A Calabi-Yau algebra with E 6 symmetry and the Clebsch-Gordan series of sl(3)
aa r X i v : . [ m a t h . R T ] M a y A CALABI–YAU ALGEBRA WITH E SYMMETRYANDTHE CLEBSCH–GORDAN SERIES OF sl (3) NICOLAS CRAMP´E † , LO¨IC POULAIN D’ANDECY ‡ , AND LUC VINET ∗ Abstract.
Building on classical invariant theory, it is observed that the polarised traces generatethe centraliser Z L ( sl ( N )) of the diagonal embedding of U ( sl ( N )) in U ( sl ( N )) ⊗ L . The paper thenfocuses on sl (3) and the case L = 2. A Calabi–Yau algebra A with three generators is introduced andexplicitly shown to possess a PBW basis and a certain central element. It is seen that Z ( sl (3)) isisomorphic to a quotient of the algebra A by a single explicit relation fixing the value of the centralelement. Upon concentrating on three highest weight representations occurring in the Clebsch–Gordan series of U ( sl (3)), a specialisation of A arises, involving the pairs of numbers characterisingthe three highest weights. In this realisation in U ( sl (3)) ⊗ U ( sl (3)), the coefficients in the definingrelations and the value of the central element have degrees that correspond to the fundamentaldegrees of the Weyl group of type E . With the correct association between the six parameters ofthe representations and some roots of E , the symmetry under the full Weyl group of type E ismade manifest. The coefficients of the relations and the value of the central element in the realisationin U ( sl (3)) ⊗ U ( sl (3)) are expressed in terms of the fundamental invariant polynomials associatedto E . It is also shown that the relations of the algebra A can be realised with Heun type operatorsin the Racah or Hahn algebra. Contents
1. Introduction 22. A Calabi–Yau algebra with three elements. 42.1. Parameters and specialisations. 62.2. Algebras with a Calabi–Yau potential 62.3. The Casimir element of A gen . 83. Polarised traces and the diagonal centraliser of U ( sl ( N )) 83.1. The Lie algebra sl ( N ) 83.2. Polarised traces 83.3. Results from classical invariant theory 94. The diagonal centraliser in two copies of sl (3) 104.1. Generators of Z ( sl (3)). 104.2. Automorphisms. 114.3. Choice of generators of Z ( sl (3)). 114.4. A realisation of A gen in U ( sl (3)) ⊗ U ( sl (3)) 134.5. The complete description of Z ( sl (3)). 145. Highest-weight specialisation of Z ( sl (3)) and E symmetry 155.1. A highest-weight specialisation of A . 155.2. The Weyl group of type E . 17 E symmetry of Z ( sl (3)). 196. A connection between A gen and the Racah or Hahn algebras 206.1. Racah algebra. 206.2. Hahn algebra. 217. Conclusion 22Appendix A. Parameters of A gen for the realisation in terms of Heun–Racah operators 23References 231. Introduction
In a nutshell, this paper introduces a Calabi–Yau algebra A gen (according to the definition usedin [10, 13]) with striking features. The algebras which are going to play central roles are denoted: A gen A A spec . The algebra A is the particular case, of special interest for our studies, of the generic Calabi–Yaualgebra A gen and the algebra A spec is a specialisation of A corresponding to the choice of threehighest weights of sl (3).Consider the centraliser of the diagonal embedding of U ( sl (3)) in U ( sl (3)) ⊗ which is non-Abelianand accounts for the degeneracies in the Clebsch–Gordan series. These degeneracies ( i.e. multi-plicities) are the Littlewood–Richardson coefficients of SL (3). We shall see that A surjects to thiscentraliser and shall observe moreover that the central parameters and the relation defining thekernel are mapped to elements whose degrees coincide with the fundamental degrees of the E Weyl group. Furthermore, we shall see that this numerical observation is in fact the shadow of anactual E symmetry occurring in the study of the direct sum decomposition of tensor products oftwo sl (3)-representations. Indeed, when the construction is restricted to triplets of highest weightsrepresentations of sl (3) that are related in the Clebsch–Gordan series, the specialisation of A isfound to be invariant under an action of the Weyl group of type E on the parameters defining thehighest weights. This action is described as the usual reflection representation once the parametersare correctly identified with some roots of E , and the specialised algebra can be entirely defined interms of fundamental invariant polynomials of type E . Now before we get into the details of this,let us offer some background as a way of introduction.A practical question in the decomposition of the tensor product of Lie or quantum algebrarepresentations in irreducible components is the labelling of basis states; this is typically doneby diagonalising a complete set of commuting operators that includes the total Casimir elements.Through the choice of a maximal Abelian subalgebra, the centraliser of the diagonal embedding inthe tensor product of the universal algebra is the natural provider of labelling operators.Interesting quadratic algebras have thus been identified by looking at the Racah problems, i.e. thecase of three-fold products, for U q ( sl (2)) and sl (2). Loosely speaking the centralisers of the diagonalembeddings in those cases are known as the Askey–Wilson [15] and Racah [12] algebras (see also[8]) which were originally introduced to encode the bispectral properties of the hypergeometricpolynomials whose names they bear . The latter connection has led to the definition of Heun–Askey–Wilson [1] and Heun–Racah [6] algebras which have their two generators realised by one ofthe bispectral operators and the other by the “Heun” operator given as the most general bilinearexpression in the two bispectral operators [16]. CALABI–YAU ALGEBRA WITH E SYMMETRY AND THE CLEBSCH–GORDAN SERIES OF sl (3) 3 With an eye to generalisations, here we consider sl (3) instead of the rank one algebras looked atso far. We examine the Clebsch–Gordan problem, that is we study the centraliser of the diagonalembedding of U ( sl (3)) in two copies of this algebra. For U q ( sl (2)) and sl (2), this problem is trivial;for sl (3) however, it is not since the Clebsch–Gordan series exhibits degeneracies. This is oneinstance of “missing label problems” where the centraliser must be called upon to complete theset of commuting operators. In the present case, the extra label will distinguish between identicalmodules occurring in the Clebsch–Gordan series. Our undertaking has led to the findings mentionedin the incipit. Of note is the identification of the cubic algebra A - a particular case of our masteralgebra A gen - which maps surjectively to the centraliser of interest. Remarkably, this same algebraalbeit with different central terms, has been identified by Lehrer and Racah [20] quite some time agoin the context of another well known “missing label problem” involving sl (3) or more precisely su (3)in this case. This other “missing label problem” (see for instance [17]) has to do with providing acomplete labelling for the states that transform under an irreducible representation of su (3) andare eigenvectors of the Casimir elements associated to the chain of subalgebras su (3) ⊃ o (3) ⊃ o (2).A simple count shows that there is indeed one operator missing which has to be taken from thecentraliser of o (3) in U ( su (3)). This is probably indicative of the larger relevance of A gen .In general, the study of the diagonal centraliser of U ( g ), where g is a semisimple or reductive Liealgebra, into an L -fold tensor product of U ( g ) can be put in perspective with the usual descriptionof the centre of U ( g ). Indeed, the centre corresponds to the situation L = 1, and it is described,through the Harish–Chandra isomorphism, as a (commutative) polynomial algebra with generatorsassociated to the fundamental invariants of the Weyl group of g . Except if g = sl (2) and L = 2,the diagonal centraliser will not be commutative, due to the degeneracies appearing in the Clebsch–Gordan series. In the situation studied in detail in this paper ( g = sl (3) and L = 2) but also inthe situation g = sl (2) and L = 3, an interesting picture is starting to emerge. In both cases, aCalabi–Yau algebra naturally appears instead of a commutative polynomial algebra (note that apolynomial algebra is in a sense the first example of a Calabi–Yau algebra). Another additionaldifficulty compared to the centre is that the diagonal centraliser is not generated by algebraicallyindependent elements. This accounts for the need to take a quotient of the Calabi–Yau algebrasin order to reach a complete description. Again in both cases, the same phenomenon appears andleads to a rather natural quotient. Namely the quotient simply amounts to fix the value of thecanonical central element of the Calabi–Yau algebra.As a final intriguing remark, we note that, as for the centre, the final description of the diagonalcentraliser involves the fundamental invariants of a certain reflection groups, whose relevance to theproblem was, at least for the authors, not easy to predict beforehand. For the situation consideredhere ( g = sl (3) and L = 2), the type E appears through a certain embedding of the root system oftype A × A × A into the root system of type E . Once this is done, it turns out quite surprisinglythat the natural symmetry (under the Weyl group W ( A ) × W ( A ) × W ( A )) actually extends toa symmetry under the full Weyl group of type E . For the case g = sl (2) and L = 3, a similarsymmetry also appears. Indeed, the Weyl group W ( D ) appears through a certain embedding ofthe root system of type A × A × A × A into the root system of type D [14].The paper will unfold as follows. The Calabi–Yau algebra A gen with three generators aroundwhich the article revolves is defined in Section 2. It is seen to be filtered with an appropriatedegree assignment to the generators and shown explicitly to have a PBW basis. The Calabi–Yaupotential from which it derives is given together with the Casimir element. The polarised tracesin U ( sl ( N )) ⊗ L are introduced in Section 3 where it is explained that their algebra denoted by N.CRAMP´E, L.POULAIN D’ANDECY, AND L.VINET Z L ( sl ( N )) is the centraliser of the diagonal embedding of U ( sl ( N )) in its L -fold tensor product.In Section 4 we restrict to two copies of sl (3) and use natural symmetry conditions to choosethe generators of Z ( sl (3)), the diagonal centraliser of U ( sl (3)). Furthermore, a particular case A of A gen is specified and shown to be realised in U ( sl (3)) ⊗ U ( sl (3)) via a surjective degreepreserving morphism to Z ( sl (3)). The kernel of this map is then identified so as to completelycharacterise the diagonal centraliser Z ( sl (3)) as the quotient of A by an additional relation. Itwill have been observed that the central parameters of A and the additional relation have degreesthat correspond to the fundamental degrees of the Weyl group of type E . The specialisationof the centraliser Z ( sl (3)) that follows from picking three weights for sl (3) labelled by the pairsof numbers ( m , m ) , ( m ′ , m ′ ) , ( m ′′ , m ′′ ) is considered in Section 5. The effect is that the centralparameters are replaced by expressions involving these numbers. An action of the Weyl group of E on the parameters ( m , m ) , ( m ′ , m ′ ) , ( m ′′ , m ′′ ) is obtained by associating these with some roots of E . It is then seen that the specialised polarised trace algebra Z L ( sl (3)) spec is invariant under theaction of the Weyl group of E corresponding to its reflection representation. Indeed the expressionsarising from the central terms are all given in terms of the fundamental invariant polynomials of E . Last in Section 6, it is shown that the relations of the algebra A gen are satisfied by operatorsof the (generalised) Heun type in the Racah and Hahn algebras. The detailed relations betweenthe parameters involved have been relegated to Appendix A. The paper ends with Section 7 thatcomprises some concluding remarks.2. A Calabi–Yau algebra with three elements.
Let P = { a , a ′ , a , a , a , a , a , a , a , a } be a set of indeterminates (which we also call centralparameters) and let C [ P ] be the polynomial algebra in these indeterminates: C [ P ] = C [ a , a ′ , a , a , a , a , a , a , a , a ] . Definition 2.1.
The algebra A gen is the algebra over C [ P ] generated by the elements: A, B, C , with the following defining relations: (1) [
A, B ] =
C , [ A, C ] = a B + a { A, B } + a A + a B + a A + a , [ B, C ] = a ′ A − a B − a { A, B } + a A − a B + a A + a , where { A, B } = AB + BA stands for the anticommutator. Recall that an algebra F is filtered if it has an increasing sequence of subspaces: F ⊂ F ⊂ F ⊂ · · · ⊂ F n ⊂ . . . such that F = [ n ≥ F n , compatible with the multiplication in the sense that F n · F m ⊂ F n + m . Defining F − = { } , theassociated graded algebra is: gr ( F ) = M n ≥ F n /F n − , with well-defined multiplication ( x + F n − )( y + F m − ) = xy + F n + m − , for x ∈ F n and y ∈ F m . TheHilbert–Poincar´e series of F records the dimensions of the graded components F n /F n − . CALABI–YAU ALGEBRA WITH E SYMMETRY AND THE CLEBSCH–GORDAN SERIES OF sl (3) 5 The algebra A gen is filtered when we define the degree of each generator in the following way: deg ( A ) = 3 , deg ( B ) = 4 , deg ( C ) = 6 . The subspace A genn consists of elements of degree less or equal to n . The chosen values of the degreesof A, B, C are natural in view of the later realisation of A gen in U ( sl (3)) ⊗ U ( sl (3)). Implicitly here,one can see the elements in the ground ring C [ P ] as being of degree 0. When we shall specialisethe algebra in the following, the coefficients will acquire the degree that explains their labelling.From the defining relations, one sees immediately that the graded algebra gr ( A gen ) is commu-tative. As a consequence of the next proposition, gr ( A gen ) is in fact isomorphic to the polynomialalgebra over C [ P ] on three commuting variables A, B, C . Proposition 2.1.
The algebra A gen has a “PBW basis”, that is, the algebra A gen is free over C [ P ] with basis: (2) { A α B β C γ } α,β,γ ∈ Z ≥ . Its Hilbert–Poincar´e series as a filtered C [ P ] -algebra is: − t )(1 − t )(1 − t ) . Proof.
The Hilbert–Poincar´e series follows from the knowledge of the basis. To prove that (2) formsa basis, we use the standard diamond lemma approach from [5]. First we define a partial orderingon the set of words in the generators of A gen and order the three generators by A < B < C .
Then for a given word X . . . X k in the generators, we already have its degree and we define also itsmisordering index as n ( X . . . X k ) = Card { ( i, j ) | ≤ i < j ≤ k and X j < X i } . Finally for two words X and Y in the generators, we state that X < Y if either deg ( X ) < deg ( Y )or Y is obtained from X by permuting the letters and satisfies n ( X ) < n ( Y ).This defines a partial order on the set of words in the generators which satisfies the two naturalconditions required in [5]. Namely, that there is only a finite number of words smaller than a givenword X and that for two given words such that X < Y , we have
ZXZ ′ < ZY Z ′ for any two words Z, Z ′ .Then we interpret the defining relations (1) as instructions for rewriting a given word as a linearcombination or ordered words in the generators. Explicitly, the instructions are: BA = AB − C , CA = AC − ( a B + a (2 AB − C ) + a A + a B + a A + a ) ,CB = BC − (cid:0) a ′ A − a B − a (2 AB − C ) + a A − a B + a A + a (cid:1) . This set of instructions is compatible with the partial order < , in the sense that when applying oneof these instructions to a subword of a word, we obtain a linear combination of words which arestrictly smaller in the partial ordering.So according to [5], to show that the set of ordered words in (2) is a basis, we only need to checkthat the ambiguities are resolvable, and in our situation, the only non-trivial verification to makeis the following. Starting from the word CBA , we have different possibilities of rewriting it (infact two, depending on whether we start with reordering CB or BA ), and we must check that they N.CRAMP´E, L.POULAIN D’ANDECY, AND L.VINET result in the same linear combination. This is a straightforward verification for which we omit thedetails. (cid:3)
Remark 1.
The algebra A gen is linked to algebras studied previously: the specialisation a = a ′ = a = 0 corresponds to the Racah algebra [14] and the specialisation a = 0 is a particular caseof the Heun–Racah algebra [6] . There exists also a realisation of the algebra A gen in terms of thegenerators of the Racah algebras: relations (1) of A gen are the ones between two different Heun–Racah operators (see Section 6.1). There is another realisation in terms of the Hahn generators. Inthis case, the relations of the algebra A gen encode the relations between a Heun–Hahn operator [22] and a more generic operator (see Section 6.2). Parameters and specialisations.
Equivalently, the algebra A gen can be defined directlyover C , by promoting the parameters in P to generators with the additional defining relations thatthey are central. As a consequence of Proposition 2.1, a basis of A gen over C consists of orderedmonomials in elements in P and A, B, C , such as: { p i . . . p i A α B β C γ } i ,...,i ,α,β,γ ∈ Z ≥ , for any ordering of P as { p , . . . , p } . In this point of view, one can assign also a degree to thecentral parameters in P , such that the algebra A gen over C remains filtered with a commutativepolynomial algebra as its graded algebra. It is immediate to see that the degrees of the parameters a ′ and a i must then be such that the r.h.s. of the defining relations are of degree equal at mostto the degree of the corresponding l.h.s. minus 1. One finds that the maximal values for thedegrees of the parameters a i must be i . The degree of a ′ must be 0. It justifies a posteriori thenames of the parameters. These are the degrees which will appear naturally in the realisation in U ( sl (3)) ⊗ U ( sl (3)).Now, let R be any complex algebra and consider any map ϑ from P to R . This is uniquelyextended to a morphism of algebra: ϑ : C [ P ] → R .
Thus, corresponding to ϑ , there is a specialisation of A gen , that we denote A ϑ , which is an algebraover R . It is simply obtained by replacing any central parameter p in P by its chosen value ϑ ( p ) in R . From Proposition 2.1, we have at once that A ϑ is free over R with basis consisting of orderedmonomials in A, B, C : { A α B β C γ } α,β,γ ∈ Z ≥ . In particular, we are going to consider the situation where R is another polynomial ring (possiblywith less indeterminates), and also the situation where R = C (for which the central parameters in P are simply replaced by complex numbers).2.2. Algebras with a Calabi–Yau potential.
We wish to point out that the algebra A gen can bederived from a potential and that it shares in this respect a property of most Calabi-Yau algebras[13], [2] of dimension 3. Let F f = C [ x , x , . . . x n ] be a free associative algebra with n generatorsand F cycl = F f / [ F f , F f ]. F cycl has the cyclic words [ x i x i . . . x i r ] as basis. The map ∂∂x j : F cycl → F f is such that(3) ∂ [ x i , x i , . . . x i r ] ∂x j = X { s | i s = j } x i s +1 x i s +2 . . . x i r x i x i . . . x i s − CALABI–YAU ALGEBRA WITH E SYMMETRY AND THE CLEBSCH–GORDAN SERIES OF sl (3) 7 and is extended to F cycl by linearity on combinations of cyclic words. Let Φ( x , . . . x n ) ∈ F cycl . Analgebra whose defining relations are given by(4) ∂ Φ ∂x j = 0 , j = 1 , . . . n is said to derive from the potential Φ. Now let x = A , x = B and x = C and take(5) Φ = [ ABC ] − [ BAC ] − a ′ A ] − a A ] + a B ] + a [ A B ] + a [ AB ] − a A ] + a [ AB ] + a B ] −
12 [ C ] − a [ A ] + a [ B ] . With this, it is not hard to see that the relations (1) of A gen are given by ∂ Φ ∂C = 0, ∂ Φ ∂B = 0, ∂ Φ ∂A = 0. Remark 2.
As mentioned in Remark 1, A gen is a generalisation of the Racah algebra. Therefore,from the result stated above, the relations of the Racah algebra can be derived from the Calabi–Yaupotential (5) with a = a ′ = a = 0 . PBW basis and central element.
We still consider an algebra derived from a potential withthree generators x = A , x = B and x = C , and we give a degree to these generators: deg ( A ) = a , deg ( B ) = b , deg ( C ) = c , and d := a + b + c . We take a potential of the form:Φ = Φ ( d ) + Φ A ∂ Φ Remark 3. It is suggested in [10] that the quotient of the algebra derived from a Calabi–Yau poten-tial by the ideal generated by the central element Ω can be viewed as a non-commutative analogueof a Poisson algebra. It is remarkable that the realisation that we have later in U ( sl (3)) ⊗ U ( sl (3)) of the algebra A gen factors precisely through such a quotient. The Casimir element of A gen . As a consequence of the PBW basis, working with the algebra A gen is an algorithmically straightforward procedure. We can use the defining relations as rewritinginstructions for ordering any word in the PBW basis. Using this straightforward approach, we checkthat the following element is central in A gen :(6) Ω = x A + x B + x A + x { A, B } + x B + x A + x ABA + x BC − x AB + x A + x B + C where x = (12 a + 3 a a a ′ + 2 a a − a a ′ a − a a ) ,x = ( − a − a a + a a a ′ + a a + 6 a a ) ,x = (6 a + 3 a a ′ + 3 a a a ′ − a a − a ′ a ) ,x = ( − a + a a + 3 a a + a a ′ a ) ,x = ( − a − a a + a a ′ + 6 a ) ,x = (2 a − a ′ a ) , x = − a + a a ′ ,x = 2 a , x = a ′ , x = − a . We call the element Ω the Casimir element of A gen . By direct investigations, we prove that it is the(non-scalar) central element of A gen of minimal degree (its degree is 12).This central element can be rewritten to underscore the link with the Calabi–Yau potential (5).Indeed, by replacing the cyclic words in the Calabi–Yau potential φ We consider the tensor product of L copies of U ( gl ( N )). For a ∈ { , . . . , L } ,we introduce the notation:(8) e ( a ) pq = 1 ⊗ a − ⊗ e pq ⊗ ⊗ L − a ∈ U ( gl ( N )) ⊗ L . The following map δ : U ( gl ( N )) → U ( gl ( N )) ⊗ L (9) e pq L X a =1 e ( a ) pq CALABI–YAU ALGEBRA WITH E SYMMETRY AND THE CLEBSCH–GORDAN SERIES OF sl (3) 9 extends to an algebra homomorphism. The map δ is called the diagonal map and its image is thediagonal embedding of U ( gl ( N )) in U ( gl ( N )) ⊗ L .Let us define the following elements of U ( gl ( N )) ⊗ L (from now on the summation over repeatedindices will be implicit): T ( a ,...,a d ) = e ( a ) i i e ( a ) i i . . . e ( a d ) i i d , a , . . . , a d ∈ { , . . . , L } . By a direct computation, we can show that these elements commute with δ ( e pq ). Therefore they arein the centraliser of the diagonal embedding of U ( gl ( N )) in U ( gl ( N )) ⊗ L . The elements T ( a ,...,a d ) are called polarised traces. Definition 3.1. We denote Z L ( gl ( N )) (respectively, Z L ( sl ( N )) ) the subalgebra of U ( gl ( N )) ⊗ L (re-spectively, of U ( sl ( N )) ⊗ L ) generated by all polarised traces T ( a ,...,a d ) with d ≥ and a , . . . , a d ∈{ , . . . , L } . Results from classical invariant theory. For the classical results on invariant theory thatwe recall below, we refer to [3, 19, 21] (see also [9, 11] and references therein).The first fundamental theorem on the algebra of polynomial functions on gl ( N ) × · · · × gl ( N ) ( L times) which are invariant under simultaneous conjugation by GL ( N ) asserts that it is generatedby the functions:(10) ( M , . . . , M L ) Tr( M a M a . . . M a d ) , with d ≥ a , . . . , a d ∈ { , . . . , L } .In general, it is furthermore known that a set of generators is obtained by restricting to thesefunctions with d ≤ N . And it is conjectured that it is enough to take d ≤ N ( N + 1). Thisconjecture is known to be true if N ∈ { , , } .With a standard reasoning that we sketch now, these results provide some information on thealgebra of polarised traces in the deformed setting of universal enveloping algebras.Take A = U ( gl ( N )) ⊗ L . It is a filtered algebra (the degree of each generator e ( a ) pq is 1) and recallthat its associated graded algebra is the commutative algebra of polynomials in e ( a ) pq . So we cannaturally identify the algebra of polynomial functions on gl ( N ) × · · · × gl ( N ) with this gradedcommutative algebra: gr ( A ) = M n ≥ C n [ e ( a ) pq ] , where C n [ e ( a ) pq ] is the space of homogeneous polynomials of degree n . The identification sends e ( a ) pq to the linear form giving the ( q, p ) coordinate of the a -th matrix.The Lie algebra gl ( N ) acts diagonally on A = U ( gl ( N )) ⊗ L (namely, by composing the diagonalmap δ followed by the commutator) and this respects the filtration, gl ( N ) · A n ⊂ A n , so it inducesan action on gr ( A ). Under the above identification, a direct verification shows that this action of gl ( N ) on gr ( A ) corresponds to the derivative of the conjugation action of GL ( N ) on polynomialfunctions. So in particular, invariant functions correspond to elements of gr ( A ) commuting withall elements δ ( e pq ). Moreover, the generating invariant functions (10) correspond to the images in gr ( A ) of the polarised traces denoted T ( a ,...,a d ) .We call the diagonal centraliser, for brevity, the centraliser of the diagonal embedding of U ( gl ( N ))in U ( gl ( N )) ⊗ L . Using the result on classical invariants, what we just explained is that the image ofthe diagonal centraliser in gr ( A ) is generated by the images of the elements T ( a ,...,a d ) . So in otherwords, for any element z of degree n in the diagonal centraliser, we have that modulo A n − the element z can be obtained as a certain polynomial in T ( a ,...,a d ) . By an easy induction, this givesthat any element in the diagonal centraliser is generated by the polarised traces T ( a ,...,a d ) .Besides we have also noted in the preceding subsection that the polarised traces T ( a ,...,a d ) commutewith the diagonal embedding of U ( gl ( N )). So we can conclude that we have the following corollaryof the classical invariant theory. Corollary 3.1. The algebra Z L ( gl ( N )) of polarised traces coincides with the centraliser of thediagonal embedding of U ( gl ( N )) in U ( gl ( N )) ⊗ L . Further, with the same arguments, we can also extract a finite set of generators for Z L ( gl ( N )),namely the set of polarised traces T ( a ,...,a d ) with d ≤ N . Moreover, for N ∈ { , , } , it is enoughto take d ≤ N ( N + 1). Remark 4. All this subsection can be formulated in an obvious way for Z L ( sl ( N )) with similarstatements (one just has to remove the polarised traces of degree d = 1 from the set of generators). The diagonal centraliser in two copies of sl (3)From now on, we will mainly focus on the situation N = 3 and L = 2 of the preceding subsection.4.1. Generators of Z ( sl (3)) . From the results in classical invariant theory, we have that Z ( sl (3))is generated by the polarised traces of degree less or equal to 6. More precisely, from [9] (see alsoreferences therein), one can extract the following information. As a subalgebra of U ( sl (3)) ⊗ U ( sl (3)),the algebra Z ( sl (3)) is filtered. The Hilbert–Poincar´e series of Z ( sl (3)) is:(11) 1 − t t (1 − t )(1 − t t )(1 − t )(1 − t )(1 − t t )(1 − t t )(1 − t )(1 − t t )(1 − t t ) . This is the Hilbert–Poincar´e series of a subalgebra of the polynomial algebra in the generators e ( a ) pq .The coefficient in front of t k t l is the dimension of the graded part of Z ( sl (3)) in degree k in thegenerators e (1) pq and l in the generators e (2) pq .A set of generators of Z ( sl (3)) is(12) T (1 , , T (1 , , T (2 , , T (1 , , , T (1 , , , T (1 , , , T (2 , , , T (1 , , , and T (1 , , , , , . Moreover, in the graded commutative algebra, we have that the first eight generators are alge-braically independent, and thus, looking at the numerator of the Hilbert–Poincar´e series, we seethat there is an algebraic relation among the generators in degree 12 (more precisely, in degree(6 , , 3) can serve as generator. For example the choice T (1 , , , , , is not possible (the rule is that at least two consecutive indices must be equal but not three). In oursetting this ambiguity disappears as we can simply remove the last generator from the list since itturns out that T (1 , , , , , can be expressed in terms of the other generators. Indeed, one can checkthat: 2 T (1 , , , , , + [ T (1 , , , T (1 , , , ] + 13 T (1 , , T (2 , , − T (1 , , T (1 , , − T (1 , , , T (1 , has no term of degree 6, and thus can be expressed in terms of the first eight generators (the explicitexpression is T (1 , T (1 , , − T (2 , T (1 , , − T (1 , , , + 2 T (1 , T (2 , − T (1 , , + T (1 , , ) − T (1 , ). CALABI–YAU ALGEBRA WITH E SYMMETRY AND THE CLEBSCH–GORDAN SERIES OF sl (3) 11 Remark 5. It is interesting to note that in our non-commutative setting, Z ( sl (3)) is generated bypolarised traces of degree up to , since the last generator is obtained by taking a commutator of twoother generators (which is a new phenomenon compared to the classical setting). Automorphisms. Consider the following maps on the generators of U ( sl (3)) ⊗ U ( sl (3)):(1 , 2) : e (1) ij e (2) ij and e (2) ij e (1) ij ,τ : e (1) ij 7→ − e (1) ji and e (2) ij 7→ − e (2) ji . They clearly extend to automorphisms of U ( sl (3)) ⊗ U ( sl (3)) and, since they leave stable thediagonal embedding of U ( sl (3)), they restrict to automorphisms of Z ( sl (3)) (one can also checkdirectly that they preserve the algebra generated by the polarised traces).Then, consider also the map on the set of the first eight generators in (12) given by the followingreplacement: (2 , 3) : e (1) ij e (1) ij and e (2) ij 7→ − e (1) ij − e (2) ij . We insist that we define it (by these formulas) only on these eight polarised traces. It is clear thatit takes values in the algebra Z ( sl (3)). For example, we have, under the map (2 , T (1 , T (1 , , T (1 , , 7→ − T (1 , , − T (1 , , , T (1 , , , T (1 , , , + T (1 , , , + T (1 , , , + T (1 , , , . Remark 6. • The map (2 , is more mysterious than (1 , or τ . Indeed, it is clear that (2 , does not extend to an automorphism of U ( sl (3)) ⊗ U ( sl (3)) . Further, had we defined it by the aboveformulas on all polarised traces, it would not have extended to an automorphism of Z ( sl (3)) (forexample, the relation expressing T (1 , , , , , in terms of the other generators would not have beenpreserved). However, we defined it only on the first eight traces in (12) which are, as we haveseen, generators of Z ( sl (3)) . With this definition, we will show later that (2 , does extend to anautomorphism of Z ( sl (3)) , once we have a presentation of this algebra (see Corollary 4.1). Fornow, we use it only as a guide to choose convenient generators. • When U ( sl (3)) ⊗ U ( sl (3)) acts on a tensor product of representations, the automorphism of Z ( sl (3)) induced by (2 , is naturally interpreted as the symmetry between looking at the represen-tation V ⋆ in V ⊗ V , and looking at the representation V ⋆ in V ⊗ V (here ⋆ denotes the dual of arepresentation). Indeed, the map (2 , formally exchanges the action on the second component ofa tensor product with the diagonal action composed by the antipode. Assume for now that (2 , 3) extends to an automorphism of Z ( sl (3)), as do (1 , 2) and τ . Wedenote Aut the group generated by these three automorphisms. It is easy to see that it has thefollowing structure: Aut = h (1 , , (2 , , τ i = h (1 , , (2 , i × h τ i ∼ = S × Z / Z , where S is the symmetric group which permutes e (1) ij , e (2) ij , − e (1) ij − e (2) ij and commutes with τ .4.3. Choice of generators of Z ( sl (3)) . First we analyse the action of Aut on the small degreecomponents of Z ( sl (3)). We have: • In degree 2, S acts through its natural permutation representation and τ acts by the identity. • Including degree 3, it adds for S a copy of the permutation representation and a copy ofthe sign representation; for τ , it adds four times the eigenvalue − • Then, bringing in degree 4, it adds for S two copies of the permutation representation (com-ing from the square of the representation in degree 2) and a copy of the trivial representation;for τ , it adds only trivial representations.Therefore, it is natural to take for generators of Z ( sl (3)) the set of elements: k , k , k , l , l , l , X, Y , such that the degrees are respectively 2 , , , , , , , 4, and moreover such that the action of Aut is as follows:(13) π ∈ S : k i k π ( i ) , l i l π ( i ) , X sgn ( π ) X , Y Y , (14) τ : k i k i , l i 7→ − l i , X 7→ − X , Y Y . Remark 7. Up to some normalisation and trivial addition of constants, these requirements uniquelyfix the choice of k , k , k , l , l , l , X . They also almost fix the choice of Y , the only unavoidableremaining freedom comes from the fact that k + k + k is invariant under Aut . So Y is uniquelyfixed up to a degree 2 polynomial in k + k + k . We will give now an explicit expression for our generators. First, recall a standard definition ofthe Casimir elements of U ( sl (3)): C (2) = e i i e i i and C (3) = e i i e i i e i ,i . We will use also a slightly different version:(15) C (2) = e i i e i i and C (3) = e i i e i i e i ,i . The relation between the two versions in U ( sl (3)) is:(16) C (2) = C (2) , C (3) = C (3) − C (2) . With these notations, we set:(17) k = C (2) ⊗ , k = 1 ⊗ C (2) , k = δ ( C (2) ) ,l = 12 ( C (3) + C (3) ) ⊗ , l = 1 ⊗ 12 ( C (3) + C (3) ) , l = − δ ( 12 ( C (3) + C (3) )) ,X = 12 ( T (1 , , − T (1 , , ) + 13 ( l − l ) ,Y = T (1 , , , + 32 ( T (1 , , + T (1 , , ) − 112 ( T (1 , ) − T (1 , T (2 , + 52 T (1 , ,Z = [ X, Y ] . Proposition 4.1. The elements k , k , k , l , l , l , X, Y given by (17) are generators of Z ( sl (3)) satisfying the symmetry conditions (13)-(14).Proof. Everything can be checked easily by straightforward calculations. (cid:3) Remark 8. In terms of polarised traces, it is not difficult to verify that: k = T (1 , , k = T (2 , , k = T (1 , + T (2 , + 2 T (1 , ,l = T (1 , , + 32 k , l = T (2 , , + 32 k , l = − l − l − T (1 , , + T (1 , , ) − T (1 , . CALABI–YAU ALGEBRA WITH E SYMMETRY AND THE CLEBSCH–GORDAN SERIES OF sl (3) 13 A realisation of A gen in U ( sl (3)) ⊗ U ( sl (3)) . We start by introducing a particular case A of the algebra A gen of Section 2, which will turn out to be the one relevant for the description of Z ( sl (3)).In the definition below, we consider k , k , k , l , l , l as indeterminates (central parameters). Thesymmetric group S acts on C [ k , k , k , l , l , l ] by permuting { k , k , k } and permuting { l , l , l } .We denote: Sym + := 16 X π ∈ S π and Sym − := 16 X π ∈ S sgn ( π ) π . Definition 4.1. The algebra A is the algebra over C [ k , k , k , l , l , l ] generated by A, B, C withthe following defining relations: (18) [ A, B ] = C , [ A, C ] = − B + a A + a A + a , [ B, C ] = − A − a { A, B } − a B + a A + a , where: a = 12 ( k + k + k ) , a = Sym − (2 k l ) ,a = Sym + (cid:16) ( − k + 2 k k − k k k ) + ( k − k k ) + l − l l (cid:17) ,a = Sym + (cid:16) ( k − k k + 6 k k + 4 k k k ) − (3 k − k k + 2 k k k ) − k ( l − l + 4 l l − l l ) + l + l l (cid:17) a = Sym − (cid:16) l (cid:0) − l + ( k − k k + k k + 9 k k ) + ( k − k k ) (cid:1)(cid:17) , Clearly, the algebra A can be seen as a specialisation of the generic algebra A gen and as suchhas a PBW basis as explained in Section 2. The use of the same names as some generators of Z ( sl (3)) is a convenient slight abuse of notation and is justified by the next proposition. Below, itis understood that through φ , the parameters k , k , k , l , l , l are sent to the generators sharingthe same names. Proposition 4.2. The following map φ extends to a surjective morphism of algebras from A to Z ( sl (3)) : (19) φ : A X , B Y , C Z . Proof. We first note that the elements k , k , k , l , l , l are all central in Z ( sl (3)). Indeed, k , k , l , l are already central in U ( sl (3)) ⊗ U ( sl (3)) since they are of the form c ⊗ ⊗ c with c a centralelement of U ( sl (3)). Concerning k and l , we see from their definitions (17) that they are of theform δ ( c ) with c ∈ U ( sl (3)). Those elements δ ( c ) being in the image of the diagonal map commutewith all elements of Z ( sl (3)).It remains to calculate in U ( sl (3)) ⊗ U ( sl (3)) the commutators [ X, Z ] and [ Y, Z ] to see thatthey satisfy the defining relations of A gen when the images of the central elements are given asin the statement of the Proposition. This is a straightforward but very lengthy calculation whichcan be done with the help of a computer, by rewriting all elements involved in the PBW basis of U ( sl (3)) ⊗ U ( sl (3)), and comparing both sides of the given relations. (cid:3) Remark 9. In the explicit realisation of A in U ( sl (3)) ⊗ U ( sl (3)) given above, the images of thegenerators of A (as a complex algebra) have the following degrees: (20) deg ( X ) = 3 , deg ( Y ) = 4 , ( and thus deg ( Z ) = 6) , deg( a i ) = i . It is remarkable that the degrees of the non vanishing central elements a i are 2,5,6,8 and 9 which areexactly the first five fundamental degrees of type E . The missing degree is 12, but such a degree 12polynomial will also appear naturally in the complete description of Z ( sl (3)) (see next subsection).This “coincidence” will be largely developed in the next section. Remark 10. The parameters a , a and a of the generic algebra A gen are sent to in the special-isation in A and therefore also in Z ( sl (3)) . However, a different choice of generators X, Y wouldresult in non-zero values for a and a . Let us also point out that a non-zero coefficient a wouldalso appear if we were looking at Z ( gl (3)) . Remark 11. As mentioned in the Introduction, another specialisations of the algebra A gen has beenused in [20] in a study of the centraliser of so (3) in su (3) . The complete description of Z ( sl (3)) . At this point, we know that the algebra Z ( sl (3)) isisomorphic to a quotient of the algebra A . Our goal now is to complete the description of Z ( sl (3))by identifying the kernel of the map φ .To guide us, we have on one side the PBW basis of the algebra A , which asserts that as a vectorspace, this is a polynomial algebra. And on the other side, we have the Hilbert–Poincar´e series of Z ( sl (3)) given in (11). The comparison of both descriptions tells us that we must have one morerelation satisfied by the images of the generators of A in Z ( sl (3)). Moreover, this relation has tobe found at the degree 12 (or more precisely at degree (6,6) in U ( sl (3)) ⊗ U ( sl (3))).Recall that the algebra A inherits from A gen a Casimir element, which was given in (6). In thespecialisation A , its expression simplifies to:(21) Ω = x A + x B + x A + x { A, B } + x B + x ABA − A + 4 B + C where (with the parameters a , a , a , a , a given in Definition 4.1):(22) x = 6 a + 2 a , x = − a − a , x = 6 a + a , x = − a , x = 8 a − , x = − a + 12 . Remarkably, the image of this central element Ω in Z ( sl (3)) is of degree 12 and it turns out thatwe have in Z ( sl (3)): φ (Ω) = a ( k , k , k , l , l , l ) , where the polynomial a ( k , k , k , l , l , l ) is equal to:(23) Sym + (cid:16) k − k (1 + 2 k ) + k ( − 39 + 6 k + 5 k + 3 k k ) − k ( − − k +6 k + 5 k − k k + 6 k k + 2 l + 8 l l − l + 140 l l ) − k (2592 k + 702 k +468 k k + 42 k k + k k − l − k l + 416 l l − k l l + 144 k l l − l +56 k l − k l − l l − k l l ) − k ( − k k − l + 68 k l + 17 k k l − l l − k l l − k l l − k k l l − l − k l + 1392 l l ) + l ( − l + l − l + 24 l l − l l + 244 l l l ) (cid:17) . CALABI–YAU ALGEBRA WITH E SYMMETRY AND THE CLEBSCH–GORDAN SERIES OF sl (3) 15 This formula is checked by a straightforward calculation directly in U ( sl (3)) ⊗ U ( sl (3)). This canbe done using a computer to express both sides in a PBW basis of U ( sl (3)) ⊗ U ( sl (3)). Theorem 4.1. The algebra Z ( sl (3)) is the quotient of the algebra A by the following additionalrelation: (24) Ω = a ( k , k , k , l , l , l ) . Proof. Recall that we have the surjective morphism φ from Proposition 4.2 from A to Z ( sl (3)).Moreover, as already said, the relation (24) is satisfied by the image of Ω in U ( sl (3)) ⊗ U ( sl (3)).So the element Ω − a ( k , k , k , l , l , l ) does belong to the kernel of the morphism φ .Both algebras A and Z ( sl (3)) are filtered complex algebras. The degrees in A are given by: deg ( k i ) = 2 , deg ( l i ) = deg ( A ) = 3 , deg ( B ) = 4 , deg ( C ) = 6 , and the degrees in Z ( sl (3)) come naturally from the degrees in U ( sl (3)) ⊗ U ( sl (3)) (the generators e ( a ) ij are of degree 1). Then it is immediate that φ is a morphism of filtered algebras (the degreesare preserved).Thus it remains only to show that the Hilbert–Poincar´e series of the quotient of A by the addi-tional relation (24) coincides with the Hilbert–Poincar´e series of Z ( sl (3)). From the PBW basis of A , we have that its Hilbert–Poincar´e series in one variable is:1(1 − t ) (1 − t ) (1 − t )(1 − t ) . Adding the relation (24) has the effect of multiplying the numerator by a factor (1 − t ). Indeed,the relation expresses C in the form x + yC where x, y involves only the other generators. Thus,using the PBW basis of A , it is easy to see that a basis of the quotiented algebra consists of allordered monomials in k , k , k , l , l , l , A, B, C with the restriction that the power of C must be 0or 1.So we end up with the Hilbert–Poincar´e series of Z ( sl (3)) given in (11) when the bidegree isreduced to a single degree. The proof is concluded. (cid:3) Corollary 4.1. The maps forming the group Aut and given by Formulas (13)-(14) extend toautomorphisms of Z ( sl (3)) .Proof. Now that we have a presentation by generators and relations of Z ( sl (3)), the verification isalmost immediate (we recall that only the verification for the transposition (2 , 3) of S remained tobe done). (cid:3) Highest-weight specialisation of Z ( sl (3)) and E symmetry A highest-weight specialisation of A . Consider a highest weight representation V m ,m of U ( sl (3)). It is parametrised by two complex numbers m , m , and generated as a U ( sl (3))-moduleby a highest weight vector v m ,m satisfying(25) h p v m ,m = ( m p − v m ,m with p = 1 , ,e pq v m ,m = 0 with 1 ≤ p < q ≤ , where we set h p = e pp − e p +1 ,p +1 . Remark 12. The standard convention would denote ( m − , m − the highest weight of therepresentation V m ,m since m − and m − are the coefficients in front of the fundamental weights.For example, if m − , m − ∈ Z ≥ and V m ,m is the corresponding finite-dimensional irreduciblerepresentation of sl (3) , then it is usually represented with a Young diagram with m + m − boxesin the first row and m − boxes in the second row.We find our parametrisation more convenient for the purpose of this paper since it includes fromthe beginning the translation by the half sum of the positive roots of sl (3) . In the representation V m ,m , the Casimir elements C (2) and C (3) of U ( sl (3)) defined in (15) areproportional to the identity matrix, their values are given by: c (2) ( m , m ) = 23 ( m + m + m m ) − , (26) c (3) ( m , m ) = 19 ( m + 2 m − m + m + 3)( m − m − . (27)Now pick three pairs of complex numbers ( m , m ), ( m ′ , m ′ ) and ( m ′′ , m ′′ ), and look at the tensorproduct of representations V m ,m ⊗ V m ′ ,m ′ . In this space, consider all the highest-weight vectorsunder the diagonal action of U ( sl (3)) corresponding to ( m ′′ , m ′′ ), that is, all vectors satisfying forthe diagonal action Formulas (25) with m , m replaced by m ′′ , m ′′ . The subspace generated by allthese vectors as a U ( sl (3))-module is denoted by: M m ′′ ,m ′′ m ,m ,m ′ ,m ′ . Example 5.1. In particular, if the three pairs are integers in Z ≥ and V m ,m , V m ′ ,m ′ , V m ′′ ,m ′′ arethe corresponding finite-dimensional irreducible representations, then the subspace M m ′′ ,m ′′ m ,m ,m ′ ,m ′ isthe isotypic component of V m ′′ ,m ′′ in the tensor product V m ,m ⊗ V m ′ ,m ′ . In this case, we have: M m ′′ ,m ′′ m ,m ,m ′ ,m ′ = D m ′′ ,m ′′ m ,m ,m ′ ,m ′ V m ′′ ,m ′′ , where the positive integer D m ′′ ,m ′′ m ,m ,m ′ ,m ′ is the multiplicity of V m ′′ ,m ′′ in V m ,m ⊗ V m ′ ,m ′ which iscalled the Littlewood–Richardson coefficient. As a subalgebra of U ( sl (3)) ⊗ U ( sl (3)), the diagonal centraliser Z ( sl (3)) acts on V m ,m ⊗ V m ′ ,m ′ .Moreover, since it centralises the diagonal action of U ( sl (3)), it leaves invariant the subspace M m ′′ ,m ′′ m ,m ,m ′ ,m ′ , which thus becomes naturally a Z ( sl (3))-module.In this representation M m ′′ ,m ′′ m ,m ,m ′ ,m ′ of Z ( sl (3)), the central parameters k , k , k , l , l , l takedefinite complex values, since they are expressed in terms of the Casimir elements. Indeed recallthat they were defined as: k = C (2) ⊗ , k = 1 ⊗ C (2) , k = δ ( C (2) ) ,l = ( C (3) + 32 C (2) ) ⊗ , l = 1 ⊗ ( C (3) + 32 C (2) ) , l = − δ ( C (3) + 32 C (2) ) . This motivates the following definition, which entails a specialisation of A and Z ( sl (3)) acting onthe space M m ′′ ,m ′′ m ,m ,m ′ ,m ′ . CALABI–YAU ALGEBRA WITH E SYMMETRY AND THE CLEBSCH–GORDAN SERIES OF sl (3) 17 Definition 5.1. The algebra A spec is the specialisation of A corresponding to the following valuesof the central parameters: k = c (2) ( m , m ) , k = c (2) ( m ′ , m ′ ) , k = c (2) ( m ′′ , m ′′ ) ,l = c (3) ( m , m ) + 32 c (2) ( m , m ) , l = c (3) ( m ′ , m ′ ) + 32 c (2) ( m ′ , m ′ ) ,l = − c (3) ( m ′′ , m ′′ ) − c (2) ( m ′′ , m ′′ ) . Similarly, we denote by Z ( sl (3)) spec the same specialisation of Z ( sl (3)) . In simple terms, this specialisation means that in Definition 4.1 of A and in the description of Z ( sl (3)) in Theorem 4.1, the central elements k , k , k , l , l , l are replaced by their expressionsabove in terms of m , m , m ′ , m ′ , m ′′ , m ′′ . Note that these specialisations are well-defined since both A and Z ( sl (3)) are free modules over C [ k , k , k , l , l , l ]. In the algebra A spec , the parameter a i becomes a polynomial in m , m , m ′ , m ′ , m ′′ and m ′′ of degree i . Similarly a becomes a polynomialof degree 12 in Z ( sl (3)) spec .We can summarise the results obtained so far in the following proposition: Proposition 5.1. The algebra Z ( sl (3)) spec is generated by X, Y, Z with the defining relations: (28) [ X, Y ] = Z , [ X, Z ] = − Y + a X + a X + a , [ Y, Z ] = − X − a { X, Y } − a Y + a X + a ,x X + x Y + x X + x { X, Y } + x Y + x XY X − X + 4 Y + Z = a , where x i are given by (22) and a i are given in Definition 4.1 and by (23) in which we replace k , k , k , l , l , l by their expressions in terms of m , m , m ′ , m ′ , m ′′ , m ′′ given in Definition 5.1. Remark 13. Consider the situation of Example 5.1 and denote by X the endomorphism giving theaction of the generator X on the highest weight vectors in M m ′′ ,m ′′ m ,m ,m ′ ,m ′ . The matrix X has somehowbeen studied previously in [18] where a matrix S related to X through S = − X is introduced. Ithas been shown in [18] that the eigenvalues of S are non-degenerate and allow to distinguish thedifferent highest-weight vectors in M m ′′ ,m ′′ m ,m ,m ′ ,m ′ thereby providing a solution to the missing labelproblem for the tensor product of two irreducible representations of sl (3) . Let us also mention [7] where an operator Θ [0 , = − X + l − l has been studied in the case where the Littlewood–Richardsoncoefficient D m ′′ ,m ′′ m ,m ,m ′ ,m ′ is 2. The explicit expressions of the polynomials a i in terms of m , m , m ′ , m ′ , m ′′ , m ′′ are quite compli-cated but it turns out that these polynomials are related to the fundamental invariant polynomialsof the Weyl group of type E . A first hint in this direction comes from observing that the degreesof the polynomials in Proposition 5.1 are 2 , , , , E . We shall explain that in detail in the next subsection.5.2. The Weyl group of type E . Let us consider a root system of type E and choose the simpleroots α , α , α , α , α , α with the numeration accordingly to the following Dynkin diagram: W ( E ) the corresponding Weyl group and by s i = s α i the Weyl reflections associatedto the roots α i satisfying, for 1 ≤ i, j ≤ s i = 1(29) s i s j = s j s i if i and j are not connected in the Dynkin diagram(30) s i s j s i = s j s i s j if i and j are connected in the Dynkin diagram . (31)Let us associate the parameters m , m , m ′ , m ′ , m ′′ and m ′′ with the simple roots as follows:(32) m = α ,m = α , m ′ = α ,m ′ = α , m ′′ = Θ ,m ′′ = − α , where Θ is the longest positive root of E . The explicit expression of Θ is such that: α = 13 (cid:0) m ′′ + 2 m ′′ − ( m + 2 m ) − ( m ′ + 2 m ′ ) (cid:1) . Note that ( m , m ), ( m ′ , m ′ ) and ( m ′′ , m ′′ ) are three subsystems of type A which are pairwiseorthogonal. In other words, they generate a subsystem of type A × A × A .The Weyl group acts on the set of roots by its usual reflection representation, and it is elementaryto calculate the induced action of the generators of W ( E ) on the parameters. We have:(33) s : ( m 7→ − m ,m m + m , s : ( m m + m ,m 7→ − m , (34) s : ( m ′ 7→ − m ′ ,m ′ m ′ + m ′ , s : ( m ′ m ′ + m ′ ,m ′ 7→ − m ′ , (35) s : ( m ′′ m ′′ + m ′′ ,m ′′ 7→ − m ′′ , s : m m + α ,m ′ m ′ + α ,m ′′ m ′′ − α , where the actions that are not given are trivial and the explicit expression of α is provided above.It is elementary that linear transformations on roots can alternatively be seen as linear transfor-mations of coordinates in the dual space, which is here the space of weights since E is simply-laced.To be explicit, consider the weight lattice of type E , that is: P E = Z ω ⊕ Z ω ⊕ Z ω ⊕ Z ω ⊕ Z ω ⊕ Z ω , where ω , ω , ω , ω , ω , ω are the fundamental weights corresponding to the simple roots. It comesnaturally with an action of the Weyl group W ( E ), which reads explicitly as: s ( ω ) = − ω − ω , s ( ω ) = − ω − ω − ω , s ( ω ) = − ω − ω − ω − ω ,s ( ω ) = − ω − ω − ω , s ( ω ) = − ω − ω , s ( ω ) = − ω − ω , and the unspecified actions are trivial since s i ( ω j ) = ω j if i = j . CALABI–YAU ALGEBRA WITH E SYMMETRY AND THE CLEBSCH–GORDAN SERIES OF sl (3) 19 Now, we parametrise a vector of the space spanned by ω , ω , ω , ω , ω , ω as follows: ω = m ω + m ω + 13 (cid:0) m ′′ + 2 m ′′ − ( m + 2 m ) − ( m ′ + 2 m ′ ) (cid:1) ω + m ′ ω + m ′ ω − m ′′ ω . In other words, we define ( m , m , m ′ , m ′ , m ′′ , m ′′ ) as the coordinates in the new basis:( ω − ω , ω − ω , ω − ω , ω − ω , ω , − ω + 23 ω ) . Then, it is immediate that the action of the Weyl group on weights transforms the parameters m , m , m ′ , m ′ , m ′′ , m ′′ as per the formulas (33)–(35).5.3. Fundamental invariants and E symmetry of Z ( sl (3)) . Consider now the polynomialalgebra S ( V ) associated to the reflection representation of W ( E ). The roots associated to the pa-rameters m , m , m ′ , m ′ , m ′′ and m ′′ in (32) form a basis of the space of the reflection representation,and thus S ( V ) can be identified with the algebra of polynomials in m , m , m ′ , m ′ , m ′′ , m ′′ .The Weyl group W ( E ) acts on S ( V ) and the subalgebra of invariant polynomials for W ( E ) isgenerated by 6 algebraically independent polynomials, which can be chosen homogeneous of degrees,respectively, 2,5,6,8,9,12 (the fundamental degrees of E ).The order of W ( E ) is 51840 and we can define an averaging operator over this group(36) h . i = 151840 X s ∈ W ( E ) s . Then a choice of six fundamental homogeneous invariant polynomials for W ( E ) is given by p = 32 h m i , p = 83 h ( m ′ ) m ′ m ′′ m ′′ i , p = 10 h m m m ′ m ′ m ′′ m ′′ i , (37) p = 53 h ( m ) m ( m ′ ) m ′ m ′′ m ′′ i , p = 4027 h ( m m m ′ ) m ′ m ′′ m ′′ i ,p = 203 h ( m m m ′ m ′ m ′′ m ′′ ) i . We can now state the following theorem about Z ( sl (3)) spec : Theorem 5.1. The algebra Z ( sl (3)) spec is invariant under the action of the Weyl group of type E given by (33) - (35) . The polynomials a , a , a , a , a and a can be expressed in terms of thefundamental invariant polynomials as follows a = p − , a = − p , a = p + p p − p , (38) a = − p + p 54 + p p 12 + p 18 + p p − p , (39) a = − p − p (cid:18) p 27 + p − (cid:19) , (40) a = − p + 35 p 12 + p 36 + 17 p p − p p − p p 18 + p − p p p p − p − p 108 + 13 p − p p − p − p − p 12 + 11 p − . Proof. By direct (but long) computations, one can show that the polynomials a i ( i = 2 , , , , , Z ( sl (3)) spec follows. (cid:3) Remark 14. There exists an additional automorphism r of Z ( sl (3)) spec with r : X → − X , r : Y → Y and r : m 7→ − m , m 7→ − m , m ′ 7→ − m ′ , m ′ 7→ − m ′ , m ′′ 7→ − m ′′ , m ′′ 7→ − m ′′ . This transformation r on the parameters and all the s i generate a group of order 103680 which isthe complete symmetry of the root system of E : the Weyl group generated by the Weyl reflectionsand the central symmetry. Remark 15. Recall that we have described a group of automorphisms Aut of Z ( sl (3)) (beforespecialisation) of order 12. Of course, all these automorphisms can be pushed to the specialisation Z ( sl (3)) spec . Their action on the parameters is as follows: we have all six permutations of the threepairs ( m , m ) , ( m ′ , m ′ ) and ( m ′′ , m ′′ ) , together with their compositions with the map τ transposing m ↔ m , m ′ ↔ m ′ and m ′′ ↔ m ′′ . Among them, the following six automorphisms leave X and Y invariant: Aut +0 = { Id, (1 , , , (1 , , , (1 , τ, (1 , τ, (2 , τ } . They are included in the symmetry described in Theorem 5.1 and corresponds to the action of someelements of the Weyl group W ( E ) .So, modulo the symmetry under W ( E ) which leaves X and Y invariant, the automorphisms in Aut provide one more non-trivial automorphism (sending X to − X and Y to Y ). Modulo W ( E ) ,this additional symmetry is exactly the central symmetry r of the preceding remark. For example, itis easy to check that the composition r ◦ τ corresponds to the element of W ( E ) which is the longestelement of the subsystem of type A × A × A . A connection between A gen and the Racah or Hahn algebras As mentioned in Section 2, special cases of the algebra A gen cover the known Racah and Hahnalgebras. These special cases correspond to taking some of the central parameters to be 0. Inthis section, we offer a somewhat different connection between the algebra A gen and the Racah orHahn algebras. Namely we explain that the defining relations of A gen are satisfied by some generaloperators of Heun type in the Racah algebra and also in the Hahn algebra. This suggests someconnections of the algebra A gen with the Heun–Racah [6] and the Heun–Hahn algebras [22].Due to several free parameters in the Heun operators, many possible values of the central pa-rameter of A gen can be realised this way. Nevertheless we do not investigate here the possibility ofrealising A gen with arbitrary central parameters in the Racah or Hahn algebras.6.1. Racah algebra. The Racah algebra R is the algebra generated by the following elements[12]: R , R , R and d, e , e , Γ , CALABI–YAU ALGEBRA WITH E SYMMETRY AND THE CLEBSCH–GORDAN SERIES OF sl (3) 21 with the defining relations:(42) d, e , e , Γ are central ,[ R , R ] = R , [ R , R ] = R + { R , R } + dR + e , [ R , R ] = R + { R , R } + dR + e , Γ = { R , R } + { R , R } + R + R + R + ( d + 1) { R , R } + (2 e + d ) R + (2 e + d ) R . The following polynomials in terms of the generators of the Racah algebra R :(43) A = z + z R + z R + 2 R , B = z + z ( z + z )2 R + z ( z + z )2 R + z R + 2 { R , R } , satisfy relations (1) of the algebra A gen with the parameters a = − a ′ = − a , a , a , a , a , a , a and a functions of z , z , z , z , z , δ, ǫ , ǫ and Γ. Theseexpressions are obtained by a straightforward computation. We provide their explicit expressionsin Appendix A.The generators A and B given by (43) are the Heun–Racah operators. The so-called Heun–Racahalgebra is realised by taking A or B as one generator and for the other, one of the generators of theRacah algebra R or R . The above result means that the algebra A gen can be obtained from twodifferent Heun–Racah operators.6.2. Hahn algebra. The Hahn algebra H [23, 22] is the algebra generated by the followingelements: H , H , H and δ, ǫ , ǫ , Λ , with the defining relations: δ , δ , ǫ , ǫ , Λ are central ,(44) H = [ H , H ] , (45) [ H , H ] = − { H , H } + δ H + δ H + ǫ , (46) [ H , H ] = 2 H − δ H + H + ǫ , (47) H = Λ + 2 { H , H } − δ { H , H } + 2 ǫ H − (4 + δ ) H + H − ǫ − δ ) H . (48)Let us define the following Heun–Hahn operator by(49) A = z + z H + z H + z H + z { H , H } . We define also the operator B as follows B = z + z H + z H + z H + z { H , H } + z H + z H H + z H H H , (50) where z = ∓ δ ∓ ( z + 1)( z − z + 1) , z = ± , (51) z = ǫ (cid:18) δ ± z (cid:19) δ δ z + 1)(2 z ± z − − z z + 1)( z ∓ z + 1) , (52) z = 14 (2 z ∓ z ∓ z ∓ − z ± δ z , z = z z ∓ z ∓ ± δ z , (53) z = δ ∓ z , z = δ − z ( z + 1) , z = − − z . (54)The elements A and B satisfy relations (1) of the algebra A gen with the parameters a = − a ′ = − a = 0 and the other parameters a , a , a , a , a , a and a functions of z , z , z , z , δ , δ , ǫ , ǫ and Λ. In Subsection 6.1, these functions are obtained by a straightforward but lengthy computa-tion. In this case they are quite complicated and are not provided.The element A given by (49) is the Heun–Hahn operator whereas the element B is a generalisation(we recover the Heun–Hahn operator for z = z = z = 0). The so-called Heun–Hahn algebra isrealised by A and one of the generators of the Hahn algebra that is H or H .7. Conclusion This study has unraveled elegant algebraic structures associated to the Clebsch–Gordan series(or the Littlewood–Richardson coefficients) of sl (3) which suggest various fascinating perspectives.Let us mention a few to conclude. It should be possible to throw more light on the solution of thecorresponding “missing label problem” by exploiting the connection with quadratic algebras andtheir representations. The Bethe ansatz could be brought to bear on the diagonalisation of thegenerators of Z ( sl (3)), the labelling operators, viewed as Hamiltonians. This will be the subjectof a forthcoming publication.Besides, the results presented here provide the groundwork to identify the full algebra associatedto the Clebsch–Gordan problem for sl (3) which should have for generators the complete set ofoperators labelling both the direct product states and the recoupled ones. The determination ofthis algebra would provide for sl (3) the analogue of the Hahn algebra attached to the Clebsch–Gordan problem for sl (2).The Calabi–Yau properties of the algebra A gen and their applications deserve further scrutiny.Also, the symmetry of the specialised diagonal centraliser of sl (3), Z ( sl (3)), under the action of theWeyl group of E lifts the veil over striking facts that stand to significantly inform the representationtheory of potential algebras such as those of Racah and Askey–Wilson. It should be recalled thatthe particular case of A gen whose quotient is Z ( sl (3)), had also been found in connection with thelabelling of representation basis vectors associated to the chain su (3) ⊃ o (3) ⊃ o (2). This is awaitingan analysis similar to the one carried here. Many other generalisations also come to mind. On theone hand there are additional well known missing label problems such as the ones correspondingto the subalgebra chains: su (4) ⊃ su (2) ⊕ su (2), o (5) ⊃ su (2) ⊕ u (1) , o (5) ⊃ o (3) , g ⊃ o (3) etc.What is the structure of the corresponding centraliser? What symmetries will operate? On theother hand, the same questions can be asked when considering diagonal embeddings in multifoldtensor products. We plan on examining these issues in the future. CALABI–YAU ALGEBRA WITH E SYMMETRY AND THE CLEBSCH–GORDAN SERIES OF sl (3) 23 Appendix A. Parameters of A gen for the realisation in terms of Heun–Racahoperators The generators A and B given by (43) satisfy relations (1) with the parameters given by a = − ,a = 4 d + 2( z z − d − e − e ) − z ( z + z ) − z z + 12 z ,a = − ( z + z + 3 z + 3 z z ) − z ( z + z ) + 2 d − ,a = − z + 2 z + 3 z ) z − dz + ( z + 3 z z + 4 z z + z + 4 z z + 3 z + 4) z +(2 z − z z ( z + z ) − z z z ) d + 8( z + z ) e + 8( z + z ) e − d z ,c = 2(2 z + 2 z + 3 z ) z z − z + z ) e z − z + z ) e z + 16( e + e ) z − e e − z +(2 d − − ( z + z + 3 z z + 3 z ) − z + z ) z ) z + dz z z ( z + z )+ d ( z z − z )( z z + 2 d − 2) + ( z − z z − z + 4 d ) e + ( z − z z − z + 4 d ) e +2Γ( z − z z + z − d ) ,a = 2 z + 2 z + 3 z ,a ′ = − ,a = − ( z + z )(3 z z + 8) − z + 3 dz + 6 z − z − ( z − z z − z ) z − ( z + z ) z ,a = ( z − z z ( z + z z + z z + z z ) − z z ) d − ( z + 3 z z + z + 4( z + z ) z + 3 z + 4) z +2( z + 4 z z + 2 z + 4) e + (( z + z + 3 z z + z ) z + z z + z z ) z +2( z + 4 z z + 2 z + 4) e − z − d z − dz z + 4 dz +((3 z + 4)( z + z ) + 6 z ) z + 8Γ ,a = d z z − dz z − z ( z − z + z )( z − z ) e + z ( z − z + z )( z − z ) e − d z z + z ( z − de + z ( z − de + 8( z + z ) e z + 8( z + z ) e z − e e z − ( ( z + 3 z z z + z z + z z + z z + z z ) + 3 z + z ( z + z ) + 2( z + z )) z − (2 z + 2 z + 3 z ) z + 2 z + 3 dz z + ( z z ( z z + z z + z + z z ) + 2 z z − z ) dz − z + 4 z z + 2 z + 4) e z − z + 4 z z + 2 z + 4) e z − z z ( z + ( z + z )) dz +( z + 3 z z + 4 z z + z + 4 z z + 3 z + 4) z z + 2 dz z +Γ( z z ( z + z ) − z + ( z − z z + z − d ) z ) . Acknowledgements. The authors warmly thanks Rupert Yu and Alexei Zhedanov for stimu-lating discussions. N.Cramp´e and L.Poulain d’Andecy are partially supported by Agence Nationalde la Recherche Projet AHA ANR-18-CE40-0001. L.Poulain d’Andecy is grateful to the Centre deRecherches Math´ematiques (CRM) for hospitality and support during his visit to Montreal in thecourse of this investigation. The research of L.Vinet is supported in part by a Discovery Grant fromthe Natural Science and Engineering Research Council (NSERC) of Canada. References [1] P. Baseilhac, S. Tsujimoto, L. Vinet, and A. Zhedanov, The Heun–Askey–Wilson Algebra and the Heun Operatorof Askey–Wilson Type , Annales Henri Poincar´e 20 (2019) 3091–3112.[2] G. Bellamy, D. Rogalski, T. Schedler, J. T. Stafford, and M. Wemyss, N oncommutative algebraic geometry,Cambridge University Press, 2016. [3] A. Berele, and J.R. Stembridge, Denominators for the Poincar´e series of invariants of small matrices , Israel J.Math. 114 (1999) 157–175.[4] R. Berger, and R. Taillefer, Poincar´e–Birkhoff–Witt deformations of Calabi–Yau algebras , J. Noncommut. Geom.1 (2007) 241–270.[5] G. Bergman, The diamond lemma for ring theory , Adv. Math. 29 (1978), 178–218.[6] G. Bergeron, N. Crampe, S. Tsujimoto, L. Vinet, and A. Zhedanov, The Heun–Racah and Heun–Bannai–Itoalgebras, arXiv:2003.09558 .[7] R. Campoamor-Stursberg, Some empirical formulae for the degeneracy separation in the Clebsch–Gordan prob-lem of su (3) , J. Phys.: Conf. Ser. 1194 (2019) 012019.[8] N. Crampe, L. Poulain d’Andecy, and L. Vinet, Temperley–Lieb, Brauer and Racah algebras and other central-izers of su (2) , Accepted in Trans. A.M.S. and arXiv:1905.06346 .[9] V. Drensky, Computing with matrix invariants , arXiv:math/0506614 .[10] P. Etingof, and V. Ginzburg, Noncommutative del Pezzo surfaces and Calabi–Yau algebras , J. Eur. Math. Soc12 (2010) 1371–1416.[11] E. Formanek, The ring of generic matrices , J. Algebra 258 (2002) 310–320.[12] V. X. Genest, L. Vinet, and A. Zhedanov, Superintegrability in two dimensions and the RacahWilson algebra ,Lett. Math. Phys. 104 (2014) 931–952.[13] V. Ginzburg, C alabi-Yau algebras, arXiv: math/0612139 .[14] Ya. A. Granovskii, and A.S. Zhedanov, Nature of the symmetry group of the 6j-symbol, JETP 67 (1988) 1982–1985.[15] Ya. I. Granovskii, and A. S. Zhedanov, Hidden Symmetry of the Racah and Clebsch-Gordan Problems for theQuantum Algebra sl q (2), hep-th/9304138 .[16] F.A. Gr¨unbaum, L. Vinet, and A. Zhedanov. Algebraic Heun operator and band-time limiting, Comm. Math.Phys. 364. (2018) 1041–1068.[17] B.R. Judd, W. Miller, J. Patera, and P. Winternitz, Complete sets of commuting operators and O (3) scalars inthe enveloping algebra of SU (3) , J. Math. Phys. 15 (1974) 1787–1799.[18] Z. Pluhaˇr, Yu.F. Smirnov, and V.N. Tolstoy, Clebsch–Gordan coefficients of SU (3) with simple symmetryproperties, J. Phys. A 19 (1986) 21.[19] C. Procesi, The invariant theory of n × n matrices , Adv. Math. 19 (1976) 306–381.[20] G. Racah, Lectures on Lie Groups in “Group Theoretical Concepts and Methods in Elementary Particle Physics”(Istanbul Summer School, 1962).[21] Yu.P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero (Russian), Izv. Akad.Nauk SSSR, Ser. Mat. 38 (1974) 723–756. Translation: Math. USSR, Izv. 8 (1974) 727–760.[22] L. Vinet, and A. Zhedanov, The Heun operator of Hahn-type, Proc. Amer. Math. Soc. 147 (2019) 2987–2998 .[23] A. Zhedanov, Hidden symmetry algebra and overlap coefficients for two ring-shaped potentials, J. Phys. A 26(1993) 4633. † Institut Denis-Poisson CNRS/UMR 7013 - Universit´e de Tours - Universit´e d’Orl´eans, Parc deGrandmont, 37200 Tours, France. E-mail address : [email protected] ‡ Laboratoire de math´ematiques de Reims UMR 9008, Universit´e de Reims Champagne-Ardenne,Moulin de la Housse BP 1039, 51100 Reims, France. E-mail address : [email protected] ∗ Centre de recherches math´ematiques, Universit´e de Montr´eal, P.O. Box 6128, Centre-villeStation, Montr´eal (Qu´ebec), H3C 3J7, Canada. E-mail address ::