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Solutions of the T-system and Baxter equations for supersymmetric spin chains

We propose Wronskian-like determinant formulae for the Baxter Q-functions and the eigenvalues of transfer matrices for spin chains related to the quantum affine superalgebra U_{q}(hat{gl}(M|N)). In contrast to the supersymmetric Bazhanov-Reshetikhin formula (the quantum supersymmetric Jacobi-Trudi formula) proposed in [Z. Tsuboi, J. Phys. A: Math. Gen. 30 (1997) 7975], the size of the matrices of these Wronskian-like formulae is less than or equal to M+N. Base on these formulae, we give new expressions of the solutions of the T-system (fusion relations for transfer matrices) for supersymmetric spin chains proposed in the abovementioned paper. Baxter equations also follow from the Wronskian-like formulae. They are finite order linear difference equations with respect to the Baxter Q-functions. Moreover, the Wronskian-like formulae also explicitly solve the functional relations for Backlund flows proposed in [V. Kazakov, A. Sorin, A. Zabrodin, Nucl. Phys. B790 (2008) 345 [arXiv:hep-th/0703147]].


Introduction
The T -system plays important roles in the study of quantum integrable systems (see for example, earlier papers [1,2], which include the T -system related to U q ( sl(M))). It is a system of fusion relations for a commuting family of transfer matrices of solvable lattice models. There is a large class of models whose R-matrices satisfy the graded Yang-Baxter equation [3,4]. In general, their symmetries are described by the quantum affine superalgebras [5]. In a series of papers [6,7,8], we proposed the T -system for solvable lattice models associated with U q ( sl(M|N)) or U q ( gl(M|N)). In an appropriate normalization, it has the following form 1 : where a, s ∈ Z ≥1 is assumed; the factor ε b ∈ C depends on the definition of the transfer matrices (see, (3.53) for (m, n) = (M, N)). x ∈ C is a multiplicative spectral parameter whose origin goes back to the evaluation map from U q ( gl(M|N)) to U q (gl(M|N)), or an automorphism of U q ( gl(M|N)). T

(a)
s (x) is a transfer matrix (or its eigenvalues) whose auxiliary space (the space where the (super)trace is taken) is an evaluation representation of U q ( gl(M|N)) based on a tensor representation of U q (gl(M|N)) labeled by a rectangular Young diagram with a hight of a and a width of s in the (M, N)-hook (see Figure 1). In particular, T 1 (x) corresponds to the transfer matrix of the Perk-Schultz-type model [10]. In contrast to the bosonic algebra U q ( gl(M)) case, the index a can take arbitrary non-negative integer values. For N = 0, the "duality relation" (1.4) becomes trivial T (M ) b (x) = ε −1 b , which means that the M-th antisymmetric tensor representation of U q (gl(M)) is trivial; while this is not the case for N > 0. The functions T (a) 0 (x), T (0) s (x) (a, s ∈ Z ≥0 ) appearing in the boundary depend on each model and the normalization. We sometimes chose a normalization where these functions are just 1 (this corresponds to the normalization of the universal R-matrix). As was pointed out in [6], the above T -system is a reduction of the so-called Hirota bilinear difference equation [11]. Due to the commutativity of the transfer matrices, the eigenvalues of the transfer matrices obey the same functional relations as the transfer matrices. In this paper, the eigenvalue of the transfer matrix will be called T-function. These equations (1.1)-(1.4) were transformed into the form of the Y -system and used [12] to derive the thermodynamic Bethe ansatz equations. We also used [13] these equations (1.1)-(1.4) to derive nonlinear integral equations with a finite number of unknown functions, which are equivalent to the thermodynamic Bethe ansatz equations. We remark that T -systems closely related to (1.1)-(1.4) also appeared recently in the study of the AdS/CFT correspondence in particle physics [14,15,16]. In particular, two copies of a gl(2|2)-like T -system coupled 2 each other appeared as the "left wing" and the "right wing" of the AdS/CFT T -system (in the form of the Y -system) [16,17].
The so-called Bazhanov-Reshetikhin formula [2] is a determinant expression of the eigenvalue of the transfer matrix for the fusion [18] model whose auxiliary space is labeled by a general Young diagram for a tensor representation of U q ( sl(M)) (or U q ( gl(M))). This formula also has a tableaux sum expression. The Bazhanov-Reshetikhin formula allows a supersymmetric generalization for U q ( sl(M|N)) or U q ( gl(M|N)), which may be called "the quantum supersymmetric Jacobi-Trudi formula" or "the supersymmetric Bazhanov-Reshetikhin formula" [6,7,8] (cf. (2.35) and (2.36); see also a recent paper [19], and also [20] for U q (B (1) r ) case). This formula or its tableaux sum expression for the Young diagram of rectangular shape gives [6,7,8] the solution of the T -system (1.1)- (1.4). The supersymmetric Bazhanov-Reshetikhin formula is a quantum affine superalgebra analogue (or the Yang-Baxterization) of the second Weyl formula for the transfer matrices since this formula reduces to the supersymmetric Jacobi-Trudi formula on the supercharacter of gl(M|N) [21] if the spectral parameter dependence is dropped (this corresponds to the limit x → 0 in our normalization of the spectral parameter x). Then, it is natural to consider an analogue of the first Weyl formula. The first Weyl formula for the superalgebra gl(M|N) is often called "Sergeev-Pragacz formula" in mathematical literature [22] (see (A.10)). In addition, the Sergeev-Pragacz formula has a nice Wronskian-like determinant expression [23]. In this paper, we propose Wronskian-like determinant expressions of the T-functions for U q ( gl(M|N)) (or U q ( sl(M|N))) for any M, N (cf. (3.15)- (3.16)). These include formulae in our previous paper [24] on U q ( sl(2|1)) and also similar formulae for the bosonic case U q ( gl(M)) [25,26,27,28] (see also papers from a different approach [29,30]). In particular for the Young diagram of rectangular shape, these formulae give Wronskian-like determinant solutions of the T -system (1.1)-(1.4) for U q ( gl(M|N)) (cf. Theorem 3.3). We also remark that these Wronskian-like determinants can be viewed as generalization of the ninth variation of Schur function in terms of the first Weyl formula [31,32].
The so-called Baxter Q-operators were introduced by Baxter in his pioneering work on the eight-vertex model [33], and have attracted interest from various point of views (see for example, [34,26,27,29,28,24,30]). They belong to the same commuting family of operators as the transfer matrices. In this paper, the eigenvalue of the Baxter Q-operator will be called the Baxter Q-function. Zeros of the Baxter Q-function correspond to the roots of the Bethe ansatz equation. We have 2 M +N kind of the Baxter Q-functions. And two of them are "trivial" in the sense that they can be normalized to just 1. Thus we have 2 M +N − 2 kind of non-trivial Baxter Q-functions. But they are not independent as they obey the functional relations (2.17)-(2.18) (cf. [35,27,37,38,36,29,39,40,41,24]). As remarked many times [42,7,37,38,29,39,41,24], there are equivalent, but different, forms of the Bethe ansatz in the model. In our case, there are (M + N)! different Bethe ansatz. They are connected by these functional relations among the Baxter Q-functions (2.17)- (2.18). In [7], we discussed equivalence of the Bethe ansatz in relation to the Weyl (super) group for any M, N ∈ Z ≥0 . We also find Wronskian-like determinant expressions of the Baxter Q-functions, which solve the functional relations (2.17)-(2.18) (cf. Theorem 3.2). These determinant expressions are similar to the ones for the T-functions, but labelled by an empty Young diagram. In [26], the Baxter Q-operators are defined as trace of the universal R-matrix over q-oscillator representations of the quantum affine algebra U q ( sl (2)). This construction of the Baxter Q-operators was generalized in the subsequent papers [27,28,24]. In particular, importance of boundary twists or horizontal fields to regularize the trace over the infinite dimensional space was recognized in [26] for the first time. Although we do not discuss operator realization of our formulae on the Baxter Q-functions, we design our formulae with the construction of the Baxter Q-operators in [26] in mind.
In [25], it was shown for gl(M)-related elliptic models that T-functions for quantum integrable models can be treated by methods of classical theory of solitons and discrete integrable equations. Moreover a part of discussions in [25] was generalized, in the recent papers [39,40], to gl(M|N)-related rational models, where (M + 1)(N + 1) kind of Baxter Q-functions were treated. In particular, Bäcklund transformations among solutions of the gl(m|n)-type T -systems (0 ≤ m ≤ M, 0 ≤ n ≤ N) were proposed in [39,40] (3.37), (3.38)). We find that our Wronskian-like determinant formulae explicitly solve all these functional relations (cf. Theorems 3.3,3.4).
In the representation theoretical context, the T-functions can be interpreted as the so-called q-characters [43]. Originally, the q-character is defined [43] as a partial trace of the universal R-matrix for some representation of the quantum affine algebra. Consequently, the eigenvalue formula of the transfer matrix by the Bethe ansatz has a similar 3 form as the q-character if the Baxter Q-functions are defined in the normalization of the universal R-matrix (this corresponds to case where the vacuum part is formally put 4 to 1). In this sense, the T-functions by the analytic Bethe ansatz in [6,7,8] are prototypes of the q-(super)characters for the quantum affine superalgebra U q ( sl(M|N)) or U q ( gl(M|N)). Thus our new Wronskian-like expression of the T-function is yet another form of the q-(super)character. In this expression, the Weyl group invariance of the q-(super) character becomes manifest.
The size of the Wronskian-like determinant for the T-function is less than (for atypical representations) or equal to M + N (for typical representations). On the other hand, the size of the determinant for the supersymmetric Bazhanov-Reshetikhin formula depends on the representation and has no upper bound. In addition, the number of the terms in the tableaux sum expression of the T-function is the dimension of the auxiliary space. Thus our new Wronskian-like determinant formulae will be particularly useful to analyze eigenvalues of the transfer matrices for large dimensional representations in the auxiliary space.
In section 2, we reformulate formulae on the analytic Bethe ansatz in our previous papers [6,7,8]. In section 3, we propose the Wronskian-like determinant expressions of the T-functions and the Baxter Q-functions, and give solutions of the T -system for U q ( gl(M|N)). These are our main results in this paper. We will also mention relation among these Wronskian-like formulae and the formulae in section 2. In section 4, we will briefly comment on how to obtain the T-functions for typical representations. The factorization property of the T-functions straightforwardly follows from the Wronskian-like determinant formulae. In [8], we showed that the T-function for the typical representation can be obtained as a deformation of the tableaux sum expression of the T-function. We rederive this result for the typical representation from our new Wronskian-like determinant formula. In section 5, we see that Baxter equations for U q ( gl(M|N)) straightforwardly follow from the Wronskian-like determinant formulae. They are finite order difference equations and linear with respect to the Baxter Q-functions. Section 6 is devoted to discussions. In Appendix A, we briefly mention representations of the superalgebra gl(M|N) and their characters. It will be good to see how the T-function in this paper reduces to the character formula (in particular, the determinant formula in [23]) of gl(M|N) (or its subalgebras) in the limit x → 0. In Appendix B, we will discuss, the normalization of the Baxter Q-functions. We will prove our main theorems in Appendix C. In Appendix D, we will derive conserved quantities, and obtain determinant formulae of generalized Baxter equations for U q ( gl(M|N)). In this paper, we assume that q is not a root of unity.
2 Analytic Bethe ansatz for the Perk-Schultz-type models and their fusion hierarchies revisited The Perk-Schultz model [10] is a trigonometric vertex model related to the (M + N) dimensional evaluation representation of U q ( gl(M|N)). It is just a representative of an infinite set of fusion models. The fusion models related to U q ( gl(M|N)) were studies in [45,46]. In this section, we briefly reformulate formulae for the T-functions of the fusion hierarchies of the Perk-Schultz-type models from the analytic Bethe ansatz in our previous papers [6,7,8], keeping in mind recent developments [39,40,24]. Let us introduce the following sets . . , f N )). Let us take the first m-components (0 ≤ m ≤ M) of the tuple B M (resp. the first n-components (0 ≤ n ≤ N) of the tuple F N ) and write it as a m-tuple B m = (b 1 , b 2 , . . . , b m ) (resp. a n-tuple F n = (f 1 , f 2 , . . . , f n )). We will use a symbol B m × F n := (b 1 , . . . , b m , f 1 , . . . , f n ). We will denote I 0 , I M +N , B 0 , F 0 as ∅. We also use the symbol I to denote any one of the tuples I 0 , I 1 , . . . , I M +N . In case we need not mind order of the components of these tuples I a , B m , F n , we will treat these as just sets, but will use the same symbols. forms a chain as a family of sets: where Card(J) is the number of the elements of the set J. We also have an opposite family of sets: We will consider functions {Q Ia (x)} M +N a=0 of a multiplicative spectral parameter x ∈ C labeled by {I a } M +N a=0 . These are the so-called Baxter Q-functions 5 , which will be discussed in what follows. We will use a notation (2.4) We will treat the tuple I a just a set for Q Ia (x) and a=0 . These (M + 1)(N + 1)-Q-functions (for rational case) essentially correspond to the ones considered in [39,40].
The eigenvalue formula of the transfer matrix of the Perk-Schultz-type model by the Bethe ansatz has the following form 6 [47]: are defined in the following way. Let us take a K-tuple I = (γ 1 , γ 2 , . . . , γ K ) whose components are mutually distinct elements of I, where 1 ≤ K ≤ M + N. We also introduce (K − 1)-tuplesÎ = (γ 1 , γ 2 , . . . , γ K−1 ) anď I = (γ 2 , γ 3 , . . . , γ K ). Then we define functions labeled by I as and . (2.8) We note that the relation X I a−1 (x) = X Ia (x) holds due the relation j∈Ia p j + j∈Ia p j = M −N and (2.4). The parameters z a ∈ C correspond to boundary twists or horizontal fields; namely we are considering a transfer matrix with a twisted boundary condition. The functions Q I (x) for I = ∅ or I = I M +N are special. In this paper, we will normalize these as In the normalization of the universal R-matrix, the conformal field theoretical models or the q-(super)character, we impose both (2.9) and (2.10), at least for the case M = N (cf. [26,27,28,24]; see Appendix B). For the Perk-Schultz model, we may use the normalization (2.10) and where L is the number of the lattice site; w j is an inhomogeneity on the spectral parameter on the j-th site of the quantum space (the homogeneous case corresponds to w j = 1). (2.11) corresponds to the so-called quantum Wronskian condition. In this case, are polynomials of x for finite L: where {x Ia k } are roots 7 of the Bethe ansatz equation: for k ∈ {1, 2, . . . , n Ia } and a ∈ {1, 2, . . . , M + N − 1}. 7 Here we treat the tuple I a just a set for x Ia k .
The condition that the poles of (2.6) from the zeros of Q Ia (x) (1 ≤ a ≤ M + N − 1) chancel each other: produces the Bethe ansatz equation (2.13). One can obtain a Bethe ansatz system, which has one-to-one correspondence with the above mentioned one for the Perk-Schultz model by the normalization (2.9) and The rational case of this system corresponds to the one considered in [39,40]. In the context of the theory of the q-character [43], (2.6) corresponds to the q-(super)character 8 for the M + N dimensional evaluation representation of U q ( gl(M|N)), and X I 1 (x) is called the "highest weight monomial". Let us write the symmetric group over the components of the tuple I (or elements of the set I) as S(I). We assume that σ ∈ S(I) acts on a ∈ I \ I as σ(a) = a. The action on F , then one sees that these are compatible with the definitions (2.6) and (2.7). Note that σ[Q J (x)] = Q J (x) for any σ ∈ S(I) such that I ⊂ J or I ⊂ I \ J. The direct product of the symmetric groups S(B) × S(F) corresponds to the Weyl group of gl(M|N) if the parameters {z a } are interpreted as in Appendix A. The Q-functions Q I (x) with the same "level" Card(I) are in the same "S(I)-orbit".

17)
8 Difference between the supercharacter and the character is not essential here as the supercharacter becomes the character by the transformation: z a → −z a for a ∈ F. In the theory of the q-characters, one usually includes the parameters {z a } into the Baxter Q-functions and writes the q-characters in terms of the variables of the form Y Ia (x) = Q Ia (xq)/Q Ia (xq −1 ). 9 Functional relations similar to (2.17) appeared in [35] for rational vertex models related to to Q I 4 (x) = Q {1,2,3,4} (x), where these paths must not contain more than two functions Q I (x), Q J (x) of the same level Card(I) = Card(J). In particular, they do not depend on the paths. and also the following (cf. [37,38,29,39,40,24]): In fact, one can easily show the relation (2.16) based on the functional relations (2.17) and (2.18). Functional relations corresponding to (2.18) were discussed in [39,40] from a point view of a classical theory of the soliton, and also to (2.17) in [41]. In addition, (2.17) was proved based on decompositions of q-oscillator representations of U q ( sl(2)) [26], U q ( sl (3)) [27] and U q ( sl(2|1)) [24]. (2.18) was also proved similarly for U q ( sl(2|1)) [24]. For U q ( gl(2)) case, (2.17) reduces to the quantum Wronskian condition. One may also unify the above two relations.
depending on their convergence regions. Or instead, one may expand the Baxter Q- {i} ) and plug this into (2.22). Then we obtain I⊔{i} k q p i into (2.17) with (2.12), we obtain Let us divide these two equations by one another:   Let us introduce notations for the partitions and Young diagrams (see [49] for details). A partition is a non increasing sequence of positive integers µ = (µ 1 , µ 2 , . . . ): µ 1 ≥ µ 2 ≥ · · · ≥ 0. We often write this in the form µ = (r mr , (r − 1) m r−1 , . . . , 2 m 2 , 1 m 1 ), where r = µ 1 , and m k = Card{j|µ j = k} is called the multiplicity of k in µ. Two partitions are regarded equivalent if all their non-zero elements of positive multiplicity coincide. Elements of the multiplicity zero should be abbreviated. For example, (4, 3, 2, 1, 1, 0, 0) = (4, 3, 2, 1, 1) = (5 0 , 4, 3, 2, 1 2 ) = (4, 3, 2, 1 2 ). We will also express the partitions µ as Young diagrams, and will use the same symbol µ for the Young diagrams. The Young diagram µ, corresponding to a partition µ, has µ k boxes in the k-th row in the plane (see Figure 3). Each box in the Young diagram is specified by a coordinate (i, j) ∈ Z 2 , where the row index i increases as one goes downwards, and the column index j increases as one goes from left to right. The top left corner of µ has coordinates (1, 1). The partition µ ′ = (µ ′ 1 , µ ′ 2 , . . . ) is called conjugate of µ, and is defined as µ ′ j = Card{k|µ k ≥ j} (see Figure 4). The Young diagram µ ′ is obtained by the transposition of rows and columns of the Young diagram µ. Let λ = (λ 1 , λ 2 , . . . ) and µ = (µ 1 , µ 2 , . . . ) be two partitions such that µ i ≥ λ i : i = 1, 2, . . . and λ µ ′ 1 = λ ′ µ 1 = 0. We will express a skew-Young diagram defined by these two partitions as λ ⊂ µ. This is the domain given by the subtraction µ − λ (see Figure 5). If λ is an empty set ∅, then λ ⊂ µ coincides with µ. Each box on the skew-Young diagram λ ⊂ µ is specified by its coordinate on µ. We will also use the 180 degrees rotation of the skew-Young   Figure 5 is the skew Young diagram with shape λ ⊂ µ = (3 2 , 2, 1) ⊂ (4 3 , 3, 2). diagram λ ⊂ µ, and denote it as λ ⊂ µ (see Figure 6): Let us take a K-tuple I = (γ 1 , γ 2 , . . . , γ K ) whose components are mutually distinct elements of I, where 1 ≤ K ≤ M +N. We also introduce k-tuple D k (I) = (γ 1 , γ 2 , . . . , γ k ) and (K −k+1)-tuple D k (I) = (γ k , γ k+1 , . . . , γ K ), where 1 ≤ k ≤ K. Next, let us define a space of admissible tableaux Tab I (λ ⊂ µ) for the K-tuple I on a (skew) Young diagram λ ⊂ µ. We assign an integer t ij in each box (i, j) of the diagram. An admissible tableau t ∈ Tab I (λ ⊂ µ) is a set of integers t = {t jk } (j,k)∈λ⊂µ , where all t jk ∈ {1, 2, . . . , K} satisfy the conditions The fusion models are described by the Young diagrams. Here we supposed that the fusion is performed in the auxiliary space of the transfer matrix. In particular, the T-functions of the fusion models from the analytic Bethe ansatz can be written as summation over Young tableaux with spectral parameters [6,7,8] (see also [2] for M > 0, N = 0 case, and [20] for U q (B (1) r )). Here we introduce slightly generalized versions of such T-functions 10 F I λ⊂µ (x) and F I λ⊂µ (x). We remark that the following discussion on the T-functions is essentially independent of the "vacuum part", and thus quantum space, where the transfer matrices act, is arbitrary. For any skew Young diagram λ ⊂ µ, define functions 11 : where the summations are taken over all admissible tableaux, and the products are taken over all boxes of the Young diagram λ ⊂ µ; m := card(I ∩ B), n := card(I ∩ F). 10 The Young diagram µ in F We also set F I ∅ (x) = F I ∅ (x) = 1, and F ∅ µ (x) = F ∅ µ (x) = 0 and for the non-empty Young diagram µ. Note that the admissible tableaux Tab I (λ ⊂ µ) becomes an empty set if the Young diagram λ ⊂ µ contains a rectangular sub-diagram of a height of m + 1 and a width of n + 1, and consequently (2.32) and (2.33) vanish for such Young diagram 12 . After inspection of (2.7), (2.8), (2.32) and (2.33), one observes the following relation: where s is any shift of the Q-functions in F I λ⊂µ (x), and I = (γ K , . . . , γ 2 , γ 1 ) is a reversed tuple of I.
Let us take an integer K (0 ≤ K ≤ M + N). The tableau sum formula (2.32) has determinant expressions.
We also have the same type of formulae for (2.33). These determinant expressions for K = M + N correspond to the supersymmetric Bazhanov-Reshetikhin formulae [6,7] (see also a recent paper [19]; and for U q (B (1) r ) case, see [20]), which are supersymmetric extension of the Bazhanov-Reshetikhin formulae [2]. The determinants (2.35) and (2.36) for U q ( gl(M|N)), M, N > 0 superficially look same as the ones for M = 0 or N = 0 case. However, one must note that their properties for M, N > 0 case are different from the ones for M = 0 or N = 0 case since the conditions of the admissible tableaux are different, and that the matrix elements of the determinants are different.
The above combinatorial tableau sum expressions of the T-functions (2.32) and (2.33) are convenient for computer calculations. Next, we consider other expressions of the T-functions. One can rewrite the above tableau sum expressions for the Young diagrams of one row or one column in the form of the non-commutative generating series [6,7] (see also, [25,39,40]).
where X is a shift operator Xf (x) = f (xq −2 )X for any function of x; m := card(I K ∩B), We also have the inverse of the above generating series.
They obey the following recurrence relations (cf. [39,40]): These generating series (2.37) and (2.39) for K = M + N (resp. (2.38) and (2.40) for K = 0), which generate eigenvalue formulae of "real" transfer matrices appeared in [6,7]. In particular, , corresponds to eq. (3.11) in [7]. In addition, from this relation, one can derive the following relation for s ∈ Z ≥0 : This is a generalization of eq. (E.6) in [6]. On the other hand, "Intermediate" (0 < K < M + N) generating series (2.37) and (2.39) were considered 13 in [39,40]. Some "intermediate" generating series were also written in Appendix E in [6]. We remark that the non-commutative generating series of the T -functions similar to the ones in [6,7] were also used recently in the study of the AdS/CFT correspondence in particle physics [38,14,15,41,16]. We define the action of σ ∈ S(I) on the T-functions as . As was pointed out in [39,40], compatibility of this type of recurrence relations (2.41)-(2.42), which corresponds to a discrete "zero curvature condition", reproduces the functional relations among the Baxter Q-functions (2.17)- (2.21). This corresponds 14 to W I a+1 (x, X) = W τ (I a+1 ) (x, X) in the notation of (2.16). Therefore τ [F We can also consider 15 a non-commutative generating series of the T-functions for the general skew-Young diagrams λ ⊂ µ: is the symmetric group of order µ ′ 1 and sgn(σ) is the signature of a permutation σ; {X j } are commutative (X i X j = X j X i ) shift operators: in the right hand side of (2.44), and write it as . We have also derived a similar formula for F In general, the function (2.32) for I = I K (resp. (2.33) for I = I K ) has poles from zeros of the Baxter Q-function Q I K (x) (resp. Q I K (x)). For example, the function F I K (1) (x) under (2.9), (2.12) and (2.15) has poles 16 for 1 ≤ a ≤ K. And these poles for 1 ≤ a ≤ K − 1 chancel each other as in (2.14) under the Bethe 14 The compatibility condition in [39,40] corresponds to the one for the rational case with p ia = −p ia+1 . The compatibility condition for the rational case with p ia = p ia+1 was discussed in [41]. 15 See eq. (3.14) in [50] for "classical" case: x = 0, λ = ∅, I K = (1, 2, . . . , M + N ). 16 For K = M + N case, the poles for a = M + N (poles from the vacuum part) are located at ansatz equation (2.13). But the poles for a = K still survive. To kill 17 this kind of pole, we introduce the following transformations for any (non-skew) Young diagram µ.
under the normalization (2.9) and the functional relations (2.17)-(2.18), and In particular for a = 0 or s = 0 case, we interpret them 18 as: For a, s ∈ Z, F (x) = 0 if a < 0, or a > 0 and s < 0, or a > m and s > n. There are "duality relations" among the functions 19 .
. 17 There are cases where poles from the vacuum part still survive (see (B.11) in Appendix B). This depends on the normalization of the Baxter Q-functions. In any case, we expect that poles from the roots of the Bethe ansatz equation disappear. 18 . And then we formally put µ 1 − µ ′ 1 = s for a = 0 and µ 1 − µ ′ 1 = −a for s = 0 in (2.45) and (2.46). We included non-vanishing terms on s < 0 for the case a = 0 as in [39]. 19 In the context of the analytic Bethe ansatz, the duality relation for (m, n) = (M, N ) appeared in [6] first. This relation with the factor γ∈IK p γ z pγ γ a for (m, n) = (M, N ) appeared in [13] (the parameters {z a } were used as chemical potentials). This factor was also written as a superdeterminant in [40].
The T-functions of fusion models obey functional relations called the T -system. For a, s ∈ Z ≥1 , F (a),I K s (u) satisfies the U q (gl(m|n))-type T -system: and F (a),I K s (u) satisfies the U q (gl(m|n))-type T -system: This type of "boundary condition" (2.47), (2.48), (2.49), (2.50) was discussed in [39,40] (for N = 0 case, see [25]). For K = M + N (resp. for K = 0), the right hand side of (2. (x) also satisfies 20 the following functional relations, which correspond to the Bäcklund transformations 21 [39,40]. 20 This should follow from similar discussions given in [39] for rational case. As we already remarked, the Baxter Q-functions in [39] correspond to the rational case of (M + 1)(N + 1) subset of 2 M+N Baxter Q-functions. Thus we have to consider actions of the Weyl group S(B) × S(F) to get the whole set of the Baxter Q-functions, and also the whole set of F (a),I s (x) and F (a),I s (x). 21 For these relations for gl(M ) related elliptic models were proposed in [25]. They were generalized to gl(M |N ) related rational models in [39] (without twist parameters case); [40] (with twist parameters case). The normalization of the spectral parameter of the functions in [39,40] are closer to the following (functions with prime): ). In fact, (2.59)-(2.62) have the same type of shift of the spectral parameter as in [39,40] if they are written in terms of The functional relations for p i K = 1 have the following form: (2.60) and the functional relations for p i K = −1 have the following form: There are also relations for F (a),I K s (x). The functional relations for p i K = 1 have the following form: and the functional relations for p i K = −1 have the following form: ) connect the solution of the U q (gl(m|n))-type T -system (resp. U q (gl(m|n))-type T -system) and that of the U q (gl(m − 1|n))-type T -system or the U q (gl(m|n − 1))-type T -system (resp. the U q (gl(m+1|n))-type T -system or the U q (gl(m|n+1))-type T -system). Thus an original problem on U q (gl(M|N)) is connected to the one on U q (gl(0|0)).
Before closing this section, we would like to mention the T-functions for conjugate representations (cf. Appendix B in [6]; eq. (3.11) in [24]). We conjecture that they are obtained by a manipulation 22 C for the T-functions: for any shift s of the spectral parameter of the Q-functions in the T-functions and for any a ∈ I and J ⊂ I. For example, (2.6) is transformed to

Wronskian-like formulae for the T -and Qfunctions
In this section, we propose Wronskian-like formulae for the T -and Q-functions (cf. (3.15), (3.16)), and claim that they satisfy the functional relations. Theorems 3.2, 3.3, 3.4 are our main results of this paper. We will also mention relation among the Wronskian-like formulae and the formulae in the previous section (cf. (3.67), (3.68)). In Proposition 3.6, we will rewrite the Wronskian-like formulae (3.15) and (3.16) as summations over a direct product of symmetric groups, which suggest T-functions for infinite dimensional representations. There are Wronskian-like formulae for the Tfunctions for the bosonic case N = 0 [25,26,27,28,30]. However, our new formulae here are not straightforward generalization of the ones for N = 0. The size of the matrices for the Wronskian-like formulae for the T -functions depends on the representation for M, N > 0 case. The upper bound of the size is M + N. Note that the size is constantly M for N = 0 case. There are papers related to rational models for sl(2|1) [29] and trigonometric models for U q ( sl(2|1)) [24]. 22 The Q-function itself is also a function of {z a } and q: Q I (x) = Q I (x, q, z 1 , . . . , z M+N ). But we need not touch these in our normalization of the spectral parameter of the Q-function. Namely, C[Q I (xq s , q, z 1 , . . . , z M+N )] = Q I (xq −s , q, z 1 , . . . , z M+N ). If one applies C to (2.47)-(2.66), one will obtain the T -system and the Bäcklund transformations for the conjugate representations. There are cases where T-functions for both fundamental and its conjugate representations are necessary to analyze physically interesting models (see for example, [51]).
where (0) α×β is a α × β zero matrix. Note that {r k , s l } are supposed to be integers, but this determinant are still well defined even if they are complex numbers. And this fact will be used in section 4. For m, n ∈ Z ≥0 and Young diagram µ, we introduce a number, called (m, n)-index [23].
. . , f n ), x ∈ C, and Young diagram µ, we introduce the following functions.

(3.18)
This is a generalization of the Cauchy identity. This represents a correlation function of the bc-system of the conformal field theory on P 1 , and is related to the "boson-fermion correspondence". (3.18) satisfies the following relations. For generic 24 Z k,l , X k,l , Y k,l , W k,l in (3.1), (3.15) is a generalization of the ninth variation of Schur function in terms of the first Weyl formula [31,32]. In fact, T B M ,∅ µ (x) corresponds to eq. (1.7) in [32]. It is interesting to reformulate discussions in this paper based on the Gauss decomposition of a sub-matrix of A(x) along the line in [32].
Lemma 3.1. If the Young diagram µ contains a rectangle with a hight of (m + 1) and a width of (n + 1), the functions (3.15) and (3.16) vanish.
This can be proved by the fact that the (m, n)-index (3.6) becomes larger than or equal to m + 2 in this case. We impose the normalization (2.9) (resp. (2.10)) when we use (3.15) (resp. (3.16)) from now on. Now we mention one of the main results in this paper. A proof of this theorem is given in Appendix C.1. Let us write the above formulae explicitly. For m ≥ n, we have 23 In the context of the conformal field theory, the author found the identity (3.17) in [53] first. Then he heard from Yasuhiko Yamada that this identity naturally follows from a method, for example, in [54]. One may say that the Baxter Q-functions or Q-operators are the Yang-Baxterization of correlation functions of the conformal field theory. There are higher genus analogues of correlation functions of the conformal field theory. They were used to calculate string amplitudes. Whether such correlation functions also allow the Yang-Baxterization as Q-functions or Q-operators (or more generally, T-functions or T-operators) will be an interesting question. 24 without assuming (3.3) since ξ m,n (∅) = m − n + 1. And also for m ≤ n, we have We have normalized the Baxter Q-functions as in (2.5). This normalization is convenient when we realize the Baxter Q-operator as super-trace over a q-oscillator representation of U q ( gl(M|N)) [55] since the final expression of the Q-operator does not depend on choice of the vacuum of the Fock space where the super-trace is taken. In this normalization, one can also easy to see that the T-function reduces to the (super)character formula of gl(M|N) (or its subalgebras) in the limit x → 0. Of course, one may adapt different normalizations of the Baxter Q-function. Let us consider the following transformation (cf. eq. (65) in [40]): (3.26) It will be good to put to simplify the formulae. For a I = 1, (2.17) and (2.18) become for U q ( sl(2|1)) case [24]. In the case of the Baxter Q-operators, the parameters S i are certain linear combinations of the Cartan generators of U q ( gl(M|N)) and external field parameters; and in the case of the Baxter Q-functions (eigenvalues), they are conserved quantum numbers (edge occupation numbers). In addition to (3.27), one may require Thus one obtains:  26 Here we used a notation: I × (i, j) = (i 1 , i 2 , . . . , i a , i, j) etc. for I = (i 1 , i 2 , . . . , i a ). 27 The second functional relation in the coefficient free form (3.33) is discussed in [39].
where the summation is taken over all the subset I of B m such that Card(I) = a. We remark that the right hand side of (3.41) and (3.42) is well defined as a function of x even if s is not non-negative integer. Based on this fact, one can derive T-functions for conjugate representations by considering negative s (as examples for U q ( sl(2|1)) in [24]). In fact, applying (2.67) to (3.41), we obtain the following relation: where the summation is taken over all the subset J of F n such that Card(J) = s. We remark that the right hand side of (3.44) and ( for a ≥ m. In [39], this type of factorization formulae for the T-functions were treated as "boundary conditions" of the Hirota equation. These formulae (3.46)-(3.49) are related to the Tfunctions for typical representations, which will be commented in section 4. Now we mention our main theorem.  A proof of this theorem is given in Appendix C.2.
Our new determinant formulae also satisfy the Bäcklund transformations [39,40]. The following theorem is also one of our main results.
A proof of this theorem is given in Appendix C.3. Due to Lemma 3.1, some terms of the above functional relations (3.58)-(3.65) vanish for large s, a. As remarked in [39], this type of functional relations contain TQ relations as special cases. Due to (3.38) and (3.40), the functional relations (3.58), (3.60), (3.62) and (3.64) reduce to TQ relations for a = 0 and s ∈ Z ≥1 . Due to (3.37) and (3.39), the functional relations (3.59), (3.61), (3.63) and (3.65) also reduce to TQ relations for s = 0 and a ∈ Z ≥1 . For example, (3.58) reduces to Considering the action of S(B) × S(F), one sees that the above equation (3.66) essentially corresponds to eq. (4.6) in [39]. These are difference equations on the Baxter Q-functions {Q I (x)} or {Q I (x)} for different "levels" Card(I). In section 5, we will mention different type of TQ relations (we will call them "Baxter equations"): difference equations on the Baxter Q-functions for the same level. Let us mention relation among the formulae in the previous section and the ones in this section. Our claim is: 29 But, in our case, we need numerous case study (similar to the ones in Appendix C) depending on the values of m, n. (3.70) where l m,n = µ ξm,n + 1, r m,n = n − m + ξ m,n (µ) − l m,n . Then we find  [23]), where the following relation will be used: where b ∈ B, f ∈ F. This relation follows from (2.22). There is a similar relation . The finite dimensional modules can be written in terms of a direct sum of infinite dimensional highest weight modules due to the Bernstein-Gel'fand-Gel'fand (BGG) resolution. In this respect, the function (3.69) and (3.70) divided by (3.18) for (m, n) = (M, N) may be interpreted 30 as T-functions for infinite dimensional highest weight representations of U q (gl(M|N)) (compare the above formula with eq. (5.21) in [27] for U q (gl(3)), and also formulae in [26,29,24,30]). Of course, this needs further research in view of the fact that the BGG-type resolution of representations of superalgebras is more involved than that of bosonic algebras (cf. [57]). Let us introduce linear operators, which act on the variables {z 1 , z 2 , . . . , z M +N } as where one should write any term like 1 z k f before the actions of these operators. For example, for b ∈ B, f ∈ F and a ∈ I (a = b, f ), we have  A merit of this type of formula is that it contains only one term, and thus is easy to evaluate. The traditional form of the eigenvalue formula by the Bethe ansatz (such as the one in section 2) contains an infinite number of terms if the auxiliary space is an infinite dimensional space, and thus is not always easy to evaluate. Some discussions on T-functions for infinite dimensional representations (in relation to the AdS/CFT correspondence) can be seen, for example, in Appendix B of [38] and section 6.4 of [14]. and {B a (x)}. The q-characters, which correspond to the T-functions, appear in the kernel of the Frenkel-Reshetikhin screening operators [43]. The transfer matrices for Y (gl(M|N)) were also written in terms of a certain "group derivative" [19] on the (super)characters of representations of gl(M|N). It will be interesting to investigate 31 connection among our operators and the operators in [43,19].

T-functions for typical representations
The type 1 quantum superalgebra U q (gl(M|N)) (or U q (sl(M|N))) admits a oneparameter family of finite-dimensional irreducible representations, which correspond to typical representations (see Appendix A). Thus the evaluation representations of U q ( gl(M|N)) (or U q ( sl(M|N))) based on the above mentioned representations depend not only on the spectral parameter but also on another continuous parameter. There are R-matrices for this family of representations [58]. In [8], we proposed T-functions for a wide class of such representations from the analytic Bethe ansatz (see also, [9,59,39]). Here we briefly comment on how to obtain the T-functions for such representations in the auxiliary space from our Wronskian-like formulae.

Baxter equations
The Baxter equations are difference equations whose solutions give the Baxter Qfunctions. The Baxter equations in question are the equations for the Q-functions {Q I (x)}, {Q I (x)} of the same "level" Card(I) = 1. Discussions on the Baxter equations for superalgebra related models can also be seen in [29,60,15]. Once Wronskian-type determinant formulae are established, the Baxter equations can be derived easily. In essence, the following Baxter equations are based on a basic theorem on the linear algebra: if two different lows or columns of a matrix coincide, the corresponding determinant vanishes. Thus one can derive the Baxter equations directly from the Wronskian-like formulae (3.15)-(3.16) related 32 to typical representations as follows:   [25,28]).
One can also factor out overall factors in (5.1)-(5.4) based on the formula (4.1) (and the same type of formula for T Bm,Fn µ (x)) as follows: and m a=0 There is another type of Baxter equations, which follows from the kernel of the non-commutative generating series (2.37)-(2.40). In contrast to the above equations, this type of Baxter equations has an infinite number of terms. This is because that the non-commutative generating series of the T-functions are infinite order difference operators. This situation is similar to other quantum affine (super)algebras cases such as U q (B In this paper, we have overcome this difficulty for U q ( gl(M|N)) case. Thus our example on U q ( gl(M|N)) will be a good reference to find [55] finite order Baxter equations and Wronskian-like formulae for T-functions for more general quantum affine (super)algebras.
We have also derived a determinant form of finite order Baxter equations based on conserved quantities in Appendix D.

Discussions
In this paper, we assume that the deformation parameter q is generic. If q is a root of unity, the Baxter Q-function becomes a periodic function of x. Thus, through our new Wronskian-like determinant formulae of the T-functions, one sees that the quantum group truncation occurs in a very different way than for the bosonic algebra case. Namely, the T-function T  the twist parameters are also proportional to appropriate roots of unity. Note that truncation occurs only on the variable s for n = 0 case.
We have proposed various formulae for the Baxter Q-functions and the T-functions. It is important to realize these as operators. Of course, the same formulae hold true for both operators and their eigenvalues since the Baxter Q-operators and the T -operators belong to the same commuting family of operators. So far, realizations of the Baxter Q-operators based on the q-oscillator representations of the quantum affine (super)algebra, which are relevant to our formulation in this paper, are available for U q ( sl(2)) [26], U q ( sl (3)) [27], U q ( sl(M)) [28], U q ( sl(2|1)) [24], and the general case U q ( gl(M|N)) (or U q ( sl(M|N))) is under investigation [55]. Although the identities among T and Q-functions in this paper are combinatorial ones and hold true independent of the representation theory, it is also important to establish connection to the universal R-matrix and to prove our new formulae from the point of view of the representation theory.
The supersymmetric Bazhanov-Reshetikhin formulae (eqs. (2.35)-(2.36) for K = M + N) [6,7,8] played an important role to derive [13] (see also [64]) a system of nonlinear integral equations (NLIE) for the free energy of integrable spin chains at finite temperature, which is equivalent to the thermodynamic Bethe ansatz (TBA) equation 34 , from the T -system (1.1)-(1.4) [6,7,8]. Then a natural question is whether the Wronskian-like determinant formulae are also useful to derive the NLIE. There is another type of NLIE [66], which is equivalent to the TBA equation. To generalize this type of NLIE to models whose underlying algebras have arbitrary rank is not always an easy problem as one needs considerable trial and errors to find auxiliary functions with good analytical properties which are needed to derive the NLIE. Although the Y -system itself is not truncated to a finite set of functional relations for generic q, an introduction of appropriate auxiliary functions modifies [67] the Y -system to a finite set of functional relations, which is suited for the NLIE. So far this type of NLIE is known for at most rank 4 case [68]. We expect that some of our formulae in this paper are also useful to construct auxiliary functions for this type of NLIE for arbitrary rank. One can easily transform the T -system into the Y -system based on a transformation on the T-functions (to the "Y -functions"), as in [12]. This type of transformation (applied for (3.58)-(3.65)) also induces a system of Bäcklund transformations for the Y -system. Thereby Bäcklund transformations for the TBA equations are obtained. Our formulae for the T-functions T Our new formulae in this paper have potential applicability to the analysis of the AdS/CFT T -system (or the Y -system) [16,17], after some modifications. Note that many formulae in this paper do not depend on precise function form of the Baxter Q-functions. The modifications should be done so that the Baxter Q-functions are compatible with the Bethe ansatz equation in [69] (for large L). We also expect that this is also the case with the Hubbard model.
To generalize our formulae to mixed representations is also an important problem. So far, the supersymmetric Bazhanov-Reshetikhin type formula for a simplest case was written long time ago (cf. eq. (4.1) in [6]), but more general case is under investigation [55]. We expect that the (super)character formulae for the mixed representations of gl(M|N) also allow the Yang-Baxterization by the Baxter Q-functions (or Q-operators).
The representation theory of the quantum affine superalgebras is not always well understood even in finite dimensional representations case. In this concern, to generalize our formulae in this paper to other quantum affine superalgebras such as U q ( osp(M|2N)), U q (D (1) (2, 1; α)) etc. is interesting not only in the study of the integrable system but also in the representation theory in mathematics. Tableaux sum expressions for T-functions, some supersymmetric Bazhanov-Reshetikhin-like formulae for fusion models and the T -system are available for U q ( osp(M|2N)) [63] case, and this will be a starting point to establish [55] Wronskian-like formulae for T-functions or q-(super)character formulae for the general quantum affine superalgebras.
where ǫ 1 , . . . , ǫ M , δ 1 , . . . , δ N are the basis of the dual space of the Cartan subalgebra with the bilinear form ( | ) such that {α i } i =M are even roots and α M is an odd root with (α M |α M ) = 0. Any weight can be expressed in the following form: There is a class of irreducible tensor representations of gl(M|N) whose highest weight Λ is characterized by the Young diagram µ = (µ 1 , µ 2 , . . . ), (µ 1 ≥ µ 2 ≥ · · · ≥ 0, µ M +1 ≤ N) in the (M, N)-hook (see Figure 1): Let Λ be a real dominant weight. An irreducible representation of gl(M|N) with the highest weight Λ is called atypical if and typical if there is no such (i, j). Here ρ is the graded half sum of the positive roots: Thus the aforementioned irreducible tensor representation is typical if µ M ≥ N, and atypical if µ M < N.
There is a large class of finite dimensional typical representations, which is not tensor-like. For example, for the above mentioned typical irreducible tensor representations with the highest weight Λ, there is a one parameter family of irreducible typical 36 representations with the highest weight Λ(c) = Λ + cω 1 , (A.9) where ω 1 = ǫ 1 + ǫ 2 + · · · + ǫ M . Note that the M-th Kac-Dynkin label b M = µ M + η 1 + c is not an integer if the parameter c is not an integer. One may generalize the above representation to the one with the highest weight Λ(c 1 , c 2 ) = Λ + c 1 ω 1 + c 2 ω 2 , where ω 2 = δ 1 + δ 2 + · · · + δ N . The Kac-Dynkin label of Λ(c 1 , c 2 ) depends only on c 1 + c 2 , and coincides with that of Λ(c 1 + c 2 ). In this sense, this representation essentially has only one parameter. These parameters c 1 and c 2 correspond to the ones in section 4. There is a gl(M|N) analog of the first Weyl formula, called the Sergeev-Pragacz formula [22]. It is an alternative representation of the supersymmetric Jacobi-Trudi formula, which corresponds to a gl(M|N) analog of the second Weyl formula. Let us take a non-skew Young diagram µ. Then the Sergeev-Pragacz formula has the following form 37 where the third product in the numerator is taken over all boxes (i, j) ∈ µ of the Young diagram µ. We assumed that x i = 0 if i ≥ M + 1 and y j = 0 if j ≥ N + 1. . 36 If the atypicality condition (A.7) holds, this representation will be reducible (but in general indecomposable). 37 We note that the sign of the variables y 1 , . . . , y N in [22] is opposite to our definition.
The representation theory of the quantum superalgebra U q (gl(M|N)) for generic q is similar (see for example, [71]) to that of gl(M|N). The atypicality conditions for both cases are the same. And the (super)characters are also the same.
To make discussions complete, we should mention the representation theory of U q ( gl(M|N)) (or its Borel subalgebra). This is realized in combination of the evaluation map from (a Borel subalgebra of) U q ( gl(M|N)) to U q (gl(M|N)) and representations of U q (gl(M|N)). We expect that the functions T B M ⊔F N µ (x) and T B M ⊔F N µ (x) come from two different evaluation maps (see eqs. (4.14)-(4.18) in [27], and also [28]). q-oscillator representations are also needed to describe the Baxter Q-operators (cf. [26,27,28,24]). We would like to postpone these issues to a separate publication [55], where the functions in this paper will be realized as operators.

Appendix B. On the normalization of the Baxter Q-functions
Let us renormalize the Baxter Q-function for any I ⊂ I as: Here the function φ I (x) (vacuum part), whose precise form will be fixed in what follows, does not contain the roots x I j of the Bethe ansatz equation. In our normalization, it satisfies φ I (0) = 1. Then the functional relations (2.17)-(2.18) are modified to 38 :   [47,4,72,8,48,39]) where the function Φ a (x) has, for example for the Perk-Schultz model [47,10], the following form 39 Here L is the number of the lattice site and w k is an inhomogeneity on the spectral parameter. Thus φ Ia (x) must satisfy the following functional equations.
for a ∈ {1, 2, . . . , M + N − 1}. (B.9) 39 For more general quantum space with the highest weight j+M δ j for each site (k = 1, 2, . . . , L) of the quantum space, we expect that the function has the following form: There will be an another form of the vacuum part, where the left hand side of (B.7) has the form − The left hand side of the Bethe ansatz equation for Φ a (x) is speculated by applying a method in [20] to a (non-skew) Young diagram, where Drinfeld polynomials can be obtained; while the one for Φ a (x) is speculated by applying a method in [20] to 180 degrees rotation of this Young diagram. See a remark in section 7 in [39]. We expect these come from two different evaluation representations of U q ( gl(M |N )), commented in Appendix A, for the quantum space of the models. A precise description of these issue will need detailed Bethe ansatz calculations based on the representation theory of the quantum affine superalgebra, and is beyond the scope of this paper (cf. [48]).
(C.5)-(C.8) are specialization of the so-called Plücker identity, and (C.9) is the Jacobi identity. We remark that some Plücker identities on supersymmetric polynomials were also discussed in [73].

C.1 Proof of Theorem 3.2
We will prove Theorem 3.2 for (3.22). The proof for (3.23) is similar to the one for (3.22). Let us introduce a notation: We have to consider the following nine cases since the (m, n)-index Let us expand the determinants 41 in the left hand side of the above relation: where we introduced a notation B m+2 \b β = (b 1 , b 2 , . . . , b β−1 , b β+1 , . . . , b m+2 ), and used the functional relation (2.22). The determinant · · · in the first summand in the right hand side of (C.17) vanishes for β = m + 1, m + 2 as it has two identical rows. The determinant in the second summand vanishes as it has two identical columns. Then the right hand side of (C.17) reduces to the right hand side of (C.16). This is nothing but the Plücker identity (C.8).
(iii.2) (C.13) for the case m = n In this case, (C.13) has the form: where we used the functional relation (2.22). The determinant · · · in the first summand in the right hand side of (C.23) vanishes for α = m + 1. The determinant in the second summand vanishes for β = m + 1. Thus the right hand side of (C.23) coincides with the right hand side of (C.22). This is nothing but the Plücker identity (C.7).

C.2 Proof of Theorem 3.3
We will prove Theorem 3.3 for (3.35). The proof for (3.36) is similar to the one for (3.35). For a, s ∈ Z ≥1 , we will prove that (3.     where we introduced notations B m \b α = (b 1 , . . . , b α−1 , b α+1 , . . . , b m ), F n \f β = (f 1 , . . . , f β−1 , f β+1 , . . . , f n ), and used the functional relation (2.22) . The determinant · · · in the first summand in the right hand side of (C.32) vanishes as it has two identical columns. Subtracting the α-th row from the (m + 1)-st row in the determinant · · · in the second summand in the right hand side of (C.32), we get: The determinant · · · in the first summand in the right hand side of (C.41) vanishes for γ = a as it has two identical columns. The determinant · · · in the second summand in the right hand side of (C.41) vanishes for γ = s. Then the right hand side of (C. The determinant · · · in the right hand side of (C.47) vanished for γ = 1. Then we obtain (C.44).
(iii) The case a − s + 2 ≤ m − n ≤ a Due to (C.2), (3.58) is identical to the following relation.  (xq 2 ) with respect to the m-th row. After some calculations similar to the ones in the case (ii), we arrive at the right hand side of (C.50).
β-th column removed from M as D