
The map χ : R ep ( C ) → Z [ B ] is an injective ring homomorphism.Proof. The fact that χ : R ep ( C ) → Z [ B ] is an ring homomorphism follows from Proposition 2.7, see e.g.[FR99, Remark 2.6]. Since L ( ζ ) is of highest ℓ -weight, by Theorem 2.4, χ ( L ( ζ )) is equal to [ ζ ] plus ℓ -weights of form [ ξ ] such that ̟ ( ξ ) is strictly smaller than ̟ ( ζ ) with respect to the partial ordering on h ∗ .Therefore [ ζ ] is the leading term in χ ( L ( ζ )) . Now the injectivity of χ is clear. (cid:3) In particular, we obtain the following.
Corollary 2.9.
The Grothendieck ring R ep ( C ) is commutative. (cid:3)
3. S
KEW REPRESENTATIONS AND J ACOBI -T RUDI IDENTITY
Skew representations.
Consider the embedding of gl m ′ | n ′ into gl m ′ + m | n ′ + n sending e ij to e ij for i, j = 1 , , . . . , m ′ + n ′ . Here gl m ′ | n ′ has the standard parity and gl m ′ + m | n ′ + n has parity (2.3).Let λ and µ be an ( m ′ + m | n ′ + n ) -hook partition and an ( m ′ | n ′ ) -hook partition, respectively. Supposefurther that λ i > µ i for all i ∈ Z > . Consider the skew Young diagram λ/µ .Let µ ♮ be the gl m ′ | n ′ -weight corresponding to µ , see (2.7), and let λ ◦ be the gl m ′ + m | n ′ + n -weight corre-sponding to λ , see (2.9). We have the finite-dimensional irreducible gl m ′ + m | n ′ + n -module L ( λ ◦ ) . Consider L ( λ ◦ ) as a gl m ′ | n ′ -module.Define L ( λ/µ ) to be the subspace of L ( λ ◦ ) by L ( λ/µ ) := { v ∈ L ( λ ◦ ) | e ii v = µ ♮ ( e ii ) v, e jk v = 0 , i = 1 , , . . . , m ′ + n ′ , j < k m ′ + n ′ } . Let ϕ m ′ | n ′ : Y( gl m | n ) → Y( gl m ′ + m | n ′ + n ) be the embedding given by ϕ m ′ | n ′ : t ij ( u ) t m ′ + n ′ + i,m ′ + n ′ + j ( u ) . Recall η m | n (and η m ′ + m | n ′ + n ) from (2.18). Let ψ m ′ | n ′ : Y( gl m | n ) → Y( gl m ′ + m | n ′ + n ) be the injectivehomomorphism given by ψ m ′ | n ′ := η m ′ + m | n ′ + n ◦ ϕ m ′ | n ′ ◦ η m | n . The following lemma can be found in [Pen16, Proof of Lemma 4.2].
Lemma 3.1 ([Gow07, Pen16]) . We have ψ m ′ | n ′ ( d i ( u )) = d m ′ + n ′ + i ( u ) , ψ m ′ | n ′ ( e i ( u )) = e m ′ + n ′ + i ( u ) , ψ m ′ | n ′ ( f i ( u )) = f m ′ + n ′ + i ( u ) . (cid:3) Regard Y( gl m ′ | n ′ ) as the subalgebra of Y( gl m ′ + m | n ′ + n ) via the natural embedding t ij ( u ) t ij ( u ) for i, j = 1 , . . . , m ′ + n ′ . We have the following lemma from [Pen16, Lemma 4.3]. Lemma 3.2 ([Pen16]) . The subalgebra Y( gl m ′ | n ′ ) of Y( gl m ′ + m | n ′ + n ) commutes with the image of Y( gl m | n ) under the map ψ m ′ | n ′ . (cid:3) Recall the evaluation map π , see (2.27), the following is straightforward from Lemma 3.2. Corollary 3.3.
The image of the homomorphism π m ′ + m | n ′ + n ◦ ψ m ′ | n ′ : Y( gl m | n ) → U( gl m ′ + m | n ′ + n ) commutes with the subalgebra U( gl m ′ | n ′ ) in U( gl m ′ + m | n ′ + n ) . (cid:3) Corollary 3.3 implies that the subspace L ( λ/µ ) is invariant under the action of the image of π m ′ + m | n ′ + n ◦ ψ m ′ | n ′ . Therefore, L ( λ/µ ) is a Y( gl m | n ) -module. We call L ( λ/µ ) a skew representation . We study the skewrepresentations in the rest of this section.3.2. q -characters of skew representations. In this section we compute the q -character of the Y( gl m | n ) -module L ( λ/µ ) .Let κ i = i − if i = 1 , . . . , m and κ i = 2 m − i if i = m + 1 , . . . , m + n . Let X i,a = (cid:0) , . . . , (cid:0) u + a + κ i ) − (cid:1) s i , . . . , (cid:1) | i | ∈ B , i ∈ ¯ I. Here the only component not equal to 1 is at the i -th position.Recall that T ( i, j ) and c ( i, j ) = j − i denote the number in the box representing the pair ( i, j ) ∈ λ/µ andthe content of the pair ( i, j ) for a semi-standard Young tableau T of shape λ/µ , respectively.It is known from [CPT15] that the dimension of L ( λ/µ ) is equal to the number of semi-standard Youngtableaux of shape λ/µ . The following theorem is a refinement of this statement. ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 13
Theorem 3.4.
The q -character of the Y( gl m | n ) -module L ( λ/µ ) is given by K λ/µ ( u ) := χ ( L ( λ/µ )) = X T Y ( i,j ) ∈ λ/µ X T ( i,j ) ,c ( i,j ) , summed over all semi-standard Young tableaux T of shape λ/µ . Before proving the theorem, we recall the following proposition from [Gow05, Theorems 1 and 2] and[Tsy20, Theorem 2.43].Similar to κ i , define κ ′ i , i = 1 , . . . , m ′ + n ′ , with m and n replaced by m ′ and n ′ , respectively. Set κ ′ m ′ + n ′ + j = m ′ − n ′ + κ j , j ∈ ¯ I . Let s ′ i = ( − | i | , i = 1 , , . . . , m ′ + n ′ + m + n , such that | i | are chosenas in (2.3). Proposition 3.5 ([Gow05, Tsy20]) . The coefficients of the series Q i ∈ ¯ I ( d i ( u − κ i )) s i are central in Y( gl m | n ) .The coefficients of the series Q m ′ + n ′ + m + ni =1 ( d i ( u − κ ′ i )) s ′ i are central in Y( gl m ′ + m | n ′ + n ) . (cid:3) Lemma 3.6.
Let λ be a Young diagram. Then the operator Q i ∈ ¯ I ( d i ( u − κ i )) s i acts on the evaluation Y( gl m | n ) -module L ( λ ♯ ) by the scalar operator Y i ∈ ¯ I ( d i ( u − κ i )) s i (cid:12)(cid:12)(cid:12) L ( λ ♯ ) = Y i ∈ ¯ I (cid:16) λ ♯i , ǫ i ) u − κ i (cid:17) s i = Y ( i,j ) ∈ λ u + c ( i, j ) + 1 u + c ( i, j ) . Similarly, the operator Q m ′ + n ′ + m + ni =1 ( d i ( u − κ ′ i )) s ′ i acts on the evaluation Y( gl m ′ + m | n ′ + n ) -module L ( λ ◦ ) by the scalar operator m ′ + n ′ + m + n Y i =1 ( d i ( u − κ ′ i )) s ′ i (cid:12)(cid:12)(cid:12) L ( λ ◦ ) = Y ( i,j ) ∈ λ u + c ( i, j ) + 1 u + c ( i, j ) . Proof.
The statement follows from Proposition 3.5 and direct computations on highest ℓ -weight vector. (cid:3) Proof of Theorem 3.4.
The proof is similar to [NT98, Lemma 2.1] and [FM02, Lemma 4.7]. Following theexposition of [Mol07, Corollary 8.5.8], Theorem 3.4 can be proved in a similar way using Lemma 3.1,Proposition 3.5, and Lemma 3.6. (cid:3)
Remark . Due to Theorem 3.4, the q -character of L ( λ/µ ) relies only on the shape λ/µ and not on m ′ , n ′ .Thus we have the module L ( λ/µ ) for arbitrary skew Young diagram λ/µ and its q -character is given byTheorem 3.4. Indeed, one can enlarge m ′ such that both λ and µ are hook partitions (of different indices).Moreover, we can also let n ′ = 0 , then the parity (2.3) is a standard parity sequence and the Lie superalgebra gl m ′ + m | n ′ + n is associated to a standard parity sequence. Therefore, one can always set n ′ = 0 to simplifythe discussion. (cid:3) For the Y( gl m | n ) -module L ( λ/µ ) , we write L z ( λ/µ ) for ( L ( λ/µ )) z . It is clear that χ ( L z ( λ/µ )) = K λ/µ ( u − z ) . Remark . Recall that the pair ( i, j ) ∈ λ/µ is represented by the unit box whose south-eastern corner hascoordinate ( i, j ) ∈ Z . We can shift the whole Young diagram so that the numbers in the pair ( i, j ) ∈ R arenot necessarily integers, but real numbers. The shape of the diagram remains the same. Only the contentsof all boxes are shifted by the same number simultaneously. Hence we may consider the entire diagram isfixed and for any z ∈ C we can define the content c z in a more general way, c z ( i, j ) = j − i − z . Clearly,the q -character K λ/µ ( u − z ) is written in the same way as K λ/µ ( u ) in Theorem 3.4 by changing c ( i, j ) to c z ( i, j ) . Note the contents of all boxes are uniquely determined by the content of a single box. Therefore, inthe following, we shall sometimes specify the content of a box. If no content of a box is specified, thenwe are using the standard definition of content, namely c ( i, j ) = j − i . We also remark that for differentYoung diagrams λ, ˜ λ, µ, ˜ µ , the diagrams λ/µ and ˜ λ/ ˜ µ may have the same shape but with possibly differentcontents. (cid:3) Divisibility of q -characters. In this section, we discuss the divisibility of q -characters of skew repre-sentations associated to special skew Young diagrams. We expect these observations would be helpful tounderstand Theorem 5.12 and prove Conjecture 5.14 below, see Section 5.5.For a partition λ , denote by λ − the skew Young diagram obtained by rotating λ by 180 degrees. Byconvention, we set the content of bottom-right box of λ − to be zero. Example 3.9.
Let λ be the partition (2 , , , then λ and λ − are given by and , respectively. Herethe number stands for the content of the corresponding box. (cid:3) Note that semi-standard Young tableaux of shape λ ′ are in one-to-one correspondence with semi-standardYoung tableaux of shape λ − given by changing numbers i in boxes of λ ′ to m + n − i +1 in the correspondingboxes of λ − , see e.g. [Zha18, Lemma 2.7].Let L z ( λ ♯ ) be the second modified evaluation module of L ( λ ♯ ) twisted by τ z and set L ( λ ♯ ) := L ( λ ♯ ) . Lemma 3.10 ([Zha18, Theorem 2.4]) . We have the isomorphism of Y( gl m | n ) -modules, L z ( λ ♯ ) ∼ = L m − n + z ( λ − ) . Proof.
The lemma follows from Theorem 3.4 and Lemma 2.5, see also [Zha18, Theorem 2.4]. (cid:3)
Let Ξ be the partition whose corresponding Young diagram is a rectangle of size m × n . For a partition λ ∈ P ( m | , define W ( λ ) to be the skew Young diagram obtained by gluing Ξ and λ − so that the bottomrow of λ − is next to the bottom row of the rectangular one exactly from left. Similarly, for µ ∈ P (0 | n ) ,define S ( µ ) to be the Young diagram obtained by attaching µ to the bottom of Ξ such that the first columnof λ is exactly below the first column of Ξ . Moreover, we always assume that the box at left-upper corner of Ξ has content zero. Example 3.11.
Consider the case m = 4 and n = 3 . Let λ = (2 , , and µ = (3 , , . Then the skewYoung diagram W ( λ ) and the Young diagram S ( µ ) are as follows. Here the number zero means that the contents of the corresponding boxes are zero. We use the green colorand red color to indicate the diagrams λ − and µ corresponding to partitions λ and µ , respectively. (cid:3) Denote by χ m the q -character map of Y( gl m ) . We also have the evaluation modules for Y( gl m ) , Y( gl | n ) defined by setting n = 0 and m = 0 , respectively. Recall that K λ/µ ( u ) = χ ( L ( λ/µ )) , we use the super-scripts to indicate the underlying algebra, e.g. K m | nλ/µ ( u ) , K m | λ/µ ( u ) , and K | nλ/µ ( u ) . ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 15
We identify an ℓ -weight ζ [ m ] = ( ζ i ( u )) | | i m for Y( gl m ) with an ℓ -weight ζ = ( ζ i ( u )) | | i ∈ ¯ I for Y( gl m | n ) via the natural embedding Y( gl m ) ֒ → Y( gl m | n ) , where ζ j ( u ) = 1 for j = m + 1 , . . . , m + n . Similarly, an ℓ -weight ϑ [ n ] = ( ϑ i ( u )) p ( ϑ [ n ] )1 i n for Y( gl | n ) is identified with an ℓ -weight ϑ = ( ϑ i ( u )) p ( ϑ [ n ] ) i ∈ ¯ I for Y( gl m | n ) via the embedding ψ m | : Y( gl | n ) ֒ → Y( gl m | n ) in Lemma 3.1, where ϑ j ( u ) = 1 for j = 1 , . . . , m . Lemma 3.12.
We have the equalities of q -characters K m | n W ( λ ) ( u ) = K m | λ − ( u − m ) · K m | n Ξ ( u ) , (3.1) K m | n S ( µ ) ( u ) = K | nµ ( u − m ) · K m | n Ξ ( u ) . (3.2) Proof.
We first show (3.1). It is not hard to see that all semi-standard Young tableaux of shape W ( λ ) are obtained by independently filling numbers , . . . , m + n to the rectangle Ξ and numbers , . . . , m to λ − so that each part gives a semi-standard Young tableau. In particular, K m | n W ( λ ) ( u ) can be written as theproduct of K m | n Ξ ( u ) and the summation of monomials in X i,a associated to semi-standard Young tableauxcorresponding to λ − filled by numbers , . . . , m . Hence we obtain (3.1).Similarly, all semi-standard Young tableaux of shape S ( µ ) are obtained by independently filling numbers , . . . , m + n to the rectangle Ξ and numbers m + 1 , . . . , m + n to µ so that each part gives a semi-standardYoung tableau. The equality (3.2) is proved in a similar way. (cid:3) The lemma does not imply an equality on the representation level as not all q -characters are for Y( gl m | n ) . Corollary 3.13.
We have the equality of q -characters K m | n W ( λ ) ( u ) = χ m ( L ( λ ♯ [ m | )) · K m | n Ξ ( u ) . (3.3) Proof.
The statement follows from Lemma 3.10 and Lemma 3.12. (cid:3)
Remark . Equations (3.1) and (3.2) are related by Lemma 3.10. Namely, ignoring the contents, we havethe equality for skew Young diagrams (cid:0) W n | m ( µ ′ ) (cid:1) ′ = (cid:0) S m | n ( µ ) (cid:1) − , where S m | n ( µ ) := S ( µ ) is defined above while W n | m ( µ ′ ) is defined in the same way as W m | n ( λ ) := W ( λ ) with m and n interchanged. (cid:3) Remark . Equation (3.2) implies that the gl m | n -character of L ( S ( µ ) ♯ ) is equal to the gl m | n -character of L (Ξ ♯ ) multiplied by the gl n -character of L (( µ ′ ) ♯ [ n | ) . This fact can be understood observing that the gl m | n -module L ( S ( µ ) ♯ ) is an irreducible Kac module. Indeed, let g +1 and g − be the odd subspaces of gl m | n spanned by e i,m + j and e m + j,i for all i = 1 , . . . , m and j = 1 , . . . , n , respectively. Note the gl m -module L (Ξ ♯ [ m | ) is one-dimensional. Extend the gl m ⊕ gl n -module L (Ξ ♯ [ m | ) ⊗ L (( µ ′ ) ♯ [ n | ) to the gl m ⊕ gl n ⊕ g +1 -module by putting g +1 (cid:0) L (Ξ ♯ [ m | ) ⊗ L (( µ ′ ) ♯ [ n | ) (cid:1) = 0 , then we have the isomorphism of vector spaces byPBW theorem for gl m | n , L ( S ( µ ) ♯ ) = Ind gl m | n gl m ⊕ gl n ⊕ g +1 (cid:0) L (Ξ ♯ [ m | ) ⊗ L (( µ ′ ) ♯ [ n | ) (cid:1) ∼ = L (( µ ′ ) ♯ [ n | ) ⊗ ∧ • [ g − ] , where ∧ • [ g − ] denotes the Grassmann algebra with mn variables and hence has dimension mn . The part K | nµ ( u − m ) corresponds to L (( µ ′ ) ♯ [ n | ) while K m | n Ξ ( u ) corresponds to ∧ • [ g − ] .Equations (3.1) and (3.3) can be interpreted similarly as well. (cid:3) Jacobi-Trudi type identity.
Set S k ( u ) = K λ/µ ( u ) if λ = ( k ) and µ = (0) , and A k ( u ) = K λ/µ ( u ) if λ = (1 k ) and µ = (0) .We have the Jacobi-Trudi type identity for q -characters of skew representations. Theorem 3.16.
We have K λ/µ ( u ) = det i,j λ ′ S λ i − µ j − i + j ( u + µ j − j + 1)= det i,j λ A λ ′ i − µ ′ j − i + j ( u − µ ′ j + j − . Here we use the convention that S k ( u ) = A k ( u ) = 0 for k < and S ( u ) = A ( u ) = 1 .Proof. We give a proof of the first equality using the Lindstr¨om-Gessel-Viennot lemma, the second equalityis proved similarly. We refer the reader to [Sag01, Chapter 4.5] for a detailed description of the method.Here we only give the necessary modifications to our situation. We remark that our adjustment is essentiallythe same as that of [Mol97, Theorem 3.1].Consider the lattice plane Z (like the usual x - y coordinate plane and it should not be confused with theone defining Young diagrams) and lattice paths from one point to another. The paths consist of steps fromone point to another of unit length northward or eastward or of length √ northeastward. More precisely, thestep starting from the point ( i, j ) can end at ( i +1 , j ) or ( i, j +1) if j m and at ( i, j +1) or ( i +1 , j +1) if m < j m + n . We call an eastward or northeastward step a contributed step . For a contributed step s ,denote by s x and s y the x -coordinate and y -coordinate of the starting point of s , respectively. For a latticepath p , define a monomial X p in X i,a by X p = Y contributed s ∈ p X s y , s x . Suppose the path p is from the point ( i , j ) to the point ( i , j ) , then it is clear that X p is a term in S i − i ( u + i ) as in Theorem 3.4. Moreover, we have S i − i ( u + i ) = X p X p summed over all lattice paths p from ( i , j ) to ( i , j ) .Set l = λ ′ . Let σ be an element of the symmetric group S l permuting numbers , . . . , l . Let p σi be alattice path from the point ( µ i − i + 1 , to the point ( λ σ ( i ) − σ ( i ) + 1 , m + n + 1) for i = 1 , . . . , l . Consideran l -tuple of paths p ( σ ) = ( p σ , . . . , p σl ) . Define the monomial X p ( σ ) associated to p ( σ ) by X p ( σ ) = l Y i =1 X p σi if all p σi exist, otherwise set X p ( σ ) = 0 . Then one has det i,j l S λ i − µ j − i + j ( u + µ j − j + 1) = X ( − sign( σ ) X p ( σ ) , (3.4)where the summation is over all possible σ ∈ S l and all possible l -tuples p ( σ ) .We call a tuple p ( σ ) an intersecting tuple if p σi intersects p σj for some i = j . All monomials in (3.4)corresponding to intersecting tuples are cancelled out, see [Sag01, Chapter 4.5]. A tuple of the form p ( σ ) is non-intersecting only if σ = id . Moreover, there exists a bijection between non-intersecting tuples withsemi-standard Young tableaux of shape λ/µ . One direction of this bijection is described as follows, see anexplicit example given below. Let p (id) = ( p id1 , . . . , p id l ) be a non-intersecting tuple, then filling the i -throw of the skew Young diagram λ/µ with the numbers s y for all contributed steps s of p id i in non-decreasing ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 17 order, for all i = 1 , . . . , l , gives a semi-standard Young tableau of shape λ/µ . Note that s x corresponds tothe content of the box in the i -th row filled with the number s y . The theorem now follows from this bijectionand Theorem 3.4. (cid:3) We exhibit an example explaining the correspondence between non-intersecting tuples and semi-standardYoung tableaux.
Example 3.17.
Let λ = (4 , , , µ = (1 , , , m = n = 2 . Consider the following semi-standard Youngtableau. T = 1 2 23 42 3 In particular, the content of the box filled with number is . Then it corresponds to the following -tupleof lattice paths. xy s by the number s y , the y -coordinate of the starting point of s . Then thelabels from the first path (the rightmost path in red color) corresponds to the first row of T . Similarly, thelabels from the second (in blue color) and third (in green color) paths give rise to the numbers in the secondand third rows of T , respectively. The content of the box filled with number s y equals s x , the x -coordinateof the starting point of s .It is also very easy to write down the elements in the determinant det i,j λ ′ S λ i − µ j − i + j ( u + µ j − j + 1) from the picture above. We order the starting points and end points from east to west. Then λ i − µ j − i + j corresponds to the horizontal length of any path from the j -th starting point to the i -th end point while µ j − j + 1 is the x -coordinate of the j -th starting point. Therefore, we have det i,j λ ′ S λ i − µ j − i + j ( u + µ j − j + 1) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ( u + 1) S ( u ) S ( u − S ( u + 1) S ( u ) S ( u − S ( u − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where S − ( u +1) means there is no path from the first starting point to the third end point and S ( u ) means there is exactly one path from the second starting point to the third end point. Moreover, this uniquepath contains no contributed steps. (cid:3) Clearly, it follows from Lemma 2.8 that there are corresponding identities on the level of representationsin the Grothendieck ring R ep ( C ) . Actually, Theorem 3.16 for the case of Y( gl N ) has been shown in [Che87,Che89] by resolutions of modules for the Yangian Y( gl N ) . Ignoring the spectral parameter u , one obtainsthe Jacobi-Trudi identity for super-characters of Lie superalgebra gl m | n , see [BB81]. Corollary 3.18. If λ/µ contains a rectangle of size at least ( m + 1) × ( n + 1) , then det i,j λ ′ S λ i − µ j − i + j ( u + µ j − j + 1) = det i,j λ A λ ′ i − µ ′ j − i + j ( u − µ ′ j + j −
1) = 0 . Proof. If λ/µ contains a rectangle of size at least ( m + 1) × ( n + 1) , then there are no semi-standard Youngtableaux of shape λ/µ . Thus by Theorem 3.4 we have K λ/µ ( u ) = 0 . The statement now follows fromTheorem 3.16. (cid:3)
4. D
RINFELD FUNCTOR AND SKEW REPRESENTATIONS
Degenerate affine Hecke algebras.
Let l be a positive integer. Following [Dri86], the degenerateaffine Hecke algebra H l is the associative algebra generated by generators σ , . . . , σ l − and x , . . . , x l withthe relations given by σ i = 1 , σ i σ i +1 σ i = σ i +1 σ i σ i +1 , [ x i , x j ] = 0 ,σ i x i = x i +1 σ i − , [ σ j , σ k ] = [ σ j , x k ] = 0 if | j − k | 6 = 1 . As vector spaces, H l ∼ = C [ S l ] ⊗ C [ x , . . . , x l ] . The generators σ , . . . , σ l − generate a subalgebra iso-morphic to C [ S l ] while x , . . . , x l generate a subalgebra isomorphic to C [ x , . . . , x l ] . We shall use theseidentifications. It is well-known that the center of H l is C [ x , . . . , x l ] S l .Let σ ij be the simple permutation ( i, j ) . Let y , . . . , y l ∈ H l be defined by y = x , y i = x i − X j , N X i =1 n i = l o . Now we recall some facts from representation theory of degenerate affine Hecke algebra H l , following[Zel80, Rog85].Let r ∈ Z > . Let l = l + · · · + l r , where l i ∈ Z > . Then algebra H l ⊗ · · · ⊗ H l r is identified to asubalgebra of H l by the embedding H l k ֒ → H l , σ a σ a + l + ··· + l k − , x b x b + l + ··· + l k − , for a = 1 , . . . , l k − , b = 1 , . . . , l k , and k = 1 , . . . , r .Let M i be an H l i -module, i = 1 , . . . , r . Define the H l module M ⊙ · · · ⊙ M r via the induction functor: M ⊙ · · · ⊙ M r := Ind H l H l ⊗···⊗H lr ( M ⊗ · · · ⊗ M r ) . ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 19
For complex numbers a , b such that b − a + 1 = l , denote by C [ a,b ] := C [ a,b ] the one-dimensionalrepresentation of H l given by σ i · [ a,b ] = [ a,b ] , i = 1 , . . . , l − , (4.2) x j · [ a,b ] = ( a + j − [ a,b ] , j = 1 , , . . . , l. (4.3)For λ ∈ h ∗ N , set W ( λ ; l ) := n µ ∈ h ∗ N | λ − µ ∈ wt(( C N ) ⊗ l ) o . Take µ ∈ W ( λ ; l ) and set λ i = λ ( ǫ i ) , µ i = µ ( ǫ i ) , and l i = λ i − µ i , for i = 1 , . . . , N . Define an H l -moduleby I ( λ, µ ) = C [ µ ,λ − ⊙ C [ µ − ,λ − ⊙ · · · ⊙ C [ µ N − N +1 ,λ N − N ] . (4.4)We also set I ( λ, µ ) = 0 if µ / ∈ W ( λ ; l ) .When λ is a dominant gl N -weight and µ ∈ W ( λ ; l ) , we call I ( λ, µ ) a standard module of H l . Thestandard module can be thought as an analog of Verma modules.Set λ,µ := [ µ ,λ − ⊗ · · · ⊗ [ µ N − N +1 ,λ N − N ] , then there is an isomorphism of C [ S l ] -modules I ( λ, µ ) ∼ = C [ S l / S l × · · · × S l N ] induced by λ,µ . In particular, one has the decomposition of S l -modules, I ( λ, µ ) ∼ = S ( ν λ,µ ) ⊕ M ν>ν λ,µ S ( ν ) ⊕ k ν , where ν λ,µ is the partition obtained by rearranging the sequence ( l , . . . , l N ) in non-increasing order and > denotes the dominance order of partitions. Here k ν are non-negative integers.It is well-known, see [Zel80, Rog85], that if λ ∈ D + N , then I ( λ, µ ) is generated by the subspace S ( ν λ,µ ) over H l . Therefore, I ( λ, µ ) has a unique irreducible quotient L ( λ, µ ) containing S ( ν λ,µ ) . The S l -module S ( ν λ,µ ) appears in L ( λ, µ ) (considered as an S l -module) with multiplicity one.One has the following analog of the BGG resolution for H l -modules. Proposition 4.1. [Che87]
Let λ ∈ h ∗ N and µ ∈ W ( λ ; l ) . Suppose λ − ρ ∈ P + N and µ − ρ ∈ P + N , then thereexists an exact sequence of H l -modules, → M σ ∈ S N [ N ( N − / I ( λ, σ · µ ) → · · · → M σ ∈ S N [1] I ( λ, σ · µ ) → I ( λ, µ ) → L ( λ, µ ) → , where S N [ k ] denotes the set of all elements of length k in S N . Drinfeld functor.
In this section, we define the Drinfeld functor [Dri86] for super Yangian Y( gl m | n ) ,following the exposition in [Ara99].Let M be an H l -module. Consider the H l ⊗ U( gl m | n ) -module M ⊗ V ⊗ l , where V = C m | n ∼ = L ( ǫ ) isthe vector representation of gl m | n . For i = 1 , . . . , l , let Q ( i ) = ( P (0 ,i ) ) ⊤ = X a,b ∈ ¯ I ( − | a || b | + | a | + | b | E ab ⊗ E ( i ) ab ∈ End( V ) ⊗ End( V ⊗ l ) . There is an algebra homomorphism ℘ : Y( gl m | n ) → H l ⊗ End( V ⊗ l ) ,T ( u ) T ( u − x ) T ( u − x ) · · · T l ( u − x l ) , where T i ( u − x i ) = 1 + 1 u − x i ⊗ Q ( i ) . Thus M ⊗ V ⊗ l becomes a Y( gl m | n ) module. One can think that it is a tensor product of evaluation Y( gl m | n ) -modules with value in M , where the i -th copy of V is evaluated at x i ∈ H l , see (2.14) and (2.27).The symmetric group acts naturally on M ⊗ V ⊗ l by σ i σ i ⊗ P ( i,i +1) , i = 1 , . . . , l − , where P ( i,i +1) ∈ End( V ⊗ l ) is defined in (2.8). Set D l ( M ) := ( M ⊗ V ⊗ l ) / l − X i =1 Im( σ i + 1) , (4.5)where Im( σ i + 1) denotes the image of σ i + 1 acting on M ⊗ V ⊗ l . Lemma 4.2.
The subspace P l − i =1 Im( σ i + 1) ⊂ M ⊗ V ⊗ l is an Y( gl m | n ) -submodule. In particular, D l ( M ) is a Y( gl m | n ) -module.Proof. The proof is parallel to [Ara99, Proposition 3]. We only show [ P ( i,i +1) , Q ( i ) ] = [ Q ( i +1) , Q ( i ) ] . Thisis obtained from [ P ( i,i +1) , P (0 ,i ) ] = [ P (0 ,i ) , P (0 ,i +1) ] by applying the supertransposition ⊤ to the -th factorof End( V ) ⊗ End( V ⊗ l ) , see (2.5). (cid:3) Denote by C H l the category of finite-dimensional representations of H l . Recall that C is the category offinite-dimensional representations of Y( gl m | n ) . The functor D l is an exact functor from C H l to C . We call D l the Drinfeld functor , cf. [Dri86].In the case n = 0 we recover the standard Drinfeld functor. In that case, we have the following usefulwell-known theorem. We say that a representation of Y( sl m ) is of level l if all its irreducible componentswhen restricted as sl m -modules are submodules of ( C m ) ⊗ l . Denote by C ( l ) m the category of finite-dimensionalrepresentations of Y( sl m ) with level l . Theorem 4.3 ([Dri86, CP96]) . If l < m , then the Drinfeld functor D l is an equivalence between the category C H l and the category C ( l ) m . (cid:3) A supersymmetric version of Theorem 4.3 for quantum affine superalgebra U q ( b gl m | n ) was recently provedin [Fli20] when l < m + n .The following lemma describes the relation between M considered as an S l -module and D l ( M ) consid-ered as a gl m | n -module. Lemma 4.4.
Let M be an H l -module. Let M = L ν S ( ν ′ ) ⊕ k ν be the decomposition of M as an S l -module,where the sum is over all partitions of l and k ν ∈ Z > . Then we have the decomposition of gl m | n -modules, D l ( M ) ∼ = M ν ∈ P l ( m | n ) L ( ν ♮ ) ⊕ k ν . Proof.
The statement follows from Theorem 2.3 (Schur-Sergeev duality), see the proof of [Ara99, Proposi-tion 4]. (cid:3)
Lemma 4.5.
Let M be an H l -module and M an H l -module. Then we have the Y( gl m | n ) -module iso-morphism D l ( M ) ⊗ D l ( M ) ∼ = D l + l ( M ⊙ M ) . Proof.
The proof is similar to that of [CP96, Proposition 4.7] or [Naz99, Proposition 5.3]. (cid:3)
Lemma 4.6.
Let M be an H l -module. Then the action of Y( gl m | n ) on D l ( M ) can be written in the form t ij ( u ) = δ ij + l X k =1 u − y k ⊗ ( − | i | E ( k ) ij , where y k are given by (4.1) . Specifically, t ( a ) ij acts by P lk =1 y a − k ⊗ ( − | i | E ( k ) ij for a > . ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 21
Proof.
The proof is parallel to that of [Ara99, Proposition 6]. Note that Q ( i ) · Q ( k ) = P ( i,k ) · Q ( k ) followsfrom P (0 ,k ) · P (0 ,i ) = P ( i,k ) · P (0 ,k ) by applying supertransposition ⊤ to the -th factor of End( V ) ⊗ End( V ⊗ l ) . (cid:3) Define ω k ∈ h ∗ , k ∈ Z > , by ω k = ( ǫ + · · · + ǫ k , if k m ; ǫ + · · · + ǫ m + ( k − m ) ǫ m +1 , if k > m. Let λ ∈ h ∗ N and µ ∈ W ( λ ; l ) . Set λ i = λ ( ǫ i ) , µ i = µ ( ǫ i ) , and l i = λ i − µ i , for i = 1 , . . . , N . Recall the H l -module I ( λ, µ ) defined in (4.4). Lemma 4.7.
Let a, b be complex numbers such that b − a + 1 = l . Then we have the Y( gl m | n ) -moduleisomorphism D l ( C [ a,b ] ) ∼ = L a ( ω l ) . Moreover, we have the Y( gl m | n ) -module isomorphism M ( λ, µ ) := D l ( I ( λ, µ )) ∼ = L µ ( ω l ) ⊗ L µ − ( ω l ) ⊗ · · · ⊗ L µ N − N +1 ( ω l N ) . Proof.
From (4.1), (4.2), and (4.3) we obtain that y i · [ a,b ] = a [ a,b ] . Therefore, the first statement followsfrom Lemma 4.4 and Lemma 4.6. The second statement follows from the first statement and Lemma 4.5. (cid:3) Unlike the gl N case [Ara99, Theorem 9], when mn > , the Y( gl m | n ) -module M ( λ, µ ) is never zero.4.3. Drinfeld functor and skew representations.
In this section, we study the relations between skewrepresentations and Drinfeld functor. We need the following proposition.
Proposition 4.8.
Let M be a finite-dimensional irreducible representation of H l , then the Y( gl m | n ) -module D l ( M ) is irreducible.Proof. The proof is similar to that of [Naz99, Theorem 5.5]. (cid:3)
We investigate the Y( gl m | n ) -module D l ( L ( λ, µ )) . Note that the highest ℓ -weight vector of D l ( L ( λ, µ )) is not given by the quotient image of tensor product of highest ℓ -weight vectors of all L µ i − i +1 ( ω l i ) ingeneral and therefore the computation of highest ℓ -weight of D l ( L ( λ, µ )) is not straightforward. We use theresolution of the H l -module L ( λ, µ ) and Jacobi-Trudi identity of q -characters to show that D l ( L ( λ, µ )) is askew representation if λ i − λ i +1 ∈ Z > and µ i − µ i +1 ∈ Z > for all i = 1 , . . . , N − .For λ ∈ h ∗ N such that λ − ρ ∈ P + N , we identify the weight λ with a partition in the usual way. We denotethis partition also by λ . Theorem 4.9.
Let λ ∈ h ∗ N and µ ∈ W ( λ ; l ) . Suppose λ − ρ ∈ P + N and µ − ρ ∈ P + N , then we have D l ( L ( λ, µ )) ∼ = L ( λ ′ /µ ′ ) as Y( gl m | n ) -modules. In particular, the skew representation L ( λ ′ /µ ′ ) is irre-ducible.Proof. Applying the Drinfeld functor D l to the resolution of the H l -module L ( λ, µ ) in Proposition 4.1, wehave the exact sequence of Y( gl m | n ) -modules, → M σ ∈ S N [ N ( N − / M ( λ, σ · µ ) → · · · → M σ ∈ S N [1] M ( λ, σ · µ ) → M ( λ, µ ) → D l ( L ( λ, µ )) → . For σ ∈ S N , it is clear from Lemma 2.8 and Lemma 4.7 that χ ( M ( λ, σ · µ )) = N Y i =1 A λ i − µ σ − i ) − i + σ − ( i ) ( u − µ σ − ( i ) + σ − ( i ) − , where N > λ ′ . By Theorem 3.16 and the resolution above, we obtain that χ ( D l ( L ( λ, µ ))) = K λ ′ /µ ′ ( u ) = χ ( L ( λ ′ /µ ′ )) . Since by Proposition 4.8, D l ( L ( λ, µ )) is an irreducible Y( gl m | n ) -module, we conclude again from Lemma2.8 that D l ( L ( λ, µ )) ∼ = L ( λ ′ /µ ′ ) . In particular, L ( λ ′ /µ ′ ) is irreducible. (cid:3) Give a partition λ of length at most N and a complex number z . Define gl N -weights λ z and z by λ z ( ǫ i ) = λ i + z, z ( ǫ i ) = z, i = 1 , . . . , N. Corollary 4.10.
Suppose the same conditions as in Theorem 4.9 hold. Let z be an arbitrary complex number,then we have D l ( L ( λ z , µ z )) ∼ = L z ( λ ′ /µ ′ ) . (cid:3) Fusion procedure and skew representations.
In this section, we study further skew representationsby fusion procedure, following [Che86, NT02, Naz04].Fix partitions λ and µ such that µ ⊂ λ . Set l = | λ | − | µ | . Let Ω be a standard Young tableau of shape λ/µ . For i ∈ { , . . . , l } , denote the content of the box in Ω containing i by c i (Ω) . Consider the operator E Ω := −→ Y i Set z i = − c i (Ω) for i = 1 , . . . , l . Then the operator E Ω ◦ P is a Y( gl m | n ) -modulehomomorphism E Ω ◦ P : V z l e ⊗ · · · e ⊗ V z → V z e ⊗ · · · e ⊗ V z l . In particular, the image of E Ω in V ⊗ l is a submodule of the Y( gl m | n ) -module V z e ⊗ · · · e ⊗ V z l .Proof. The proof is similar to that of [Naz04, Proposition 4.2]. Note that our z i corresponds to − z i there.Explicitly, the proof is modified by applying the supertransposition ⊤ on the -th copy of End( V ) in End( V ) ⊗ End( V ) ⊗ l and replacing x with − u , cf. [Mol07, Section 6.5]. (cid:3) We denote by F (Ω) the Y( gl m | n ) -submodule of V z e ⊗ · · · e ⊗ V z l defined by the image of E Ω acting on V ⊗ l .Now we are ready to compare F (Ω) with L ( λ/µ ) .Define the rational function g µ ( u ) by g µ ( u ) = Y i > ( u + µ i − i )( u − i + 1)( u + µ i − i + 1)( u − i ) . Then g µ ( ∞ ) = 1 and we identify g µ ( u ) as a series in B = 1 + u − C [[ u − ]] . Recall the automorphismdefined by Γ ϑ : T ( u ) ϑ ( u ) T ( u ) from (2.17) for ϑ ( u ) ∈ B . Denote by C ϑ the even one-dimensional Y( gl m | n ) -module defined by the homomorphism ε ◦ Γ g µ : Y( gl m | n ) → End( C ϑ ) , T ( u ) ϑ ( u ) , ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 23 where ε is the counit map, see (2.14). Theorem 4.12. The Y( gl m | n ) -modules L ( λ/µ ) ⊗ C g µ and F (Ω) are isomorphic.Proof. The proof is similar to that of [Naz04, Theorem 1.6]. We only remark that the irreducibility of L ( λ/µ ) there is obtained from Olshanski’s centralizer construction [MO00] of Y( gl N ) . In this paper we obtain theirreducibility of Y( gl m | n ) module L ( λ/µ ) in a different way using Drinfeld functor, see Theorem 4.9. (cid:3) We have the so-called binary property for tensor products of skew representations of Y( gl m | n ) , cf. [NT02,Theorem 4.9]. Theorem 4.13. Let λ ( i ) and µ ( i ) be partitions such that µ ( i ) ⊂ λ ( i ) , i = 1 , . . . , k . Let z , . . . , z k be complexnumbers. Then the Y( gl m | n ) -module L z ( λ (1) /µ (1) ) ⊗ · · · ⊗ L z k ( λ ( k ) /µ ( k ) ) is irreducible if and only if L z i ( λ ( i ) /µ ( i ) ) ⊗ L z j ( λ ( j ) /µ ( j ) ) is irreducible for all i < j k .Proof. Let Ω i be the column tableau of shape λ i /µ i for i = 1 , . . . , k . Denote by F z i (Ω i ) the pull back of F (Ω i ) through τ z i . Thanks to Theorem 4.12, it suffices to show that F z (Ω ) ⊗ · · · ⊗ F z k (Ω k ) is irreducible if and only if F z i (Ω i ) ⊗ F z j (Ω j ) for all i < j k , see [NT02, Theorems 4.8 and 4.9].The argument in [NT02] using fusion procedure concerns the operators in the group algebra C [ S l ] whichcan be generalized to the super setting with very few changes. Therefore the statement follows. (cid:3) For any Y( gl m | n ) -modules M , . . . , M k , we have ( M e ⊗ · · · e ⊗ M k ) ι ∼ = M ι ⊗ · · · ⊗ M ιk . Let g ιµ ( u ) = g µ ( − u ) . Theorem 4.14. The Y( gl m | n ) -modules L ( λ/µ ) ι ⊗ C g ιµ and F (Ω) ι are isomorphic and the Y( gl m | n ) -module F (Ω) ι is a submodule of V c (Ω) ⊗ · · · ⊗ V c l (Ω) , where V is the modified evaluation vector representation.Proof. The statement follows from Proposition 4.11 and Theorem 4.12. (cid:3) Application: irreducibility of tensor products. Due to Theorem 4.3 and Proposition 4.8, many resultsfrom the representation theory of Y( gl N ) can be generalized to the case of Y( gl m | n ) . We give two suchexamples in this and next sections.The following statement should be well-known for experts. However, we are not able to find the suitablereference, cf. [Zel80]. Proposition 4.15. Let λ ( i ) and µ ( i ) be partitions such that µ ( i ) ⊂ λ ( i ) , i = 1 , . . . , k . Let z , . . . , z k becomplex numbers such that z i − z j Z for all i < j k . Then the induction product L ( λ (1) z , µ (1) z ) ⊙ · · · ⊙ L ( λ ( k ) z k , µ ( k ) z k ) is an irreducible H l -module, where l = P ki =1 ( | λ ( i ) | − | µ ( i ) | ) .Proof. Let N be sufficiently large. Applying the Drinfeld functor to L ( λ (1) z , µ (1) z ) ⊙ · · · ⊙ L ( λ ( k ) z k , µ ( k ) z k ) with m = N and n = 0 , we obtain the Y( gl N ) -module L z ( λ (1) ′ /µ (1) ′ ) ⊗ · · · ⊗ L z k ( λ ( k ) ′ /µ ( k ) ′ ) , which is known to be irreducible when z i − z j Z for all i < j k , see [NT98, Corollary 3.9]. Theproposition follows from Theorem 4.3. (cid:3) The following theorem is a direct corollary of Proposition 4.8, Proposition 4.15, and Corollary 4.10. Theorem 4.16. Let λ ( i ) and µ ( i ) be partitions such that µ ( i ) ⊂ λ ( i ) , i = 1 , . . . , k . Let z , . . . , z k be complexnumbers such that z i − z j Z for all i < j k . Then the tensor product of skew representations L z ( λ (1) /µ (1) ) ⊗ · · · ⊗ L z k ( λ ( k ) /µ ( k ) ) is an irreducible Y( gl m | n ) -module. (cid:3) Let λ and µ be two partitions. Let N be sufficiently large. Define the numbers a i = λ i − i + 1 , b i = µ i − i + 1 , i = 1 , . . . , N. For each pair ( i, j ) such that i < j N , define the subsets of Z by h a j , a i i = { a j , a j + 1 , . . . , a i } \ { a j , a j +1 , . . . , a i } , h b j , b i i = { b j , b j + 1 , . . . , b i } \ { b j , b j +1 , . . . , b i } . Note that if λ i = λ i − = · · · = λ j , then h a j , a i i = Ø . Proposition 4.17. Let λ and µ be two partitions, z and w two complex numbers. Then the induction product L (( λ ′ ) z , z ) ⊙ L (( µ ′ ) w , w ) is irreducible if and only if for each pair ( i, j ) such that i < j N , wehave b j + z − w, b i + z − w / ∈ h a j , a i i or a j − z + w, a i − z + w / ∈ h b j , b i i . (4.6) In particular, L (( λ ′ ) z , z ) ⊙ L (( λ ′ ) z , z ) is irreducible.Proof. The proof is similar to that of Proposition 4.15 using [Mol02, Theorem 1.1]. (cid:3) Theorem 4.18. Let λ and µ be two partitions, z and w two complex numbers. Suppose the condition (4.6) holds for all pairs ( i, j ) such that i < j , then the Y( gl m | n ) -module L z ( λ ♮ ) ⊗ L w ( µ ♮ ) is irreducible. Inparticular, L z ( λ ♮ ) ⊗ L z ( λ ♮ ) is irreducible.Proof. The theorem follows from Proposition 4.8 and Proposition 4.17. (cid:3) Combining Theorem 4.18 with Theorem 4.13, one is able to give sufficient conditions for a tensor productof evaluation Y( gl m | n ) -modules to be irreducible.Comparing to the Y( gl N ) case, conditions (4.6) are not necessary for L z ( λ ♮ ) ⊗ L w ( µ ♮ ) to be irreducible.It would be interesting to generalize [Mol02, Theorem 1.1] to skew representations of Y( gl m | n ) . Example 4.19. We compare the sufficient and necessary conditions for L z (2 ǫ ) ⊗ L w (2 ǫ ) to be irreducibleover Y( gl ) and Y( gl | ) . The Y( gl ) -module L z (2 ǫ ) ⊗ L w (2 ǫ ) is irreducible if and only if z − w = ± , ± , while the Y( gl | ) -module L z (2 ǫ ) ⊗ L w (2 ǫ ) is irreducible if and only if z − w = ± . Thereforethe conditions (4.6) are not necessary for the irreducibility of the Y( gl | ) -module L z (2 ǫ ) ⊗ L w (2 ǫ ) . (cid:3) We call an irreducible Y( gl m | n ) -module M real if M ⊗ M is also irreducible, see [Lec03]. Theorem 4.18implies that the evaluation module L z ( λ ♮ ) is real. Actually, it holds for all skew representations. Theorem 4.20. The skew representation L z ( λ/µ ) is real.Proof. The statement follows from [NT02, Remark (d) of Theorem 4.8], [Naz04, Theorem 1.6], Theorem4.3, and Proposition 4.8. (cid:3) ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 25 Application: extended T-systems. In this section, we apply Drinfeld functor to show that the q -characters of skew representations satisfy extended T-systems.Let ℧ = λ/µ be a skew Young diagram. We say that ℧ is a prime skew Young diagram if it can not bedivided into two parts intersecting at most one point. Example 4.21. We explain the definition with the following 3 skew Young diagrams.The first skew Young diagram is prime while the rest are not. For example, the last two diagrams can bedivided into two parts so that one part is in red color. Clearly, the two parts of the second one intersects at apoint while those of the third one are disconnected. (cid:3) Recall that different pairs ( λ, µ ) may give the same skew Young diagram. Let ℧ be a prime skew Youngdiagram (ignoring the content). We choose a λ so that λ and ℧ have the same number of columns and ℧ = λ/µ . The contents of ℧ are determined by λ , namely the box of λ at the left-upper corner has contentzero. Let l be the number of columns of ℧ , then l = λ .Suppose ℧ = λ/µ has at least two columns, namely l > . Let ℧ + and ℧ − be the prime skew Youngdiagrams obtained by deleting the leftmost column and the rightmost column of ℧ , respectively. Also let ℧ be the prime skew Young diagrams obtained by removing both the leftmost and rightmost columns of ℧ .Note that ℧ may be empty.Define two skew Young diagrams X ℧ and Y ℧ as follows, X ℧ = { ( i, j ) : µ ′ j + 1 i λ ′ j +1 − , j l − } , Y ℧ = { ( i, j ) : µ ′ j +1 i λ ′ j , j l − } . The skew Young diagrams X ℧ and Y ℧ are obtained by taking the intersection and union, respectively, of thediagram ℧ + shifted to the left by one unit and then up by one unit and ℧ − .Note that in general as ℧ is prime, we have µ ′ j λ ′ j +1 − for i = 1 , . . . , l − . Hence the j -th columnof X ℧ may be empty and X ℧ may be non-prime. However, Y ℧ is always prime.Snakes defined in [MY12a] bijectively correspond to certain skew Young diagrams via the correspondencein [MY12a, Proposition 7.3]. The skew Young diagrams X ℧ and Y ℧ correspond to the neighbouring snakesin [MY12b, Section 3.6] in this sense.Recall, if a skew Young diagram contains a rectangle of size ( m + 1) × ( n + 1) (a column of length N + 1 in the Y( gl N ) case), then the corresponding skew representation has dimension zero. Theorem 4.22 ([MY12b, Theorem 4.1]) . Suppose ℧ is a prime skew Young diagram having at least twocolumns and N is sufficiently large. Then we have the following relation in R ep (Y( gl N )) , the Grothendieckring of the category of finite-dimensional representations of Y( gl N ) , [ L ( ℧ + )] ⊗ [ L ( ℧ − )] = [ L ( ℧ )][ L ( ℧ )] + [ L ( X ℧ )][ L ( Y ℧ )] . Moreover, L ( ℧ ) ⊗ L ( ℧ ) and L ( X ℧ ) ⊗ L ( Y ℧ ) are irreducible Y( gl N ) -modules. (cid:3) Remark . Because we only care about the case when N is sufficiently large, our definitions of primediagrams and Y ℧ here are slightly different from that in [MY12b, Section 3.5] as we allow a column of aprime skew Young diagram or Y ℧ to be very long. (cid:3) Applying Drinfeld functor, we get the corresponding supersymmetric version of Theorem 4.22. Corollary 4.24. Suppose ℧ is a prime skew Young diagram having at least two columns. Then we have thefollowing relation in R ep ( C ) , [ L ( ℧ + )] ⊗ [ L ( ℧ − )] = [ L ( ℧ )][ L ( ℧ )] + [ L ( X ℧ )][ L ( Y ℧ )] . Moreover, L ( ℧ ) ⊗ L ( ℧ ) and L ( X ℧ ) ⊗ L ( Y ℧ ) are irreducible Y( gl m | n ) -modules.Proof. Using Theorem 4.22 and Theorem 4.3 (the equivalence of Drinfeld functor), then we have the corre-sponding equality for the representations of degenerate affine Hecke algebra. Applying Drinfeld functor tothis resulted equality, the first statement follows. Similarly, the second part follows from Proposition 4.8. (cid:3) Example 4.25. Given i, j ∈ Z > and k ∈ C , let ℧ ij ; k be the rectangular Young diagram of size i × j whoseleft-upper corner box has content k . Then we have ℧ + i ( j +1); k = ℧ ij ; k +1 , ℧ − i ( j +1); k = ℧ ij ; k , ℧ i ( j +1); k = ℧ i ( j − k +1 , X ℧ i ( j +1); k = ℧ ( i − j ; k , Y ℧ i ( j +1); k = ℧ ( i +1) j ; k +1 . Let T ( i ) j ( u + k − ( i − j + 1) / 2) = K ℧ ij ; k ( u ) , where K ℧ ij ( u ) is the q -character of L ( ℧ ij ; k ) , see Theorem 3.4.Setting k = ( i − j ) / and ℧ = ℧ i ( j +1); k , one obtains the T-systems from Corollary 4.24, T ( i ) j ( u − ) T ( i ) j ( u + ) = T ( i ) j − ( u ) T ( i ) j +1 ( u ) + T ( i − j ( u ) T ( i +1) j ( u ) . The boundary conditions are given by • T ( i ) j ( u ) = 0 if i < or j < or both i > m and j > n ; • T ( i ) j ( u ) = 1 if i, j ∈ Z > and ij = 0 .Hence we may regard Corollary 4.24 as extended T-systems. (cid:3) We remark that our extended T-systems are different from that of [Zha18, Theorem 3.3].Suppose ℧ is a skew Young diagram with consecutive columns (not necessarily prime) and has at least 2columns. One can define ℧ ± and ℧ in the same way. Theorem 4.26. If ℧ is not prime, then L ( ℧ + ) ⊗ L ( ℧ − ) and L ( ℧ ) ⊗ L ( ℧ ) are isomorphic and irreducibleas Y( gl m | n ) -modules.Proof. The theorem is proved in a similar way to that of Corollary 4.24 using [MY12b, Theorem 4.3]. (cid:3) 5. Q UANTUM B EREZINIAN AND TRANSFER MATRICES Quantum Berezinian. Following [MR14], we recall the quantum Berezinian and related results in thecase of Y( gl m | n ) .Let A be a superalgebra. Let a ij ∈ A with parity | i | + | j | . Suppose the inverse of the matrix AA = X i,j ∈ ¯ I a ij ⊗ E ij ( − | i || j | + | j | ∈ A ⊗ End( V ) , with values in A exists. Then we denote the entries of the inverse matrix by a ′ ij ∈ A : A − = X i,j ∈ ¯ I a ′ ij ⊗ E ij ( − | i || j | + | j | . ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 27 Define the quantum Berezinian Ber( A ) , see [Naz91], of the matrix A by Ber( A ) = X σ ∈ S m sgn( σ ) · a σ (1)1 · · · a σ ( m ) m X ˜ σ ∈ S n sgn(˜ σ ) · a ′ m +1 ,m +˜ σ (1) · · · a ′ m + n,m +˜ σ ( n ) . Let A m | n := Y( gl m | n )[[ u − ]](( τ )) be the superalgebra of Laurent series in τ whose coefficients are powerseries in u − whose coefficients are in Y( gl m | n ) with the relations ( g u k τ l )( g u k τ l ) = g g u k ( u − l ) k τ l + l , g , g ∈ Y( gl m | n ) , l , l ∈ Z , k , k ∈ Z . Thus τ is the shift operator with respect to variable u and it should not be confused with automorphism ofthe Yangian τ defined in (2.16).Let q be a formal variable which commutes with all other elements. Let A ( q ) be a matrix with elementsin A m | n [ q ] given by A ( q ) = 1 − q T ( u ) τ = X i,j ∈ ¯ I ( δ ij − q t ij ( u ) τ ) ⊗ E ij ( − | i || j | + | j | . Clearly A ( q ) is invertible. Let D ( u, τ ; q ) := Ber( A ( q )) (5.1)be the quantum Berezinian. We simply write D ( u, τ ) for D ( u, τ ; 1) .The matrix T ( u ) is also invertible. Let Z ( u ) = Ber( T ( u ) τ ) τ n − m . Note that Z ( u ) does not contain τ . Itis known the coefficients of Z ( u ) generate the center of Y( gl m | n ) and Z ( u ) = Y i ∈ ¯ I ( d i ( u − κ i )) s i , in the notation of Proposition 3.5, see [Gow05, Theorem 1] and [Gow07, Theorem 4].Recall the standard action of symmetric group S k on the space V ⊗ k where σ i acts as the graded flipoperator P ( i,i +1) , see (2.8). We denote by A k and S k the images of the normalized anti-symmetrizer andsymmetrizer, respectively, A k = 1 k ! X σ ∈ S k sgn( σ ) · σ, S k = 1 k ! X σ ∈ S k σ. Theorem 5.1 ([MR14, Theorem 2.13]) . We have D ( u, τ ; q ) = 1 + ∞ X k =1 ( − k str A k T ( u ) T ( u − · · · T k ( u − k + 1) q k τ k , (5.2) D ( u, τ ; q ) − = 1 + ∞ X k =1 str S k T ( u ) T ( u − · · · T k ( u − k + 1) q k τ k , where the supertrace is taken over all copies of End( V ) . (cid:3) Universal R-matrix and transfer matrices. The Yangian Y( gl m | n ) has a universal R-matrix. Its ex-istence and properties can be deduced from [RS11], cf. also [Dri85]. We do not provide any justification inthis paper.The universal R-matrix is an element R ( u ) ∈ u − Y( gl m | n ) ⊗ Y( gl m | n )[[ u − ]] such that for all X ∈ Y( gl m | n ) we have (id ⊗ ∆)( R ( u )) = R ( u ) R ( u ) ∈ Y( gl m | n ) ⊗ [[ u − ]] , (∆ ⊗ id)( R ( u )) = R ( u ) R ( u ) ∈ Y( gl m | n ) ⊗ [[ u − ]] , R ( u ) · (id ⊗ τ u )(∆ op ( X )) = (id ⊗ τ u )(∆( X )) · R ( u ) ∈ Y( gl m | n ) ⊗ [[ u − ]] , where τ u is the Yangian automorphism defined in (2.16) (not to be confused with the shift operator τ ). Itfollows that the universal R-matrix R ( u ) satisfies the Yang-Baxter equation R ( u − v ) R ( u ) R ( v ) = R ( v ) R ( u ) R ( u − v ) . Let M be a finite-dimensional Y( gl m | n ) -module. Denote by Θ M : Y( gl m | n ) → End( M ) the correspondingmap. The R-matrix can be normalized so that (Θ V z ⊗ id)( R ( − u )) = T ( u + z ) ∈ End( V ) ⊗ Y( gl m | n )[[ u − ]] , cf. [Naz98, Lemma 3.4 and Theorem 3.6].Define the transfer matrix T M associated to M by T M ( u ) = str M (cid:0) (Θ M ⊗ id)( R ( − u )) (cid:1) ∈ Y( gl m | n )[[ u − ]] . The following lemma is standard, see [FR99, Lemma 2]. Lemma 5.2. For any pair of finite-dimensional Y( gl m | n ) -modules M and M , we have [ T M ( u ) , T M ( u )] = 0 , T M ⊗ M ( u ) = T M ( u ) T M ( u ) . For a short exact sequence M ֒ → M ։ M , we have T M ( u ) = T M ( u ) + T M ( u ) . (cid:3) Lemma 5.2 says that the map T : R ep ( C ) → Y( gl m | n )[[ u − ]] sending a finite-dimensional Y( gl m | n ) -module M to the transfer matrix T M ( u ) in Y( gl m | n )[[ u − ]] is a ring homomorphism.We shall focus on transfer matrices associated to skew representations L ( λ/µ ) ι . When M = L ( λ/µ ) ι ,we write T λ/µ ( u ) for T M ( u ) . Then T M z ( u ) = T λ/µ ( u − z ) . Recall that the partition (1 k ) corresponds to the Young diagram consisting of a column with k boxes while ( k ) corresponds to the Young diagram consisting of a row with k boxes. We use the short-hand notation, T k ( u ) := T (1 k ) ( u ) , T k ( u ) := T ( k ) ( u ) . Applying the map T to the equality in R ep ( C ) corresponding to Theorem 3.16, we obtain the Jacobi-Trudiidentity for transfer matrices. Theorem 5.3. Let λ and µ be two partitions such that µ ⊂ λ . Then we have T λ/µ ( u ) = det i,j λ ′ T λ i − µ j − i + j ( u + µ j − j + 1)= det i,j λ T λ ′ i − µ ′ j − i + j ( u − µ ′ j + j − . Here we use the convention that T ( u ) = T ( u ) = 1 and T k ( u ) = T k ( u ) = 0 for k < . (cid:3) Theorem 5.3 was conjectured in [Tsu97] on the level of eigenvalues, and proved for the case of hookYoung diagrams in [KV08]. Corollary 5.4. If λ/µ contains a rectangle of size at least ( m + 1) × ( n + 1) , then det i,j λ ′ T λ i − µ j − i + j ( u + µ j − j + 1) = det i,j λ T λ ′ i − µ ′ j − i + j ( u − µ ′ j + j − 1) = 0 . Proof. The statement follows from Corollary 3.18. (cid:3) ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 29 Proposition 5.5. We have T k ( u ) = str A k T ( u ) T ( u − · · · T k ( u − k + 1) , T k ( u ) = str S k T ( u ) T ( u + 1) · · · T k ( u + k − . Proof. Consider the weight ω k . Then the corresponding Young diagram is a column with k boxes. Let Ω bethe column tableau. Then c i (Ω) = 1 − i , i = 1 , . . . , k , and it is well-known that E Ω = k ! A k and A k is theprojection of V ⊗ k onto the image of E Ω , which is isomorphic to L ( ω k ) as a gl m | n -module. The first formulaof the proposition now follows from Theorem 4.14. The second formula is similar. (cid:3) Harish-Chandra homomorphism. In this section, we define an analog of Harish-Chandra homomor-phism H for Y( gl m | n ) and compute the images of transfer matrices associated to skew Young diagramsunder H .Let Y( gl m | n ) h be the centralizer of h ⊂ gl m | n in Y( gl m | n ) , Y( gl m | n ) h = { X ∈ Y( gl m | n ) | [ t (1) ii , X ] = 0 , for i ∈ ¯ I } . Let J be the intersection of Y( gl m | n ) h and the right ideal of Y( gl m | n ) generated by f ( r ) j , for j ∈ I and r ∈ Z > . Note that J is also the intersection of Y( gl m | n ) h and the left ideal of Y( gl m | n ) generated by e ( r ) j ,for j ∈ I and r ∈ Z > . We have the decomposition as vector spaces, Y( gl m | n ) h = Y m | n ⊕ J. The projection H of Y( gl m | n ) h onto the subspace Y m | n along J , H : Y( gl m | n ) h → Y m | n , is an algebra homomorphism. We call H the Harish-Chandra homomorphism of Y( gl m | n ) .Clearly, from the Gauss decomposition, we have H ( t ii ( u )) = d i ( u ) .The following lemma can be proved using induction, cf. [MTV06, Lemma 4.1]. Lemma 5.6. For any X ∈ Y( gl m | n ) h of the form ∗ t ( r ) i j t ( r ) i i · · · t ( r k ) i k i k , i a , j ∈ ¯ I, r a > , a = 0 , , . . . , k, k ∈ Z > , i < j , where ∗ is an element in Y( gl m | n ) , we have H ( X ) = 0 . (cid:3) It is well-known that all coefficients of T k ( u ) and T k ( u ) belong to the centralizer Y( gl m | n ) h , cf. [MTV06,Proposition 4.7]. Hence we can compute the Harish-Chandra images of transfer matrices T λ/µ ( u ) . Lemma 5.7. We have H ( T k ( u )) = X I k Y a =1 s i a d i a ( u − a + 1) , H ( T k ( u )) = X J k Y a =1 s j a d j a ( u − a + k ) , summed over all sequences I = { i < i < · · · < i b < m + 1 i b +1 · · · i k m + n } and J = { m + n > j > j > · · · > j b > m + 1 > j b +1 > · · · > j k > } with b = 0 , . . . , k , respectively.Proof. The lemma follows from Proposition 5.5, Lemma 5.6, and an analog of [MR14, Proposition 2.3] withsequences of the form like I or J , see [MR14, Remark 2.4]. A similar computation was done in [MR14,Section 3.3]. (cid:3) The following is immediate from Theorem 5.1 and Lemma 5.7. Corollary 5.8. The Harish-Chandra image of D ( u, τ ; q ) is given by H ( D ( u, τ ; q )) = −→ Y i m + n (cid:16) − q d i ( u ) τ (cid:17) s i . (cid:3) Proposition 5.9. We have H ( T λ/µ ( u )) = X T Y ( i,j ) ∈ λ/µ s T ( i,j ) d T ( i,j ) ( u + c ( i, j )) , (5.3) where the summation is over all semi-standard Young tableaux T of shape λ/µ .Proof. The statement follows from the fact that H is an algebra homomorphism, Theorem 5.3, Lemma 5.7and the proof of Theorem 3.16 (identifying s i d i ( u + a ) with X i,a ). (cid:3) Remark . Note that if we identify s i d i ( u + a ) with X i,a for i ∈ ¯ I and a ∈ C , where s i is the parity of X i,a , then the right hand side of (5.3) is identified with right hand side of the equation in Theorem 3.4. Thisimplies that the q -character map can be thought as the composition of Harish-Chandra map and the map T ,see [FR99, Section 3]. (cid:3) Corollary 5.11. If λ/µ does not contain a rectangle of size ( m + 1) × ( n + 1) , then H ( T λ/µ ( u )) is non-zero.In particular, T λ/µ ( u ) is non-zero.Proof. As λ/µ does not contain a rectangle of size ( m + 1) × ( n + 1) , there exists at least one semi-standardYoung tableau of shape λ/µ . Hence the space L ( λ/µ ) is non-trivial by Theorem 3.4 and an irreducible finite-dimensional Y( gl m | n ) -module by Theorem 4.9. Let T be the semi-standard Young tableau correspondingto the highest ℓ -weight of L ( λ/µ ) , see Theorem 3.4. We consider the monomial Y ( i,j ) ∈ λ/µ (cid:16) d (1) T ( i,j ) u − (cid:17) in the right hand side of (5.3). Since T also corresponds to the highest gl m | n -weight in L ( λ/µ ) , this mono-mial appears only in Y ( i,j ) ∈ λ/µ s T ( i,j ) d T ( i,j ) ( u + c ( i, j )) when T = T . The statement now follows from the fact that d ( r ) i , i ∈ ¯ I and r ∈ Z > , are algebraicallyindependent. (cid:3) Rational form of quantum Berezinian. Motivated by [HMVY19, HLM19] and [LM19, Corollary6.13], we are interested in writing Ber(1 − T ( u ) τ ) as a ratio of two polynomials in τ .Let Ξ , Ξ + , and Ξ − be partitions corresponding to the rectangular Young diagrams of sizes m × n , m × ( n + 1) , and ( m + 1) × n , respectively. Introduce partitions Υ + i = ( n, . . . , n | {z } m n ′ s , i ) , Υ − j = ( n + 1 , . . . , n + 1 | {z } j ( n +1) ′ s , n, . . . , n | {z } ( m − j ) n ′ s ) , where i = 0 , , . . . , n and j = 1 , . . . , m . In particular, Υ +0 = Υ − = Ξ , Υ + n = Ξ − , Υ − m = Ξ + .Note that T Ξ ( u ) = 0 by Corollary 5.11. ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 31 Theorem 5.12. We have D ( u, τ ; q ) = (cid:16) m X i =1 ( − i q i E i ( u ) τ i (cid:17)(cid:16) n X j =1 q j G j ( u ) τ j (cid:17) − , = (cid:16) n X j =1 q j G j ( u ) τ j (cid:17) − (cid:16) m X i =1 ( − i q i E i ( u ) τ i (cid:17) . where E i ( u ) = T Ξ + / (1 m − i ) ( u + m − i ) T Ξ ( u + m + 1 − i ) , G i ( u ) = T Υ + i ( u + m + 1 − i ) T Ξ ( u + m + 1 − i ) , E i ( u ) = T Υ − i ( u − n ) T Ξ ( u − n ) , G i ( u ) = T Ξ − / ( m − i ) ( u − n + 1) T Ξ ( u − n ) , are ratios of transfer matrices. Here we can take ratio as transfer matrices commute.Proof. We only show the first equality for q = 1 . The second one and the case of general q are similar.By Theorem 5.1 and Proposition 5.5, it suffices to show that (cid:16) ∞ X k =1 ( − k T k ( u ) τ k (cid:17)(cid:16) n X j =1 G j ( u ) τ j (cid:17) = 1 + m X i =1 ( − i E i ( u ) τ i . This reduces to show that E i ( u ) = T i ( u ) + min( i,n ) X a =1 ( − a T i − a ( u ) G a ( u − i + a ) , i = 1 , . . . , m ; (5.4) T j ( u ) + min( j,n ) X a =1 ( − a T j − a ( u ) G a ( u − j + a ) , j = m + 1 , m + 2 , . . . . (5.5)Let us first show equation (5.4). By Theorem 5.3, we have T Ξ + / (1 m − i ) ( u + m − i ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T i ( u ) T m +1 ( u + m − i + 1) · · · T m + n ( u + m − i + n ) T i − ( u ) T m ( u + m − i + 1) · · · T m + n − ( u + m − i + n ) ... ... . . . ... T i − n ( u ) T m +1 − n ( u + m − i + 1) · · · T m ( u + m − i + n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.6)and T Ξ ( u + m + 1 − i ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T m ( u + m − i + 1) · · · T m + n − ( u + m − i + n ) ... . . . ... T m +1 − n ( u + m − i + 1) · · · T m ( u + m − i + n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5.7)It follows from Theorem 5.3 that T Υ + a ( u + m + 1 − i ) is equal to the minor of the matrix in (5.6) obtainedby deleting the first column and the ( a + 1) -th row. Expanding the determinant in (5.6) with respect to thefirst column and dividing both sides by the determinant in (5.7), then equation (5.4) follows from (5.7).Denote by X i ( u ) the determinant in (5.6).Equation (5.5) is proved similarly for i = m + 1 , . . . , m + n . In this case we have X i ( u ) = 0 as the firstand ( i + 1 − m ) -th columns coincide. Hence equation (5.5) is obtained by using Theorem 5.3 and expanding X i ( u ) = 0 with respect to the first column.Finally, we show equation (5.5) for i = m + n + k where k > . Let λ and µ be partitions correspondingto rectangular Young diagrams of sizes ( m + k ) × ( n + 1) and ( k − × n , respectively. Clearly, λ/µ contains a rectangle of size ( m + 1) × ( n + 1) . It follows from Corollary 5.4 that T λ/µ ( u ) = 0 . ApplyingTheorem 5.3, we have − n T λ/µ ( u − n ) = X i ( u ) . Again, equation (5.5) is obtained by expanding the determinant X i ( u ) with respect to the first column. (cid:3) Recall that C ( p )1+ u is the one-dimensional Y( gl m | n ) -module generated by a vector of highest ℓ -weight (1 + u , . . . , u ) of parity p . Corollary 5.13. We have Z ( u ) = ( − n T Ξ + ( u ) T Ξ − ( u + 1) = T C (¯0)1+ 1 u . Proof. Note that ( − q ) n − m Ber(1 − q T ( u ) τ ) → Z ( u ) τ m − n as q → ∞ . The first equality follows fromTheorem 5.12 by taking the limit q → ∞ . It is easy to see that L (cid:0) (Ξ + ) ♮ (cid:1) ∼ = L − (cid:0) (Ξ − ) ♮ (cid:1) ⊗ C (¯ n )1+ u . The second equality follows since T is a homomorphism of rings. (cid:3) Spectra of transfer matrices and divisibility of q -characters. In this section, we describe the relationbetween Theorem 5.12 and the results in [HLM19].Let M be a finite-dimensional irreducible Y( gl m | n ) -module of highest ℓ -weight ζ = ( ζ i ( u ); s ) i ∈ ¯ I . Weare interested in finding the spectra of transfer matrices acting on the space M .Let l = ( l i ) i ∈ I be a sequence of non-negative integers. Let t = ( t ( i ) j ) , i ∈ I , j = 1 , . . . , l i , be a sequenceof complex numbers. Define monic polynomials y i ( u ) = l i Y j =1 ( u − t ( i ) j ) , and set y = ( y i ) i ∈ I .The Bethe ansatz equation associated to ζ , l , ( s i ) i ∈ ¯ I is a system of algebraic equations in t given by ζ i ( t ( i ) j ) ζ i +1 ( t ( i ) j ) y i − ( t ( i ) j + s i ) y i − ( t ( i ) j ) y i ( t ( i ) j − s i ) y i ( t ( i ) j + s i +1 ) y i +1 ( t ( i ) j ) y i +1 ( t ( i ) j − s i +1 ) = 1 , (5.8)where i ∈ I and j = 1 , . . . , l i . It is known that when the Bethe ansatz equation is satisfied, one can constructthe Bethe vector B l ( t ) ∈ M which is shown to be an eigenvector (if it is nonzero) of the first transfer matrix str( T ( u )) , see [BR08]. One also expects the Bethe vector to be an eigenvector of all transfer matrices.Motivated by [HLM19], we have the following conjecture.Let y ( u ) = y m + n ( u ) = 1 . Define a rational difference operator D ( u, τ, ζ , y ; q ) by D ( u, τ, ζ , y ; q ) = −→ Y i m + n (cid:16) − q ζ i ( u ) · y i − ( u + s i ) y i ( u − s i ) y i − ( u ) y i ( u ) τ (cid:17) s i . (5.9) Conjecture 5.14. If t satisfies the Bethe ansatz equation (5.8) , then we have D ( u, τ ; q ) B l ( t ) = D ( u, τ, ζ , y ; q ) B l ( t ) . (cid:3) The conjecture was confirmed for Y( gl N ) in [MTV06, Theorem 6.1] and for Y( gl | ) in [LM19, Theorem6.5]. Conjecture 5.14 can be thought as the supersymmetric version of [FH15, Theorem 5.11] and [FJMM17, ACOBI-TRUDI IDENTITY AND SUPER YANGIAN 33 Theorem 7.5]. Namely, the eigenvalues of transfer matrix associated to a finite dimensional Y( gl m | n ) -module W acting on the finite dimensional Y( gl m | n ) -module M can be obtained by certain substitutionsto the Harish-Chandra image of T W ( u ) . For instance, applying the substitutions d i ( u ) ζ i ( u ) · y i − ( u + s i ) y i ( u − s i ) y i − ( u ) y i ( u ) (5.10)to the Harish-Chandra image H ( D ( u, τ ; q )) in Corollary 5.8, one obtains exactly the rational differenceoperator D ( u, τ, ζ , y ; q ) in (5.9).The rational difference operator on the right hand side of (5.9) can also be understood using Theorem 5.12and the divisibility of q -characters in Section 3.3. Let D ( u, τ, ζ , y ; q ) = −→ Y i m (cid:16) − q ζ i ( u ) · y i − ( u + s i ) y i ( u − s i ) y i − ( u ) y i ( u ) τ (cid:17) , D ( u, τ, ζ , y ; q ) = ←− Y m +1 i m + n (cid:16) − q ζ i ( u ) · y i − ( u + s i ) y i ( u − s i ) y i − ( u ) y i ( u ) τ (cid:17) . Then D ( u, τ, ζ , y ; q ) = D ( u, τ, ζ , y ; q ) (cid:0) D ( u, τ, ζ , y ; q ) (cid:1) − . (5.11)The rational form decomposition of D ( u, τ ; q ) in Theorem 5.12 is consistent with that of D ( u, τ, ζ , y ; q ) in(5.11) in the following sense. 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Y ORK S T ., D ENVER , CO 80208, USA E-mail address : [email protected] E.M.: D EPARTMENT OF M ATHEMATICAL S CIENCES , I NDIANA U NIVERSITY -P URDUE U NIVERSITY I NDIANAPOLIS , 402 N.B LACKFORD S T ., LD 270, I NDIANAPOLIS , IN 46202, USA E-mail address ::