Tug-the-hook symmetry for quantum 6j-symbols

We introduce a novel symmetry for quantum 6j-symbols, which we call the tug-the-hook symmetry. Unlike other known symmetries, it is applicable for any representations, including ones with multiplicities. We provide several evidences in favour of the tug-the-hook symmetry. First, this symmetry follows from the eigenvalue conjecture. Second, it is shown by several new examples of explicit coincidence of 6j-symbols with multiplicities. Third, the tug-the-hook symmetry for Wilson loops for knots in the 3d Chern-Simons theory implies the tug-the-hook symmetry for quantum 6j-symbols. An important implication of the analysis is the generalization of the tug-the-hook symmetry for the Chern-Simons Wilson loops to the case of links.

Most above mentioned physical examples involve irreducible finite-dimensional representations of classical Lie groups and algebras which correspond to the classical case q = 1. However, the eigenvalue hypothesis can be formulated only for the quantum case. In the classical case R-matrices are reduced to permutation matrices which permute two highest weight vectors and have unique eigenvalues equal to ±1. But it does not mean that corresponding Racah matrices for all representations are equal. Meanwhile, if for generic q from the eigenvalue hypothesis, we obtain the equality of Racah matrices, this equality still holds in particular case of q = 1. This fact allows one to get results for the classical case, which is of great importance in physics.
The eigenvalue conjecture has been proved for U q (sl 2 ) case in [26]. Although there is no proof for the rank N > 2, the hypothesis was checked in some cases. In the multiplicity-free case, there are exact expressions for Racah coefficients through R-matrix eigenvalues in the case of all equal incoming representations for matrices of the size up to 5 × 5 [24] and 6 × 6 [27] for three strands, and in the case when all incoming represenations can be different -for matrices of the size up to 3 × 3 for three strands [25] and for matrices of the size up to 5 × 5 for four strands [28]. The situation becomes much more complicated when multiplicities occur, but even in this case when Racah matrices can be made block-diagonal, these blocks should satisfy the eigenvalue hypothesis [29].
The eigenvalue hypothesis has lots of important applications. The one we use in our consideration is its ability to predict classes of symmetries of Racah coefficients [30] and quantum knot invariants. However, explicit symmetries of 6j-symbols have been provided only for symmetric and their conjugate representations [26,31]. Another interesting implication is its connection [32] with the one-hook scaling property of the colored Alexander polynomial, which relates these polynomials for single-hook representations and the fundamental one. This symmetry unexpectedly connects quantum knot invariants with the Kadomtsev-Petviashvili integrable hierarchy [33].
In this paper, we introduce a new symmetry for quantum 6j-symbols (for other currently known symmetries see Subsection 2.2). From the one hand, relying on the eigenvalue conjecture, we derive the tug-the-hook symmetry for Racah coefficients (see Section 3). This symmetry is proved in the above mentioned cases where the eigenvalue hypothesis is confirmed (see also Subsection 4.1). From the other hand, we present examples of coincidence of Racah matrices connected by the tug-the-hook transformation (see Subsection 4.3). Moreover, the tug-the-hook symmetry has been proved for the colored HOMFLY polynomials (which are Wilson loops for the 3d Chern-Simons theory) for knots [34]. Due to the fact that the HOMFLY polynomials are particularly constructed from Racah matrices, this symmetry should also hold for Racah coefficients (see Subsection 4.2). This independent evidence for the tugthe-hook symmetry for Racah coefficients also indirectly confirms the eigenvalue hypothesis. Another important implication we provide is the tug-the-hook symmetry for the HOMFLY polynomials for links (see Subsection 4.2).

Preliminaries
In this section, we discuss basic definitions and properties of objects we use in our analysis -quantum 6j-symbols (Subsection 2.1), R-matrices (Subsections 2.13 and 2.4) and the colored HOMFLY polynomials (Subsection 2.6). We also discuss the eigenvalue hypothesis and its origins in Subsection 2.5.

Quantum Racah coefficients and 6j-symbols
In this subsection, we introduce quantum Racah coefficient and its normalization -6j-symbol. In conformal field theories, Racah matrices are called crossing or fusion matrices and relate two types of conformal blocks coming from two different orders of operator product expansion in four point correlation function of primary fields.
Consider three finite-dimensional irreducible representations V R1 , V R2 , V R3 of the quantized universal enveloping algebra U q (sl N ). We assume that q is a nonzero complex number which is not a root of unity, thus, all finitedimensional representations are representations of highest weights, and they can be enumerated by Young diagrams. Recall that a Young diagram µ = {µ 1 ≥ µ 2 ≥ · · · ≥ µ l } is constructed from highest weights {ω 1 , ω 2 , . . . , ω l } of a representation V µ : µ i = l k=i ω k ∀ i = 1, . . . , l, and vice versa ω i = µ i − µ i+1 . Since the tensor product V R1 ⊗ V R2 ⊗ V R3 is associative, there is a natural isomorphism: (2.1) One can expand tensor products of two representations into a direct sum of irreducible components: Here M R1,R2

R23
are the subspaces of highest weight vectors with highest weights corresponding to the Young diagrams R 12 ⊢ |R 1 | + |R 2 | and R 23 ⊢ |R 2 | + |R 3 | respectively. The dimensions of the spaces M R1,R2 R12 and M R2,R3 R23 are called multiplicities of representations V R12 and V R23 respectively. Expand the second tensor products in (2.1): Then, the associativity condition (2.1) implies the following definitions.
Racah matrix is the map Wigner 6j-symbol is a normalized Racah coefficient: where qdim(R) denotes the quantum dimension of representation V R .
Racah coefficients are invariant under the transposition of all representations: where ′ denotes transposition of a Young diagram.

Currently known symmetries of quantum 6j-symbols
In this subsection we briefly review currently known symmetries of quantum 6j-symbols. 1. Quantum 6j-symbol is invariant under cyclic permutation of columns and under exchange of columns (with the corresponding conjugation of representations) up to a sign: where R is a representation conjugate to R.
3. Complex conjugation of quantum 6j-symbol is equivalent to conjugation of all representations: 4. There is the following unitarity property: (2.10) 5. The generalized Racah backcoupling rule: 2 is sl N quadratic Casimir for the representation R and signs in the sum depend on all representations R 1,2,3 , r 1,2,3 and R.

R-matrix
In this subsection, we define the second important object of our study -the so-called quantum R-matrix. In WZW conformal field theory R-matrices correspond to half-monodromy operators acting on primary fields and are usually called braiding operators.

R-matrices are invertible linear operators defined by
where P (x ⊗ y) = y ⊗ x andŘ is the universal R-matrix : Here exp q is the quantum exponent, C nm is the Cartan matrix and from (2.13) acts on V Ri ⊗V Ri+1 and generators are taken in the corresponding representations V Ri and V Ri+1 .
It is well-known that R i , i = 1, . . . , m, define a representation of the Artin's braid group B m on m strands: where σ 1 , . . . , σ m−1 are generators of the braid group B m . The eigenvalues of the universal R-matrixŘ i,i+1 are well-known [11,44]: Here index i enumerates representations, and V Ri,i+1 are irreducible components of the tensor product V Ri and V Ri+1 : where we allow repeated summands; κ(R) is defined by the following formula: and ǫ Ri,i+1 = ±1 is a sign. In the case when all representations are the same (R i = R), sings of the eigenvalues depend on whether highest weight vectors of the representations are symmetric or antisymmetric under permutation of two representations V Ri and V Ri+1 . These two types of representations V Ri,i+1 are said to belong to either symmetric, or antisymmetric squares of the representation V R .

R-matrices via Racah matrices
As we know the R-matrix eigenvalues (2.16), we would like to diagonalize all R-matrices. This can be done with the use of Racah matrices. Consider m representations V Ri and choose the basis in V R1 ⊗ · · · ⊗ V Rm in which the matrix R 1 gets a block form. Such basis corresponds to the following order in the tensor product: If we additionally rotate the components corresponding to each R 12 , we can diagonalize the matrix R 1 .
The same procedure can be made in the basis corresponding to Therefore, in order to diagonalize the matrix R 2 one should make the basis transformation with the help of Racah matrix: where Λ R2 is a diagonal matrix with eigenvalues of R 2 . The similar procedure can be made with all R-matrices.
While eigenvalues for all R-matrices can be computed explicitly (2.16), calculation of Racah coefficients is a puzzling procedure, that was done analytically only for U q (sl 2 ) case.

Eigenvalue hypothesis [24]
In this subsection, we introduce the eigenvalue hypothesis. Let us, first, describe some intuition on how the hypothesis can be formulated. The conjecture originates from the Yang-Baxter equation Consider the case when Racah matrix act in the tensor cube of a representation V R of U q (sl N ): Let us choose the basis where R 1 is diagonal. Then, following the pocedure described in Subsection 2.4, one can diagonalize R 2 using Racah matrix: R 2 = U † R 1 U , and the Yang-Baxter equation (2.23) takes the form is uniquely expressed through a set of normalized R-matrix eigenvalues. The eigenvalue hypothesis also can be formulated in the case of three different irreducible U q (sl N ) representations as follows.
As a straightforward corollary, we get that if one finds a transformation of representations which does not change quantum R-matrices eigenvalues, it correspond to a symmetry of quantum Racah coefficients with respect to the same transformation of representations.

Colored HOMFLY polynomial
In this subsection, we define the colored HOMFLY polynomial with the use of Reshetikhin-Turaev approach [13].
Alexander theorem. Any link L in R 3 can be obtained by a closure of the corresponding braid.
If a link L has one component, it is called a knot and is usually denoted K. Each link component must carry its own representation. In particular, for a knot each strand of the corresponding braid carries one and the same representation.
Let L be an oriented link with L components K 1 , . . . , K L colored by irreducible finite-dimensional representations V R1 , . . . , V RL of U q (sl N ), and β L ∈ B m is some m-strand braid which closure gives L.
The colored HOMFLY polynomial is a quantum group invariant of a link L defined as follows 1 :

26)
where q tr is the quantum trace.
One can expand the quantum trace in (2.26): It was a breakthrough when the connection with topological field theory was established [46]. Namely, the colored HOMFLY polynomial turned out to be the Wilson loop in 3d Chern-Simons theory with SU (N ) gauge group defined by representation R and knot K : where Pexp denotes a path-ordered exponential, 1 qdim(R) is the normalization factor and the Chern-Simons action is given by For links, the colored HOMFLY polynomial is defined by several Wilson loop operators, each one is associated with separate link component and carrying its own representation.

Tug-the-hook symmetry for Racah coefficients
Symmetries of 6j-symbols have been completely described for U q (sl 2 ) case only [23]. The corresponding symmetry group is S 4 × S 3 , where S 4 is responsible for the tetrahedral symmetry, and S 3 stands for the Regge symmetry. For U q (sl N ), N > 2, the full symmetry group is unknown. However, there is a work [30] on the possibility of construction of new symmetries with the use of the eigenvalue hypothesis, but explicit symmetries were provided only for the cases of symmetric and their conjugate representations in [26,31].
In this section, we introduce the tug-the-hook symmetry formulated in terms of Young diagrams transformation which leaves Racah coefficients invariant. This symmetry has a significant property. It transforms any Young diagram which can be placed inside a hook, including cases with multiplicities. This is the first found symmetry of Racah matrices which is valid beyond multiplicity-free cases.
Let us introduce the tug-the-hook transformation [47] in terms of Young diagrams. A Young diagram is placed inside an appropriate (K + M |M ) fat hook for some integers K and M . Introduce an analogue of Frobenius notations: parametrize the first K rows by their length R i , i = 1, . . . , K, the rest rows are parametrized by shifted . . , K +M . The tug-the-hook transformation T (K+M|M) ǫ pulls the Young diagram inside the fat hook: where ǫ is the corresponding shift of the diagram. A shift ǫ can be negative, what corresponds to the inverse shift. Note that this transformation is defined when the resulting shifted Frobenius variables form a Young diagram. An example is shown in Fig. 1. from the left diagram to the right one [47].
We put forward the following conjecture.

Given arbitrary irreducible finite-dimensional representations
for generic q, Racah coefficients are invariant under the tug-the-hook transformation: for K, M and integer shifts ǫ 1 , ǫ 2 , ǫ 3 for which the tug-the-hook transformation is defined.
This hypothesis follows from the eigenvalue conjecture. The proof consists of the following steps.
1. sl N Casimir invariants are invariant under the tug-the-hook transformation what was proved in our recent paper [34].
2. R-matrices eigenvalues (2.16) can be expressed through sl N quadratic Casimir invariants C 2 [11,42]: and signs ε Q for all Q are multiplied by one and the same factor ±1 under the tug-the-hook transformation, what does not change the normalized eigenvalues. Thus, normalized R-matrices eigenvalues are also invariant under the tug-the-hook transformation.
3. Thus, the eigenvalue hypothesis implies that Racah coefficients possess the tug-the-hook symmetry.
Emphasise several important peculiarities of the tug-the-hook symmetry for quantum Racah coefficients.

Remark 1.
Due to the fact that Racah coefficients do not depend on K and M , the tug-the-hook symmetry holds for any hook which the corresponding Young diagrams can fit.
Remark 2. The tug-the-hook symmetry was first observed [47] and then proved [34] for the colored HOMFLY polynomials for knots. In that case, the symmetry explicitly depends on sl N +1 algebra rank N because it acts only inside fat hooks (N + M |M ). This fact also means that lenghts of Young diagrams must be larger than N , and the tug-the-hook symmetry for the HOMFLY polynomials has sl N +M|M supergroup origins, where it is well-defined for its finite-dimensional representations. Unlike the HOMFLY case, the tug-the-hook symmetry for Racah coefficients is independent on N and holds for corresponding representations of any sl N +1 algebra.
Remark 3. The tug-the-hook symmetry is the first found symmetry of Racah matrices which acts on any representations and works for cases with multiplicities.
4 Evidence for the tug-the-hook symmetry In this section, we provide facts and examples of the validity of conjecture that quantum 6j-symbols possess the tug-the-hook symmetry. Unlike the example of the previous section, here we consider cases with multiplicities where the eigenvalue conjecture is not proved. Still, in these cases the tug-the-hook symmetry for Racah coefficients holds. This fact also indirectly confirms the eigenvalue hypothesis.

Proofs for the eigenvalue hypothesis
The eigenvalue hypothesis has been proved for several specific cases: • for U q (sl 2 ) in [26]; • in multiplicity-free U q (sl N ) case for coinciding incoming representations for matrices up to size 5 × 5 [24,50].
Thus, for these cases, the tug-the-hook symmetry for Racah matrices is also proved.

Tug-the-hook symmetry for the colored HOMFLY polynomials
In recent paper [34], the tug-the-hook symmetry for the HOMFLY polynomials for knots has been proved. It is of physical importance, as it connects Wilson loops in the 3d Chern-Simons theory (2.28) in different representations.
In that case the tug-the-hook symmetry is specified: K = N , where N is sl N +1 algebra rank: Let us examine a consequence of this symmetry. First, note that where h N (R) is the number of boxes on the shifted downwards by N diagonal of the Young diagram R. The normalized HOMFLY polynomials are constructed from traces of products of R-matrices and quantum dimensions (2.27): Thus, one expects that tr MQ (π (β K )) are preserved (up to the sign) under the tug-the-hook transformation too. For clearness, consider an example of the 3-strand braid, for which the HOMFLY polynomial with arbitrary amount n of a i , b i ∈ Z. As it is described in Subsection 2.4, we choose the basis in which R 1 is diagonal R 1 = Λ R1 . Then, R 2 is diagonalized by the corresponding U -matrix: R 2 = U † Λ R1 U . Λ R1 is preserved under the tug-the-hook transformation, and tr MQ n i=1 Λ ai R1 U † Λ bi R1 U should be invariant under this transformation for any n, a i , b i , i = 1, . . . , n. So, it is naturally to expect that Racah coefficients possess the tug-the-hook symmetry when all representations in the tensor product are equal.
From the other hand, as we have already noted, for the general case of different incoming representations, R-matrices eigenvalues conserve under the tug-the-hook transformation too. Moreover, our conjecture still holds in this general case, so Racah matrices possess the tug-the-hook symmetry. Thus, all building blocks of the link HOMFLY polynomial are invariant under the tug-the-hook transformation, and we can state that the colored HOMFLY polynomial for links possess the tug-the-hook symmetry. Note that the tug-the-hook symmetry for the HOMFLY polynomials for links was neither proved, nor observed, and here we firstly claim it.
Performing similar procedures, we also check the invariance of Racah matrices under the following tug-the-hook transformations: 2. V [8,5, These examples provide very important checks for the tug-the-hook symmetry for Racah matrices.

Conclusion
In this paper, we have introduced a new symmetry of quantum 6j-symbols. It is a very important result, as this symmetry can be applied to any representations which fit a corresponding hook, and holds even in cases with multiplicities. Previously known symmetries concerned only U q (sl 2 ) case and U q (sl N ), N > 2, for symmetric representations and their conjugate ones. The tug-the-hook symmetry for 6j-symbols follows from the eigenvalue hypothesis. However, we have provided evidence for this symmetry which do not rely on the eigenvalue conjecture. First, we have explicitly checked that for the known Racah matrices the symmetry actually holds. Second, the tug-the-hook symmetry was proved for the colored HOMFLY polynomials for knots, and the tug-the-hook symmetry for Racah matrices for equal incoming representations should follow from the one for the HOMFLY polynomials, as Racah matrix is one of building blocks of the HOMFLY polynomial. From the other hand, using the eigenvalue conjecture, one can derive the tug-the-hook symmetry for the HOMFLY polynomials for links.
This work also demonstrates the importance of the eigenvalue conjecture. Namely, one can find new symmetries of 6j-symbols and quantum knot invariants by observing invariance of R-matrices eigenvalues.