QQ-system and Weyl-type transfer matrices in integrable SO(2r) spin chains

We propose the full system of Baxter Q-functions (QQ-system) for the integrable spin chains with the symmetry of $D_r$ Lie algebra. We use this QQ-system to derive new Weyl-type formulas expressing transfer matrices in all symmetric and antisymmetric (fundamental) representations through $r$ basic, single-index Q-functions. Our functional relations are consistent with the Q-operators proposed recently by one of the authors and verified explicitly on the level of operators at small finite length.


Introduction
Baxter Q-operators play an important role in the theory of integrable spin chains [1], in 2D integrable quantum field theory and sigma models [2], in integrable examples of higher dimensional CFTs, such as QCD in BFKL limit [3][4][5], N = 4 super Yang-Mills theory and ABJM theory [6] where Q-functional approach has led to the elegant description of spectrum of the systems in terms of the quantum spectral curve (QSC) [7,8], the ODE/IM correspondence [9], the fermionic basis [10], stochastic processes [11,12] and pure mathematics [13]. In particular, they provide a natural formulation for the Bethe ansatz equations (BAE) whose solutions (Bethe roots) yield the spectrum of energy for the Heisenberg-type spin chains and are at the heart of Sklyanin's separation of variables (SoV) construction [14]. They also allow for natural representations of transfer matrices (T-operators), encoding all quantum conserved charges of the system. All these operators, T and Q, commute due to the underlying integrable structure, so that on a given eigenstate we can operate with their eigenvalues -the functions of a spectral parameter: T (x) and Q(x). For A-type spin chains all these operators can be built within the framework of the quantum inverse scattering method [15] from solutions of the Yang-Baxter equation. The transfer matrices are built from Lax matrices of finite dimension while, as noted in [2,16,17], the construction of Q-operators is related to an infinite-dimensional Hilbert space. These methods were further developed in [18][19][20][21][22][23][24][25][26][27]. For us the most relevant articles are [28][29][30] for Q-operators of A-type spin chains and the recent generalisation to some Q-operators of D-type spin chains [31]. An alternative approach, based on the formalism of co-derivatives [32], was proposed in [33] and further developed in [34] in relation to the interplay between quantum and classical integrability of A-type spin chains.
T-functions represent a quantum generalisation of the characters for the symmetry algebra of the spin chain. They depend on the representation f in the auxiliary space and, generically, on the twist τ -a group element introduced into the spin chain in the form of twisted, quasiperiodic boundary conditions or, alternatively, as generalized "magnetic" fields. That is why we will denote the T-functions as T (τ ) f . 1 Generally, one has an infinite number of different Tfunctions since there exists an infinite number of inequivalent representations. However, most of them are not independent quantities. The most constructive way to see that is to represent T-functions in terms of Baxter Q-functions since the latter always form a finite variety. Say, for A r algebra, the 2 r+1 Q-functions are usually labeled by subsets of integers I ⊂ {1, 2, . . . , r, r + 1} 1 Though the explicit τ dependence will often be omitted.
where r is the rank of the algebra (for instance I = {1, 3, 4} ⊂ {1, 2, 3, 4, 5}). This QQ-system 2 can be conveniently depicted as a Hasse diagram in the shape of an r + 1-dimensional hypercube with the vertices labeling the corresponding Q-functions [35], see Figure 1 for the example of A 2 .
As we will see, in D r algebra the labeling is similar but slightly different. Moreover, only r + 1 Q-functions are algebraically independent as in the A r case [19,[36][37][38], see also the supersymmetric generalisation [8,35,39,40]. The system of all Q-functions, which we will call here QQ-system, is endowed with a Graßmannian structure. The remaining Q-functions can thus be expressed through a chosen basis of r + 1 Q's by various Plücker QQ-relations, often in the form of Wronskian determinants (Casoratians). For the Heisenberg spin chains with A r symmetry, the most traditional representation of T-functions is given in terms of the basis of Q-functions of the type Q 1 , Q 12 , Q 123 , . . . , Q 123...r+1 (or different re-labelings of the same basis), cf. Figure 1. The same functions enter into the formulation of the standard nested system of BAE's. The T-functions in this basis are usually represented by the so-called tableaux formulas which are direct generalizations of Schur polynomials for characters [37,41,42].
The other well-known, so-called Cherednik-Bazhanov-Reshetikhin (CBR), formulas for Tfunctions in an arbitrary finite dimensional representation ℓ, are given in terms of determinants of T-functions in the simplest symmetric or antisymmetric representations [43,44]. They have been proven in [32], including the supersymmetric A r|s algebra. They represent the quantum generalization of the Jacobi-Trudi formulas for characters. For the D r algebra the corresponding determinant representations have been found in [45]. For both algebras, the CBR type formulas appear to be solutions [46], with appropriate boundary conditions, of Hirota finite difference equations for T-functions [42,47,48] (TT-system).
The goal of this article is to construct a similar QQ-system, together with a similar Weyl-type representation for T-matrices, for the D r algebra. The standard Weyl formula for D r characters is see e.g. [53][54][55][56], with ℓ j = f j + r − j and the highest weights f 1 f 2 · · · f r−1 |f r | are all integers or all half-integers (the last one can also be negative). However, in general, the situation for D r is more complicated than for A r algebra. The representations of the Lie algebra do not "quantize" trivially, i.e. cannot be lifted to the Yangian algebra (apart from the symmetric and spinorial representations), see [57] and [58] for an instructive example. Instead, in order to construct the T-functions, one has to introduce the representations acting in the so-called Kirillov-Reshetikhin modules [59]. Such modules are known only for rectangular representations (a, s). These representations have highest weights f 1 = f 2 = · · · = f a = s and f a+1 = . . . = f r = 0 for a r−2 and f 1 = . . . = f r−1 = |f r | = s/2 for a = r − 1, r. The Kirillov-Reshetikhin characters are linear combinations of the above mentioned Weyl characters. The symmetric and spinorial characters in Kirillov-Reshetikhin representation are not different from the Weyl characters (1.4) but in other representations they do differ.
The generating function for characters in symmetric representations reads so that they coincide with standard Weyl characters (1.4). On the contrary, the KR-characters for totally antisymmetric representations already differ from usual D r Weyl characters (1.4). The generating function of KR characters for these representations reads where only the coefficients of t k , up to t r−2 term, give the KR type antisymmetric characters Ψ a=k (τ ).
In this work, we propose a new form of the QQ-system, appropriate for the D r algebra, and discuss the corresponding Hasse diagram. We will also introduce a new type of QQ ′ -type conditions (in terms of r + r single-index Q-functions) leading to the Weyl-type formulas for Tfunctions in the symmetric and antisymmetric representations of Kirillov-Reshetikhin modules: T-functions will be given in terms of ratios of determinants of the basic set of r single-index Q-functions generalizing the classical Weyl-type formulas (1.5) and (1.6). We will show the equivalence of these relations to the tableau-type formulas for T-functions. The QQ-relations and QQ ′ -type conditions were checked using explicit expressions for T and Q-operators found in [31] at small lengths of the spin chain.

Lax matrix construction and eigenvalues of T-operators
We start by introducing the the fundamental R-matrix of so(2r) which was written down in [60]. It is a matrix is of size (2r) 2 × (2r) 2 and it reads (2.1) Here κ = r − 1, the letter I denotes the identity matrix and the permutation and trace operator (P, Q) are defined as the tensor products The elementary 2r × 2r matrices E ij obey the standard relations E ij E kl = δ jk E il . We use the notation i ′ = 2r − i + 1. The R-matrix in (2.1) is related by a similarity transformation to the one originally obtained in [60], cf. [31], and generates the extended Yangian X(so(2r)) [61]. It is invariant under transformations

Transfer matrix construction for first fundamental
In the following we focus on spin chains of length N with the defining representation at each site. The quantum space of the spin chain is The R-matrix (2.1) allows to construct the fundamental transfer matrix T = T 1 , i.e. with the defining representation in auxiliary space, which contains the Hamiltonian of the spin chain. It is also convenient to introduce the symmetric generalisations T s at this point. The required Lax matrix was given in [62]. It reads Here we introduce the generators J ij of so(2r) obeying the commutation relations with J ij = −J j ′ i ′ . We stress that the formula for the Lax matrix only holds for symmetric representations with generators acting on the highest weight state |hws as follows where s ∈ N for finite dimensional representations. The generators in such representation satisfy the characteristic identity 2r j,k=1 which is needed in order to satisfy the Yang-Baxter equation, see also [63] for a recent discussion of such constraints. A realisation of the generators J ij for general s in terms of oscillators can be found in [31]. The defining representation s = 1 can be realised via We recover the R-matrix L(x) = R(x − κ 2 ). The first space in (2.5) with generators J ij serves as our auxiliary space and the quantum space is built from N copies of the second one with matrix elements E ij . The transfer matrix constructed from this monodromy is defined via where L i (x) denotes the Lax matrix acting non-trivially on the ith spin chain site and the trace is taken over the representation with generators J ij . We further introduced a diagonal twist with the parameters τ ∈ C r that we already encountered in the definition of characters. Some symmetries of the transfer matrix constructed via (2.11) can be found in Appendix B.1. The Hamiltonian of the spin chain is obtained from the fundamental transfer matrix T 1 by taking the logarithmic derivative at the permutation point The Hamiltonian density is obtained from the logarithmic derivative of the R-matrix at the permutation point and it reads (2.14) and D N the twist (2.12) at site N enters via H N,N +1 = D N H N,1 D −1 N . We also remind the reader that κ = r − 1.

Diagonalisation of fundamental transfer matrix
As discussed at the end of the previous section, the fundamental transfer matrix T = T 1 with s = 1 contains the nearest-neighbor Hamiltonian and higher local charges. It has been diagonalised in [62,64] using the algebraic Bethe ansatz, see also [65] for the trigonometric case. One of the key observations is that the transfer matrix can be written as where the two terms are related via We note that the twist only slightly modifies the derivation of the spectrum of the transfer matrix in [62,64]. Following the same logic as in the references above we find the contributions of T ± to the eigenvalues of the transfer matrix (2.17) with the notation q [k] ≡ q(x + k 2 ). In (2.17) above we introduced the Q-functions along the tail of the Dynkin diagram, cf. Figure 2. This equation is valid on the level of operators. In the diagonal form the Q-functions are written in terms of the Bethe roots x (j) i at level j ∈ {1, 2, . . . , r−2, +, −} corresponding to the nodes of the Dynkin diagram as given in Figure 2. The index i takes values i ∈ {1, 2, . . . , m j }. Here m j denotes the magnon numbers m = (m 1 , . . . , m r−2 , m + , m − ). They are determined for a given state labelled by weight vector n via where m 0 = N is the length of the spin chain, see [62] and n i = f i − f i+1 for 1 i < r and n r = f r−1 + f r . The first Q-functions along the tail of the Dynkin diagram are then given by Here q 0 does not depend on any Bethe roots and plays a similar role as the Q-functions for the full sets in A-type. The last two Q-functions factor into polynomials. We have where s ± are the Q-functions that correspond to the spinorial nodes. They are polynomials of degree m ± in the spectral parameter cf. [62]. As for the first fundamental representation of A-type, the energy eigenvalues only depend on the Bethe roots of the first nesting level.

QQ-relations from Bethe ansatz equations
The Bethe equations can be read off from the eigenvalue equation of the transfer matrix which is obtained by combining (2.15) and (2.17). When demanding that the transfer matrix is regular and Bethe roots are distinct the Bethe equations arise as pole cancellation conditions. They are conveniently written in terms of Q-functions as where (. . . ) k with 1 k r − 2 indicates that the expression is taken at a root of q k and (. . . ) ± at a root of s ± . From the first equations, along the tail of the Dynkin diagram, cf. Figure 2, we induce the standard A n type Plücker QQ-relation where q k and q k are two different Q-functions at the same level of the Hasse diagram, see Section 5 for that details. The Bethe ansatz equations can be restored by shifting its argument x → x ± 1, taking each of relations at q k = 0 and divide one by another. At the fork of the Dynkin diagram, (r − 2)th node, the QQ-relation takes the form At the spinorial nodes ±, the QQ-relations are These QQ-relations for spinorial nodes have appeared in [66] in relation to the ODE/IM correspodence [9] and recently in [67]. In Section 5 we propose a more general version of the QQ-relations.

Basic (extremal) Q-functions
A construction of the Q-operators corresponding to the extremal nodes of the Dynkin diagram, cf. Figure 2, was recently proposed in [31]. The latter construction was inspired by the isomorphism A 3 ∼ D 3 , admits the expected asympotic behavior (2. 18

Q-operator construction for 1st fundamental
We construct 2r Q-operators Q i with 1 i 2r corresponding to the first fundamental node. The Lax matrix needed is of the size 2r × 2r with oscillators as entries and its leading order in the spectral parameter is quadratic. It reads The Lax matrix above contains 2(r − 1) oscillators arranged into the vectorsw and w as follows They obey the standard commutation relations The matrix J is given in (B.2). The Q-operator Q 1 is defined as the regularised trace over the monodromy of the Lax matrices (4.1) which is constructed by taking the N -fold tensor product in the matrix space and multiplying in the auxiliary oscillator space: The twist matrix D in the auxiliary space depends on the parameters τ i , cf. (2.12) for the transfer matrix. In the case of the Q-operator Q 1 it reads with the number operator N i =ā i a i . The trace is defined aŝ By construction of the Q-operators Q 1 belongs to the family of commuting operators. From Q 1 we define the remaining 2r − 1 Q-operators at the first fundamental node. For that we introduce the transformatioñ with 1 i = j r. It belongs to the class of transformations discussed in (2.3) and commutes with the R-matrix. It follows that the Q-operators defined via and also belong to the family of commuting operators. This defines us 2r Q-operators Up to the exponential prefactor, we identify the q-function q 1 with the eigenvalues of the Qoperator Q 1 . Here we could have chosen any other single-index Q.

Q-operator construction for spinor representations
Similarly we proceed for the Q-operators corresponding to the spinorial nodes ± of the Dynkin diagram in Figure 2. Here the Lax matrix is a 2 × 2 block matrix with block size r × r. It readš and contains r(r−1) The submatricesĀ and A are of the form (4.12) Similar as before we define the Q-operator as the trace of the monodromy built out of the Lax matrixĽ above as (4.13) Here we introduced the twist in the auxiliary space viǎ The remaining Q-operators at the spinorial nodes are obtained through the similarity transformation with α i = ±1, that commutes with the R-matrix, cf. (2.3), and subsequently inverting the twist parameters. For α i = 1 the matrix B( α) reduces to the identity. We define labelled by α = (α 1 , . . . , α r ) with α i = ±1. By construction the 2 r operators S α commute with one another. We choose to identify s ± with S (+1,...,+1,±1) .

The QQ-system for D-type
In this section we introduce the QQ-system. It has been verified at small finite length using the construction [31] that was reviewed in Section 4. In total we have 3 r − 2 r−1 r + 2 Q-functions, see Figure 4 and Figure 5 for r = 3, 4 examples. The QQ-relations along the tail of the Dynkin diagram have a similar structure as in A-type but the labeling of single-index functions is different. We shall say that a subset I of {1, . . . , 2r} is acceptable if for all 1 k r, the integers k and k ′ = 2r − k + 1 do not both belong to I. A Q-function Q I is associated to each acceptable I and these functions satisfy the relations J is acceptable of order at most r − 2 and does not contain i, i ′ , j or j ′ . We have excluded here the case where k and k ′ are contained in the same set as the Q-functions defined this way would not have the expected asymptotic behavior. For the D r spin chains under the consideration, the Q-operator of the empty set can be conveniently fixed as though such a choice for a generic D r QQ-system can be changed by a gauge transformation, see below in this section.
As discussed at the end of the Section 2, the Q-operators Q I with |I| = r − 1 or |I| = r factorise into the spinorial Q-functions. More precisely, and The set notation for the Q-operators S I can be mapped to the notation S α in the previous subsection using α as follows: to an acceptable set I of order r we associate α such that, for 1 i r, We thus obtain a one-to-one correspondence between S {i 1 ,...,ir} and S α as defined in (4.16). We further remark that the polynomial structure of the spinorial Q-functions allows to determine them from the quadratic relations (5.3) and (5.4). The QQ-relation (5.1) can be summarised in a Hasse diagram which reminds that of the A r case. The latter is exemplified for D 3 in Figure 3 and for D 4 in Figure 5. However, the last two levels are nontrivial: the level |I| = (r − 1) factorises according to (5.3) It follows that they must belong to the same node of the Dynkin diagram. Among them we have the QQ-relations which relate the spinorial Q-functions to the last Q-functions on the tail of the Dynkin diagram, i.e. at the r − 2'th node (|I ∩ J| = r − 2). Notice that for each level r − 2 Q-function there are two ways to obtain them from spinorial Q-functions, e.g.: when r = 4, I ∩ J = {1, 3} can come from I = {1, 3, 2, 4} and J = {1, 3, 7, 5} or from I = {1, 3, 2, 5} and J = {1, 3, 7, 4}. This relation allows us to resolve the last two levels in the D r Hasse diagram, cf. Figure 4 and Figure 5 on page 14 for the cases D 3 and D 4 , respectively. Let us note that the former is isomorphic to A 3 Hasse diagram of Figure 3, since the two algebras are isomorphic, whereas the Hasse diagram Figure 5 for D 4 is new and it can be easily generalized to any D r . Here we used the directions of the arrows in the Hasse diagram to distinguish from the QQ-relations (5.1) used for the last nodes as depicted in Figure 3. Using the QQ-relations in (5.1) we can express all Q-functions Q I along the tail in terms of Casoratian determinants of the fundamental Q-functions. We find Similar formulas exist for spinorial type: if I is an acceptable set of order k r − 2 and i k+1 , . . . , i r are such that I r = I ∪ {i k+1 , . . . , i r } is acceptable of order r then one has Gauge transformation The QQ-system as written above corresponds to a particular choice of gauge. In order to describe this gauge freedom, we draw inspiration from the r = 3 case, see Appendix E. One needs to introduce two new Q-functions S ±,∅ , (5.1) and (5.3) remain unchanged while (5.4) and (5.8) become with α associated to I according to (5.5). These QQ-relations remain unchanged if one applies the gauge transformation, depending on three arbitrary functions g, g + and g − , given by for I acceptable. In this paper we work in the "spin chain" gauge

Transfer matrix in terms of fundamental Q's
In Section 2.2 we gave the transfer matrix in terms of one single Q-function for each nesting level. We can use the Casoratian formula (5.9) to express the transfer matrix only in terms of Q ∅ and a half the number of fundamental Q-functions Q {i} . We will show in this section that the transfer matrix is then given by . This formula fulfills, at least for the fundamental T-function, one of the main purposes of our paper -to derive the Weyl-type expressions for the transfer matrices of spin chains based on D r algebra, "quantizing" in this way the classical Weyl character determinant formula. The latter one can be restored in the classical limit x → ∞. In that limit Q j ∼ x→∞ τ x j x m j while the fundamental T behaves as x 2N r j=1 (τ j + 1 τ j ).

Induction
We can prove the formula (6.1) by expressing the transfer matrix in terms of the first r fundamental Q-functions, as in (5.9), and inserting it into (2.17). We obtain The desired expression (6.1) for the transfer matrix (in the case i a = a) follows from (6.2) using the identity which can be shown by induction on r. It obviously holds true for r = 1. It remains to show that , (6.4) or equivalently (assuming the determinants are non-vanishing) The latter identity can be proven as follows: one first expands each of the (r + 1) × (r + 1) determinants with respect to their last row (i.e. the one involving Q {r+1} ). Both sides become for 1 j r+2 and one just has to check that the coefficients on each side are the same. For j ∈ {1, r+1, r+2} this is completely trivial whereas for j ∈ {2, . . . , r} this becomes 3 where C j is the transpose of the vector Q  N l , M k+1 , . . . , M r ||N 1 , . . . , N l−1 , M k , N l+1 , . . . , N r | . (6.7) In our case M = (C 1 , . . . , C j−1 , C j+1 , . . . , C r , C r+2 ) and N = (C 2 , . . . , C r+1 ) have many columns in common so that if we decide to exchange M r = C r+2 only two terms survive in the sum (when l = j − 1 or l = r) and they give exactly what we want.

Reshuffling Q-functions in the transfer matrix
Here we show that the expression for the transfer matrix (6.1) in terms of r fundamental Qfunctions is invariant under the replacement Q ia → Q i ′ a for any a. By obvious symmetry w.r.t. permutations of the functions Q i , i ∈ {1, 2, . . . , r} it suffices to show that the transfer matrix is invariant under Q ir → Q i ′ r . This is the case if Using the Jacobi identity on determinants, one can rewrite the numerator and the denominator in the previous condition aš

Bethe ansatz equations of Wronskian type
Here we present a Wronskian relation on the basis of r + 1 single-index Q-functions which can serve for finding the Bethe roots and, eventually, the energy of state. We can call it the Wronskian BAE, in the analogy to the very useful Wronskian BAE for the A r type Heisenberg XXX spin chain which has the form where Q-functions have the form (1.2). See [40,68,69], where this method of finding Bethe equations was proposed and applied.
A similar relation in D-type spin chain is non as simple. In the following we propose to use for this purpose the equation (6.14) which in the form of Wronskians reads where i a ∈ {1, . . . , 2r} and i a = i ′ b . In total there are 2 r−1 r such equations but a single one provides the needed Bethe roots. Our Q-operators are polynomials (up to exponential twist factor) with the known leading order asymptotics. For a given state it is fixed by the global charges of this state. With the notation for r global charges as Inserting, for example, r + 1 first Q-functions into (7.2) we obtain the polynomial equations on the coefficients of these Q-functions (Bethe roots) at each order of the spectral parameter. Let us look at the example of N = 1 and r = 4 with Cartan eigenvalues f = (1, 0, 0, 0). The polynomial structure of the Q-functions is then of the form We checked that the coefficients obtained from the Q-operator construction yield indeed a solution of (7.2). Vice versa, the equation (7.2) can be used to find the Bethe roots and fix the single-index Q-functions for each state. Since (7.2) follows from the first r equations of standard BAE (3.2) it can serve as an alternative to them.
But the situation with D-type algebra is more complicated than with A-type algebras. Remarkably, the solutions of (7.2) for the coefficients c i,j , and thus for the Bethe roots, should always exist in spite of the fact that the set of equations on coefficients is over-defined: Generally, if we choose there a i ≡ i we have 2 r−1 i=1 J i + 2N equations on just r−1 i=1 J i + 2N coefficients. On the other hand, the excessive power of equation (7.2) leads to the existence of non-physical, "parasite" solutions apart from the physical ones. To exclude them we have to find a good selection rule.
Once we find the physical Q {1} function for a given state, we can use it to find the energy through the formula (2.23). We can also use in this formula for energy any other function Q {i} from this solution, with the same result.

QQ ′ -type formulas for T-functions
In this section we present QQ ′ -formulas for the symmetric and spinorial T-operators. The reasoning behind our rather heuristic derivation is in analogy to [29] where the BGG resolution [70] was used. Here we give arguments on the level of characters, see also [33], which we take as hints to obtain the actual BGG-type relation for the fundamental and spinorial transfer matrices. The final formulas have been checked in several examples for small finite lengths. Further we provide a consistency check. Namely, we recover the Weyl-type expression for the fundamental transfer matrix (6.1) by reducing there the number of used single-index Q-functions from 2r to r. In the final subsection we introduce the Hirota equation and solve it using the QQ ′ -formulas for symmetric T-operators. This yields QQ ′ -type formulas for any rectangular transfer matrix T a,s .

Symmetric transfer matrices
In [31] it was argued that the product of Lax matrices can be brought to the form where the Lax operator L i are defined via L i (x) =B 1,i L(x)B 1,i for i = 1, . . . , r and L i (x) = JL i ′ (x)J for i = r + 1, . . . , 2r, see Section 4.1. The superscripts (1, 2) indicate two different families of oscillators. The letter S i denotes a similarity transformation in the oscillators space and G i a dummy matrix that does not depend on the spectral parameter and commutes with the Lax matrix L + i (x). Their precise form is given in [31]. We identify the Lax matrix L + i (x) as a realisation of (2.5). The parameter x i then plays the role of the representation label. We stress that the term linear in the spectral parameter is given by the generators J ij , cf. (2.5). In the case (8.1), the representation of so(2r) is infinite-dimensional in the oscillators space and becomes reducible for certain values of the parameter x i . The infinite-dimensional representation of so(2r) is characterised by its character. For example for i = 1 the Cartan elements are of the form The character can then be computed via . (8.4) We find similar formulas for the product of Lax matrices The finite dimensional characters are related to the one above by the sum formula (8.6) From our results for finite length and the discussion above we find that the formula can be lifted to transfer matrices and Q-operators. It reads (8.7) Notice that in the limit x → ∞ (8.7) becomes (8.6), as it should be.

Spinorial transfer matries
A similar factorisation formula as (8.1) exists for the spinorial Lax matrices (4.11). It readš Here we definedĽ α (x) = B( α)Ľ(x)B( α) and use a notation similar as above in (8.1). The similarity transformationŠ α only depends on the oscillators andǦ α is a matrix that is independent of the spectral parameter and commutes with the Lax matrixĽ + α . The latter denotes an infinite-dimensional realisation of the spinorial Lax matrix (8.9) where J ij denote the generators of a spinorial representation. Again the parameter x 1 in (8.8) has the role of the representation label. As before we compute the character of the oscillator representation. In the case α = (+, . . . , +) we find The general formula can be obtained using the relations among the spinorial Lax matrices as presented in Section 4.2. We get The characters of the finite-dimensional spinor representations ± with f = (s, . . . , s, ±s) can then be written as Here the sum is taken over all configurations { α} ± such that i α = ±1.
On the level of monodromies, we propose the formula (8.14) This formula has been verified for small finite lengths by comparing to the transfer matrices directly constructed within the quantum inverse scattering method using the Lax matrices in (8.9) for finite-dimensional spinor representations.

Derivation of Weyl-type formula for T 1,1 from QQ ′ -relations
Let us write (8.7) as where h i = r j( =i,i ′ ) (u i −u j ) −1 and u j = τ j +1/τ j . We further assume that when s ∈ {1−r, . . . , 0} the identity is still verified if one sets and T 1,s = 0 for 1 − r s −1 .
(8. 16) We show here that the conditions (8.15) and (8.16) are enough to recover the expression (6.1) giving T 1,1 in terms of only r of the single-index Q-functions, and so are consistent with it. We also show in Appendix F how to retrieve the Wronskian equation (7.2) from these conditions. One simply has to notice that (8.15) implies that there exist some Q-dependent coefficients C j,k ′ ,k (defined for 0 k ′ k r and 0 j k − k ′ ) such that 17) It suffices indeed to expand the determinants with respect to their last row and perform the sum over i. One has for instance In particular, plugging the constraints (8.16) in the previous relation when k = r gives us we recover (6.1) in the case i a = a. Notice that with this derivation, the symmetry under Q {i} ↔ Q {i ′ } is immediate because the equations we started from were already symmetric.

Transfer matrices for general rectangular representations
In this section we propose relatively simple formulas for T-functions in rectangular representations in terms of bi-linear expressions involving Wronskians of both types of single-index Q-functions, Q i and Q i ′ , where i = 1, 2, . . . , r. These formulas follow from (8.7) when solving the Hirota equations satisfied by the T-functions. These equations read as follows (s ∈ N * ): which can be written in the same form as the previous equation if one sets T r−1,s = T +,s T −,s , and T [+1] ±,s T The boundary conditions are (0 a r − 2, s ∈ N) and We shall determine here the QQ ′ -type relations for T a,s for 1 a r − 1, but not for T ±,s . For these spinorial transfer matrices, the spinorial Q-functions seem more suitable, see equation (8.14). We start from which can be also written, if the transfer matrices satisfy the Hirota equation (8.21) with boundary conditions (8.24), as follows Putting the two expressions together yields the following expression for the second row of transfer matrices: This procedure can be continued for 1 a r − 1. This yields the natural generalisation We have checked this for a 3. The general proof boils down to verifying the relation

Weyl-type formulas for T-functions from tableaux representations
The tableaux sum formulas of [46] yield the transfer matrices of any rectangular representation T a,s expressed through the single terms in the sum of the transfer matrix (2.15) as given in (2.17). In total there are 2r different terms (boxes), r for T + and r for T − . Instead of using the summands in the form (2.17) with Q's of different levels as building blocks (boxes), we use these summands in the form given in (6.2). The latter only contain r single-index Q-functions and Q ∅ and yield new expressions for totally symmetric T 1,s and totally antisymmetric T a,1 T-functions. We start from the expressions (6.2) for T ± such that where b k,r denotes a box as given in [46] for D r with index k. The expression above, in the character limit x → ∞, allows to identify b k,r in terms of the single-index Q-functions. We get for r + 1 k 2r, and we recall that k ′ = 2r − k + 1.
The simplest examples of the tableaux sum formulas beyond T 1,1 are for T 1,2 and T 2,1 . They arise when writing and identifying the terms in the brackets with T 1,0 T 1,2 and T 0,1 T 2,1 from the Hirota equation (8.21), so that and As we see, it is independent of the actual representation of the box terms b k,r . As we will see in the following, substituting (9.2) and (9.3) will yield new expressions for the transfer matrices that only depend on r single-index Q-functions and Q ∅ .

Symmetric representations
The transfer matrices for generic symmetric representations are given by [46] T 1,s = 1 where the symbol ′ stands for a sum in which we do not allow for r and r + 1 to appear at the same time. The denominator appears as a consequence of our boundary conditions for the Hirota equation.

General symmetric sum
Let us defineb For l 1, one has The two identities are equivalent, so it is enough to prove the first one. We do it by induction in r. It is trivial when r = 1. If it is true for some r 0 1 then let us show by induction on l that it is also true for r 0 + 1: the case l = 1 has been proven earlier in Section 6.1 so we assume that the identity holds for some l 0 1. We then write Since we have assumed that the identity holds for r 0 and any l, for (r 0 + 1, l 0 ) we can write . (9.14) Consequently, for (9.11) to hold for (r 0 + 1, l 0 + 1), one only has to show that . (9.15) This last relation can be proven in much the same way as (6.5) which corresponds to the case l 0 = 0.

Application to computation of transfer matrices
To use efficiently the summation formulas (9.11) and (9.12) we first rewrite equation (9.7) as This gives in virtue of (9.11) and (9.12) The terms for 1 l s − 1 of each sum can be combined thanks to a Plücker identity, to give an explicit and concise Weyl-type representation of symmetric T-functions for D r algebra

Antisymmetric representations
The transfer matrices for generic antisymmetric representations are given by [46] T a,1 = 1 j l ,r . (9.19) For 1 l r, one has {i} r (9.20) and where we used the Heaviside function θ that is 0 for negative arguments and 1 for non-negative ones. These formulas can be proven in much the same way as (9.11) and (9.12). They allow us to write For instance, when a = 2 it reads . (9.23) In principle, one can now generate, from (9.22) above, all transfer matrices or rectangular representations using Cherednik-Bazhanov-Reshetikhin type formulas written for D r symmetry in [42]. The formulas (9.18) for symmetric transfer matrices look much simpler than for the antisymmetric ones. It would be good to have the CBR type representation involving them for D r algebra, but it does not seem to exist.

Discussion
In this work, we proposed the full system of Baxter Q-functions -the QQ-system -for the spin chains with SO(2r) symmetry. This QQ-system is described by a novel type of Hasse diagram presented for various ranks on Figures 4, 5 and 6. We also found Weyl-type formulas for transfer matrices (T-functions) of symmetric and antisymmetric representations in terms of sums of ratios of determinants of r basic, single-index Q-functions. We also proposed QQ ′ -type formulas expressing the T-functions through 2r basic single-index Q-functions. These could be a powerful tool for the study of spin chains and sigma models with D r symmetry. We also reformulated the Bethe ansatz equations in the form of a single Wronskian relation on r + 1 basic single-index Q-functions. It is the analogue of a similar Wronskian relation for spin chains with A r symmetry. However, apart from the Bethe roots our equation contain extra solutions whose role has yet to be clarified.
Our main assumptions in this article are the Plücker QQ-relations (5.1) and (5.8), as well as the QQ ′ -relations (8.7) and (8.14). The QQ-relations are motivated by the asymptotics of the Q-operators and the QQ ′ -type relations by the factorisation formulas for the Lax matrices for Q-operators and the corresponding character formulas, as discussed in Section 8.1 and 8.2. Both relations remain to be proven but we have tested them explicitly for several examples of small finite length T and Q-operators. The QQ-relations (5.1) allow to express the fundamental transfer matrix, for which an expression in terms of one Q-function of each level is known from the algebraic Bethe ansatz, in terms of r single-index Q-functions and Q ∅ , cf. (6.1). This Weyl-type expression has been independently obtained from the QQ ′ -type relations (8.7), see Section 8.3. We take this as a consistency check. The new formulas for T a,s are obtained from the Hirota equation and the tableaux formulas of [46]. They can be seen as consequences of the Weyl-type formula for the fundamental transfer matrix in terms of r + 1 Q-functions and the QQ ′ -type relations.
Unlike the well understood QQ-system of A-type, in the D-type QQ-system there are still many questions left and issues to be clarified. The questions exists already on the operator level: the Yangian for SO(2r) spin chains is constructed only for "rectangular" representations and the R-matrix -the main building block for Q and T-functions is known only for the symmetric and spinorial representations [31,62,71]. A full classification of Lax matrices including the ones for the Q-operators was recently given in [72] for A-type. This may shed some light upon the transfer matrices for general rectangular representations and beyond. It would be interesting, using the tableaux formulas (which look quite involved [73]) to find the Weyl-type determinant formulas for the arbitrary rectangular representations, generalizing our formulas for symmetric and antisymmetric representations. These could also be interesting for the study of the Q-system and its relation to cluster algebras, see e.g. [74,75]. Moreover, a solid proof of our QQ-system and our Wronskian formulas for T 's in symmetric and antisymmetric representations is yet to be found on both the analytic and operator level. It may be possible using the BGG resolution or the analogue of coderivative method proposed in [32] and used for this purpose in [33], see also [76] for a review. Unfortunately, we do not know yet a suitable analogue of Baxter TQ equations (quantum spectral determinant) which appeared to be so useful for the spin chains with A r symmetry [1,2], see [40] for the modern description in terms of forms as well as [77] in terms of the quantum determinant. It is possible that the QQ ′ -type formulas for transfer matrices proposed in this work can replace the Baxter equations for D r algebra.
It would be interesting to generalize our approach to the study of spin chains with open boundary conditions and to the non-compact, highest weight and principal series representations of D algebras. For a much better studied case of these aspects in A-type integrable system we refer the reader to [50,[78][79][80][81]. One encounters non-compact representations in sigma models [82] and spin chains [83] with principal series representations of the d-dimensional conformal group SO(2, d) [83]. They recently appeared in the study of d-dimensional fishnet CFT [68,84,85] and the associated planar graphs (of the shape of regular 2-dimensional lattice) [86]. Whether as for the lowest, 4-dimensional conformal SO(2, 4) symmetry one can use its isomorphism to the A type SU (2, 2) group to construct the suitable QQ-system and Baxter TQ system for the efficient study of Fishnet CFT [87,88], for d > 4 we have to find an alternative approach which can be based on the D-type QQ-system constructed in this work. The structure of the QQ-system does not depend on the choice of real section of the orthogonal group but the Wronskian, Weyl-type formulas for T do depend. So one could try to construct the quantum spectral curve (QSC formalism for d > 4 fishnet CFT in analogy to the d = 4 case [89]. Finally, we hope that our methods can be generalized to B, C and exceptional types of algebras and their deformations, as well as to superalgebras such as osp(m|2n) where the QQsystem and T-functions are yet to be constructed. A first step could be the evaluation of the oscillators type Lax matrices for Q-operators using the results of [90,91] as done in [92] for type A.
We further note that when defining reflection matrix the twist parameters of the transfer matrix exchange:

B.2 Crossing symmetry of single-index Q-operators
In this appendix we discuss the derivation of the crossing relation for the single-index Q-operators. The corresponding Lax matrices satisfy we find that the normalised trace is independent of particle hole transformation. The extra factor above drops. We obtain Q t 1 (−x)

E QQ-system of A 3 ≃ D 3
We show in this appendix that, as is expected from the isomorphism A 3 ≃ D 3 , the known QQsystem for A 3 can be interpreted as the QQ-system for D 3 , albeit in a particular gauge. We start with a reminder of the QQ-system for A 3 : in order to avoid confusion, we shall denote Q I for I ⊂ {1, 2, 3, 4} the Q-functions for A 3 and the SL(4) twists will be z 1 , z 2 , z 3 and z 4 such that z 1 z 2 z 3 z 4 = 1. The following relations hold (neither i nor j belongs to I): I∪{i} Q Both of these equations are identified with equation (5.11). More generally, both QQ-systems are the same if one makes the following identification between the two sets of Q-functions: The previous equation shows that in identifying the two QQ-systems we had to partly fix the gauge for D 3 . This explains why in the A 3 QQ-system there are only two gauge degrees of freedom [40] while there are three of them for the D 3 one. where we only assume that for all 1 a = b r one has {i a , i ′ a } ∩ {i b , i ′ b } = ∅. This is exactly equation (7.2).