Invariants for links from classical and affine Yokonuma–Hecke algebras
We present a construction of invariants for links using an isomorphism theorem for affine Yokonuma–Hecke algebras. The isomorphism relates affine Yokonuma–Hecke algebras with usual affine Hecke algebras. We use it to construct a large class of Markov traces on affine Yokonuma–Hecke algebras, and in turn, to produce invariants for links in the solid torus. By restriction, this construction contains the construction of invariants for classical links from classical Yokonuma–Hecke algebras. In general, the obtained invariants form an infinite family of 3-variables polynomials. As a consequence of the construction via the isomorphism, we reduce the number of invariants to study, given the number of connected components of a link. In particular, if the link is a classical link with components, we show that invariants generate the whole family.
Invariants for links from classical and affine Yokonuma–Hecke algebras
1. The Yokonuma–Hecke algebras (of type GL), denoted , have been used by J. Juyumaya and S. Lambropoulou to construct invariants for various types of links, in the same spirit as the construction of the HOMFLYPT polynomial from usual Hecke algebras. We refer to  and references therein. In particular, the algebras provide invariants for classical links and the natural question was to decide if these invariants were equivalent, or not, to the HOMFLYPT polynomial. This study culminated in the recent discovery  that these invariants are actually topologically stronger than the HOMFLYPT polynomial (i.e. they distinguish more links).
In , another approach to study invariants coming from Yokonuma–Hecke algebras was developed. The starting point was the fact that the algebra is isomorphic to a direct sum of matrix algebras with coefficients in tensor products of usual Hecke algebras. This allowed an explicit construction of Markov traces on from the known Markov trace on Hecke algebras (on Hecke algebras, there is a unique Markov trace up to normalisation, and it gives the HOMFLYPT polynomial). In addition to its usefulness for the construction of Markov traces, the approach via the isomorphism also helps to study the resulting invariants. Indeed some properties of the invariants follow quite immediately from a precise understanding of the isomorphism (see paragraph 4 below).
Independently of which approach is used, another ingredient was added in : a third parameter in the invariants. While the first two parameters come from the algebra , this third parameter has its origin in the framed braid group, and corresponds to a certain degree of freedom one has when going from the framed braid group to the algebra . More precisely, we can deform the standard surjective morphism from the framed braid group algebra to its quotient into a family of morphisms (depending on ) respecting the braid relations and the Markov conditions. Another way of interpreting the parameter is that it modifies the quadratic relation satisfied by the generators of . Its existence explains (or is reflected in) the fact that different presentations for were used before. Juyumaya–Lambropoulou invariants correspond to certain specialisations of this parameter , depending on the chosen presentation. So the parameter unifies every possible choices and yields more general invariants. It is indicated in [3, Remark 8.5] that changing the presentation seems to give a non-equivalent topological invariant.
2. In this paper, we consider the affine Yokonuma–Hecke algebras (of type GL), denoted . They were introduced in  in connections with the representation theory and the Jucys–Murphy elements of the classical Yokonuma–Hecke algebras. Our main goal here is to generalise for the whole approach to link invariants via the isomorphism theorem. The invariants are in general for links in the solid torus. The classical links are naturally contained in the solid torus links and, restricted to them, the obtained invariants correspond to the invariants obtained in  from (naturally seen as a subalgebra of ). Specialising the parameter , we identify the Juyumaya–Lambropoulou invariants among them. For those invariants, we emphasize that we recover some known results  by a different method and furthermore obtain some new results already in this particular case.
We start with an isomorphism between the algebra and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras. As done in , the isomorphism can be proved repeating the same arguments as for (see  where the proof for is presented, as a particular case of a more general result by G. Lusztig [8, §34]). Here we sketch a short different proof for using the known result for . We also prove the analogous theorem for the cyclotomic quotients of (with Ariki–Koike algebras replacing affine Hecke algebras). Useful for concrete use, the formulas for the generators are simple and given explicitly.
Concerning links, S. Lambropoulou constructed invariants, analogues of the HOMFLYPT polynomial, for links in the solid torus from affine Hecke algebras . Then, it was explained in  how to obtain invariants for those links from the algebras , unifying the methods of J. Juyumaya and S. Lambropoulou for and the construction of S. Lambropoulou for affine Hecke algebras. Due to the recent results of , it is expected that the invariants obtained from are stronger than the ones obtained from affine Hecke algebras.
Here we follow the alternative approach which uses the isomorphism to construct Markov traces on the family of algebras . To sum up, the Markov traces are constructed and can be calculated with the following steps: for an element of , apply first the isomorphic map to obtain an element of the direct sum of matrix algebras; then, for each matrix, apply the usual trace which results in an element of a tensor product of affine Hecke algebras; finally apply a tensor product of Markov traces on affine Hecke algebras. Our result consists in obtaining the compatibility conditions relating the Markov traces appearing in different matrix algebras so that the preceding procedure eventually results in a genuine Markov trace on .
With the definition used here, for a given , the set of Markov traces on forms a vector space. From the isomorphism, a set of distinguished Markov traces appears naturally, which spans the set of all Markov traces constructed here. Thus, our study of Markov traces (and of invariants) is reduced to the study of these “basic” Markov traces (and of the corresponding “basic” invariants). It turns out that these basic Markov traces are indexed, for a given , by the non-empty subsets together with a choice, denoted formally by , of arbitrary Markov traces on affine Hecke algebras. We note that if we restrict to , the parameter disappears and the basic Markov traces on are indexed, for a given , only by the non-empty subsets . This recovers a result of .
3. Throughout the paper, we intended to give in details the connections between the two approaches, so that one would be able to pass easily from one to the other. This will allow in particular to specialise and translate all our results on the invariants to Juyumaya–Lambropoulou invariants as well.
Roughly speaking, J. Juyumaya and S. Lambropoulou constructed invariants from in two steps . The same approach was followed in  for . First a certain trace map, analogous to the Ocneanu trace and satisfying a certain positive Markov condition, was constructed. Then a rescaling procedure was implemented, in order to produce genuine invariants. The rescaling procedure amounts to two things: a renormalisation of the generators and a renormalisation, depending on , of the trace. In the approach presented here, the first step is included from the beginning in a more general quadratic relation for the generators. The second step is already included in the definition of a Markov trace, namely that it is a family, on , of trace maps satisfying the two Markov conditions. As a consequence, to obtain invariants here, one directly applies the Markov trace and no rescaling procedure is needed.
For the comparison, our first task is to explain that Juyumaya–Lambropoulou approach is equivalent to considering certain Markov traces (with the definition used here) and to relate their variables with the parameters considered here. Then we need to identify these Markov traces in terms of the ones constructed via the isomorphism theorem. We obtain finally the explicit decomposition of these Markov traces in terms of the basic Markov traces indexed by and as above.
In particular, for , this results in an explicit formula for the Juyumaya–Lambropoulou invariants, as studied in , in terms of the basic invariants constructed here. We note that, in this case, the parameter is not present, and that Juyumaya–Lambropoulou invariants are also parametrised, for a given , by non-empty subsets of . Nevertheless, they do not coincide with the basic invariants and the comparison formula is not trivial (see Formulas (5.5) in Section 5). In general, for , we obtain the expression of the invariants constructed in  in terms of the basic invariants constructed here.
Concerning the third parameter , we recall that it was not present in the previous approach. Actually, one need to specialise it to a certain value in our invariants to recover the Juyumaya–Lambropoulou invariants. The two different presentations of that were used, as in , correspond to two different values of that we give explicitly. Similarly, for , the invariants constructed in  correspond to a certain specialisation of .
4. We conclude this introduction by describing the main properties obtained for the invariants. As explained before, they follow quite directly from a precise understanding of the isomorphism, and are expressed easily in terms of the basic invariants defined here. The main results are:
for and a non-empty subset , the corresponding invariants coincide with invariants corresponding to and the full set . Therefore, we only have to consider the full sets for different .
further, given a number of connected components of a link, the invariants corresponding to are zero if . So, given , we only have to consider .
Moreover, with the comparison results explained in paragraph 3, it is easy to deduce the similar properties for invariants obtained via Juyumaya–Lambropoulou approach. The first item remains true as it is. The second item results in an explicit formula expressing, if , the invariants corresponding to in terms of the invariants corresponding to with . Specialising to the appropriate values and restricting to classical links, we recover with the first item a result of . The second item in this case was proved only for also in .
2 Affine Yokonuma–Hecke algebras
Let and and be indeterminates. We work over the ring . The properties of the affine Yokonuma–Hecke algebras recalled here can be found in .
We use to denote the symmetric group on elements, and to denote the transposition . The affine Yokonuma–Hecke algebra is generated by elements
where . The elements are idempotents and we have:
Let and let be a reduced expression for . Since the generators of satisfy the same braid relations as the generators of , Matsumoto’s lemma implies that the following element does not depend on the reduced expression of :
Elements of are defined inductively by
The elements commute with each other. They also commute with the generators and they satisfy if .
For , we set . The following set of elements forms a basis of :
This fact has the following consequences:
Recall that the Yokonuma–Hecke algebra is presented by generators , and defining relations those in (2.1)–(2.3) which do not involve the generator . We have that is isomorphic to the subalgebra of generated by (hence the common names for the generators).
In particular, the commutative subalgebra of generated by is isomorphic to the group algebra of .
By definition, the affine Hecke algebra (of type GL) is . We have, for any , that the quotient of by the relations , , is isomorphic to . We denote by the corresponding surjective morphism from to , and the generators of are denoted .
The subalgebra of generated by is the usual Hecke algebra, denoted . We also have .
2.2 Compositions of
Let be the set of -compositions of , that is the set of -tuples such that . We denote .
For , the Young subgroup is the subgroup of , where acts on the letters , acts on the letters , and so on. The subgroup is generated by the transpositions with .
We denote by the algebra (by convention ). It is isomorphic to the subalgebra of generated by and , with , and is a free submodule with basis .
Similarly, we have a subalgebra of the Hecke algebra . It is naturally a subalgebra of (generated only by , with ).
For , let be the index of the Young subgroup in , that is,
We define the socle of a -composition by
The composition belongs to where is the number of non-zero parts in . We denote by the set of all socles of -compositions, or in other words, is the set of -compositions whose parts belong to . We note that there is a one-to-one correspondence between the set and the set of non-empty subsets of , given by
2.3 characters of
Let be the set of roots of unity of order . A complex character of the group is characterised by the choice of for each . We denote by the set of complex characters of , extended to the subalgebra of .
For each , we denote by the primitive idempotent of associated to . Then the set is a basis of . Therefore, from the basis (2.7) of , we obtain the following other basis of :
Let . For , we let be the number of elements such that . Then the sequence is a -composition of which we denote by .
For a given , we consider a particular character such that . The character is defined by
The symmetric group acts on the set by the formula . The stabilizer of under the action of is the Young subgroup . In each left coset in , there is a unique representative of minimal length. So, for any such that , we define a permutation by requiring that is the element of minimal length such that:
3 Isomorphism Theorems
We present isomorphism theorems for the algebras and their cyclotomic quotients. We sketch a short proof, which uses the corresponding result for (see [5, Section 3.1]). We are still working over .
3.1 Isomorphism theorem for affine Yokonuma–Hecke algebras
For , we consider the algebra of matrices of size with coefficients in . We recall that , given by (2.8), is the number of characters such that . So we index the rows and columns of a matrix in by such characters. Moreover, for two characters such that , we denote by the matrix in with 1 in line and column , and 0 everywhere else.
The affine Yokonuma–Hecke algebra is isomorphic to , the isomorphism being given on the elements of the basis (2.11) by
where , and ( is the length function on ).
Sketch of a proof.
We start with explicit formulas for the images of the generators of given below in (3.2)–(3.4), and we check that the images of the generators satisfy all the defining relations (2.1)–(2.3) of . For the relations not involving the generator , this is already known from the isomorphism theorem for . We omit the remaining straightforward verifications.
Thus, Formulas (3.2)–(3.4) induce a morphism of algebras, and we check that it coincides with given by (3.1). Again, for the images of the elements of the basis of the form , this is already known from the situation. The multiplication by is straightforward.
It remains to check that is bijective. The surjectivity follows from a direct inspection of Formula (3.3) together with the already known fact that every is in the image of . The injectivity can be checked directly. Indeed, assume that a certain linear combination is in the kernel of . Then for every , we obtain that
where the sum is over such that . For satisfying this condition, we have if and only if , and therefore every coefficients in the above sum are . ∎
Formulas for the generators.
Here, we give the images under the isomorphism of the generators of . We recall that in , the sum being over . Let and set . Then, by definition of , it is straightforward to see that , where and .
Let . We have:
It follows that the image of , , is a sum of diagonal matrices; the coefficient in position is 1 if , and 0 otherwise.
Let . We have:
Let . We have:
The first line follows from if . The second line follows from if .
3.2 Isomorphism Theorem for cyclotomic Yokonuma–Hecke algebras
Let be an -tuple of non-zero parameters for a certain (equivalently, one could consider as indeterminates and work over the extended ring ). The cyclotomic Yokonuma–Hecke algebra is the quotient of the affine Yokonuma–Hecke algebra by the relation
It is shown in  that the algebra is a free -module with basis
In particular, if , is isomorphic to the Yokonuma–Hecke algebra .
Similarly, the cyclotomic Hecke algebra (or the Ariki–Koike algebra) is the quotient of the affine Hecke algebra by the relation . Equivalently, it is the quotient of the cyclotomic Yokonuma–Hecke algebra by the relations , . It is a free -module with basis .
For , we set . By definition, is the quotient of the algebra by the relations
The cyclotomic Yokonuma–Hecke algebra is isomorphic to the direct sum .
Let be the (two-sided) ideal of generated by the left hand side of the relation (3.5). For , let be the ideal of generated by the left hand sides of the relations (3.6). The corollary follows from Theorem 3.1 together with the fact that . It remains to check this fact.
The inclusion “” follows at once from Formula (3.3) for . For the other inclusion, let . Let such that , so that there is a character with and . Again, Formula (3.3) for gives . Therefore, for every generators of , we have in a matrix in with the generator as one diagonal element and everywhere else. As is an ideal, this shows that is included in . ∎
4 Markov traces on affine Yokonuma–Hecke algebras
From now on, we extend the ground ring to , and we consider our algebras over this extended ring.
4.1 Definition of Markov traces on and
A Markov trace on the family of algebras is a family of linear functions satisfying:
A Markov trace on the family of algebras is a family of linear functions satisfying:
Recall the definition of from Section 2. For each and each such that , we choose a Markov trace on . By convention, and maps of the form are identities on . Below in (4.3), each term in the sum over acts on . We skip the proof of the following theorem. It can be done exactly as in [5, Lemma 5.4].
The following maps form a Markov trace on :
Roughly speaking, to construct a Markov trace on , after having applied the isomorphism and the usual trace of a matrix, we must choose and apply a Markov trace on each component of for each . This choice of Markov traces is restricted: if and have the same number of non-zero components, the chosen Markov traces must coincide; otherwise, they can be chosen independently.
Basic Markov traces.
Recall the bijection (2.10) between and non-empty subsets of . Following the theorem, we define some distinguished Markov traces as follows:
Choose a non-empty and consider the associated . Choose a Markov trace on for each , and set , , in (4.3).
Then, in (4.3), set all other Markov traces with to be .
We denote formally the choice of Markov traces in the first item by and denote by the resulting Markov trace on . We call it a basic Markov trace. Every Markov traces constructed in the preceding theorem is a linear combination of basic Markov traces , where and vary.
A map on can be seen, up to , as acting on the direct sum of matrix algebras. This way, for a given , the maps are non-zero only on the summands such that , that is, such that if and only if . As examples:
if then is non-zero only on , for with in position . In this case, ;
we will see that it is enough to consider the situation . In this case, is non-zero only on , for with all parts different from 0.
(i) By restriction to the subalgebra of , a Markov trace on reduces to a Markov trace on (and similarly for and ). On , there is a unique Markov trace up to a normalisation factor. Therefore, the choice of the Markov traces in the theorem above reduces, for , to a choice of an overall factor for each . This is the result proved in . In other words, for , the basic Markov traces are parametrised by , or equivalently, by the non-empty subsets of .
(ii) Let be the space of Markov traces on . The space spanned by the basic Markov traces is isomorphic to
If we restrict to a subspace of of dimension , we obtain a space of Markov traces on of dimension . In particular, for , the dimension is . We note that a full description of the space does not seem to be known (and similarly for the cyclotomic quotients others than ).
5 Invariants for links
Let be another indeterminate. We work from now on over the ring and we consider now all algebras over this extended ring .
We sketch a construction of invariants with values in for -framed solid torus links. We refer to [7, 2] for definitions and fundamental results (as the analogues of Alexander and Markov theorems) concerning solid torus links and their -framed versions. Note that any invariant for -framed links is also an invariant of non-framed links, simply by considering links with all framings equal to 0.
The set of classical links is