Last update: 22 December 2013
Let be transcendental (so that we may view it as a variable when necessary). The affine Hecke algebra is the quotient of the group algebra of the affine braid group by the relations The affine Hecke algebra is an infinite dimensional algebra with a very interesting representation theory (see [KLu0862716] and [CGi1997]). With as in (3.4) the subalgebra is a commutative subalgebra of It is a theorem of Bernstein and Zelevinsky (see [RRa0401322, Theorem 4.12]), that the center of is the ring of symmetric (Laurent) polynomials in If define if is a reduced word for in terms of the generating reflections Then, with as in (3.4)
Let The cyclotomic Hecke algebra with parameters is the quotient of the affine Hecke algebra by the relation The algebra is a deformation of the group algebra of the complex reflection group and is of dimension It was introduced by Ariki and Koike [Ari1996] and its representations and its connection to the affine Hecke algebra have been well studied ([Ari1996],[Ari1996],[Gec1998]).
Fix and an infinite number of values in The affine BMW (Birman-Murakami-Wenzl) algebra is the quotient of the group algebra of the affine braid group by the relations
The cyclotomic BMW algebras have been defined and studied by [Här1673464], [Här9712030], [Här1834081] and, for all practical purposes the affine BMW algebras appear in these papers. The cyclotomic BMW algebras are quotients of the affine BMW algebras in the same way that cyclotomic Hecke algebras are quotients of affine Hecke algebras. The classical BMW algebras have been studied in [Wen1988], [HRa1995], [Mur1990], [LRa1977] and many other works. The “degenerate” version of the affine BMW algebras was defined by Nazarov [Naz1996] who called them “degenerate affine Wenzl algebras”. The relation between his algebras and the affine BMW algebras is analogous to the relation between the graded Hecke algebras (sometimes called the degenerate affine Hecke algebras) and the affine Hecke algebras (see [Lus1989]).
Goodman and Hauschild [GHa0411155] have proved that elements of the affine BMW algebra can be viewed as linear combinations of affine tangles. An affine tangle has
strands and a flagpole just as in the case of an affine braid, but there is no restriction that a strand must connect an upper vertex to a
lower vertex. Let and
In order to explicitly compute the representations of affine and cyclotomic Hecke algebras and BMW algebras which are obtained by applying the functors we need to fix notations for working with the representations of finite dimensional complex semisimple L ie algebras of classical type.
Let be a complex semisimple Lie algebra of type or and let be the corresponding Drinfeld-Jimbo quantum group. Use the notations in [Bou1968, p. 252-258], for the root systems of types and so that are orthonormal (in type also include the fundamental weights are given by and the finite dimensional modules are indexed by dominant integral weights where and, in type the sum is over instead of
Identify with the configuration of boxes which has boxes in row If put boxes in row and mark them with signs. For example and
If is a box in position of the content of is If then where is the box at the end of row in Note that may be a Also, in types and Using these formulas can easily be computed for all dominant integral weights For example
For all dominant integral weights in type and we have where the sum over is a sum over all partitions (of length obtained by adding a box to and the sum over denotes a sum over all dominant weights obtained by adding or removing a box from In type addition and removal of a box should include the possibility of addition and removal of a box marked with a sign, and removal of a box from row when changes to
Let be the simple complex Lie algebra of classical type, the corresponding quantum group and let be the irreducible module of highest weight For each let be the affine braid group representation defined in Proposition 3.1.
|(a)||If is type then is a representation of the affine Hecke algebra with (In this Type case use a different normalization of the map and set|
|(b)||If is type and if where is a dominant integral weight then is a representation of the cyclotomic Hecke algebra for any (multi)set of parameters containing the (multi)set of values as runs over the addable boxes of|
|(c)||If is type or and is a highest weight module then there are unique values depending only on the central character of such that is a representation of the affine BMW algebra with parameters|
|(d)||If is type or and where is a dominant integral weight then is a representation of the cyclotomic BMW algebra with and as in (c), and any (multi)set of parameters containing the (multi)set of values as runs over the dominant integral weights appearing in the decomposition (6.16) of Here where and are as defined in (6.13) and (6.14), respectively.|
(a) It is only necessary to show that satisfies for This is proved in [LRa1977, Prop. 4.4].
(c) The arguments establishing the relations (6.3a–c) in the definition of the affine BMW algebra are exactly as in [LRa1977, Prop. 5.10]. It remains to establish (6.3d–e). The element in the affine BMW algebra acts on as where is the unique projection onto the invariants in and is as in (6.5). Using the identity (6.9) and the pictorial equalities it follows that acts as By (2.11), this is equal to establishing the relation in (6.3e).
Since acts as on the morphism is a morphism from Since is a highest weight module this morphism is for some By the results of Drinfeld [Dri1990] and Reshetikhin [Res1990] (see [Bau1620662, p. 250]), the action of the morphism corresponds to the action of a central element of on Thus the constant depends only on the central character of
(b) Let be the addable boxes of and consider the action of on We will show that satisfies the relation where By (2.11) and (2.14) it follows that where the sum is over all partitions obtained by adding a box to is the projection onto in the tensor product and is the content of the box which is added to to get Thus is a diagonal operator with eigenvalues and so it satisfies the equation (6.2).
(d) Using the appropriate case of the decomposition rule for the proof of the relation is as in (b). The values of are determined from (6.16). To compute the value of note that in the notations of the proof of Theorem 5.1. Thus is determined by the formula in Theorem 5.1 and the decomposition of in (6.14).
Remark. The parameters in needed in Theorem 6.1c can be determined by using the formula of Baumann [Bau1620662, Theorem 1], which characterizes in terms of the values given in (5.6). To do this it is necessary to use formula (5.6) for several times: is always the highest weight of but many different will be needed. Note that the proof of the formula (5.6) for in [TWe1217386] does not require to be dominant integral.
The following theorem provides an analogue of Schur-Weyl duality for the affine Hecke algebras, cyclotomic Hecke algebras, affine BMW algebras and cyclotomic BMW algebras. Alternative Schur-Weyl dualities have been given by Chari-Pressley [CPr1996] for the case of affine Hecke algebras and by Sakamoto and Shoji [SSh1999] for cyclotomic Hecke algebras. Cherednik [Che1987] also used a Schur-Weyl duality for the affine Hecke algebra which is different from the Schur-Weyl duality given here.
Assume that is not of type Let be a dominant integral weight and let In each of the cases given in Theorem 6.1 the representation is surjective.
Part (a) is a consequence of (b) since the representation of in (a) is the composition of the representation from (b) with the surjective algebra homomorphism coming from the definition of Similarly part (c) is a consequence of part (d). The proof of the surjectivity of the representation in Theorem 6.1b and Theorem 6.1d are exactly the same as the proofs of [LRa1977, Cor. 4.15], and [LRa1977, Cor. 5.22], respectively. The case considered there is the case but all the arguments there generalize verbatim to the case when is an arbitrary dominant integral weight. In [LRa1977, §4], the elements in the affine braid group are denoted The assumption in [LRa1977] is unecessary for this theorem if the full decomposition rule given in (6.14) is used.
The main point is that the eigenvalues of separate the components of the decomposition of By induction it is sufficient to check that the eigenvalues of distinguish the components of for all By (2.10), (2.11) and (2.14), the eigenvalues of are of the form where is a dominant integral weight is as in Theorem 6.1d and runs over the components in the decomposition (6.16) of Different addable boxes for can never have the same content since they cannot be in the same diagonal. Similarly for two different removable boxes. Let be an addable box and a removable box for Unless is type and and are in row we have Thus, when is not of type and so the two eigenvalues coming from these boxes are different.
Let denote the affine Hecke algebra, the cyclotomic Hecke algebra, the affine BMW algebra or the cyclotomic BMW algebra corresponding to the case of Theorem 6.1 which is being considered. Then, as in the classical Schur-Weyl duality setting, Theorem 6.2 implies that as bimodules where is the irreducible of highest weight and is the irreducible module defined by (4.1).
The irreducible modules appearing in (6.17) can be constructed quite explicitly. All the necessary computations for doing this have already been done in [LRa1977, §4 and 5], which does the case All the arguments in [LRa1977, §4 and 5], generalize directly to the case when is an arbitrary dominant integral weight. The final result is Theorem 6.3 below. The result in part (a) of Theorem 6.3 is due to Cherednik [Che1987].
If and are partitions such that the skew shape is the configuration of boxes of in which are not in Let be a skew shape with boxes. A standard tableau of shape is a filling of the boxes of with such that
|(a)||the rows of are increasing (left to right), and|
|(b)||the columns of are increasing (top to bottom).|
For example, is a standard tableau of shape
For any two partitions and an up down tableau of length from to is a sequence of partitions such that
Let be a skew shape with boxes. Then the module
for the affine Hecke algebra is irreducible and is given by
(so that the symbols are a
action given by
is the constant
denotes the content of the box
is the box containing in
is the same filling as except and are switched, and
if is not a standard tableau.
|(b)||Let be a pair of partitions. Then the module for the affine BMW algebra is irreducible and is given by (so that the symbols are a of with action given by where both sums are over up-down tableaux that are the same as except possibly at the step and and in type and and in type|
If where is a dominant integral weight and then each of the representations (where is the affine Hecke algebra, the cyclotomic Hecke algebra, the affine BMW algebra, or the cyclotomic BMW algebra) gives rise to a Markov trace via Theorem 5.1. The parameters and the weights of these Markov traces are given by Theorems 5.1 and 5.3.
In type A case is a skew shape with boxes and the parameters and the weights of (most of) these traces have been given in terms of partitions in [GIM2000]. In [GIM2000], is a partition of a special form ([GIM2000, 2.2(*)]), and so, in their case, the skew shape can be viewed as an of partitions. Their formulas can be recovered from ours by rewriting the quantum dimension from (5.3) in terms of the partition as in [Mac1995, I §3 Ex. 1]: where, if is the box in position of then is the hook length at and for a positive integer Thus, the first formula in [GIM2000, §2.3], coincides with and so the formula for the weights of the Markov trace on cyclotomic Hecke algebras which is given in [GIM2000, Prop. 2.3], coincides exactly with the formula in Theorem 5.3. To remove the constants that come from the difference between and the affine braid group action in Theorem 6.1a should be normalized so that and Then, from Theorem 5.1, (6.16) and (6.18) it follows that the parameters of the Markov trace are and where, in the last expression, the first product is over boxes which are in the same row as the added box and the second product is over which are in the same column as Then cancellation of the common terms in the numerator and denominators of each product yields the combinatorial formulas for the parameters of the Markov traces on cyclotomic Hecke algebras which are given in [GIM2000, Thm. 2.4].
Lambropoulou [Lam1999, §4], has proved that there is a unique Markov trace on the affine Hecke algebra with a given choice of parameters A similar result is true for the affine BMW algebra.
For each fixed choice of parameters and there is a unique Markov trace on the affine BMW algebra
Consider the image of an affine braid in the affine BMW algebra. The Markov trace of this braid can be viewed pictorially as the closure of the braid Consider a string in the closure as it winds around the other strings and the pole. If the string crosses another string twice without going around the pole between these two crossings then we can use the relation to rewrite the closed braid as a linear combination of closed braids with fewer crossings between strings. By successive steps of this type we can reduce the computation of the Markov trace of a braid to a linear combination of
Remark. For computations it is helpful to note that
The original construction of the irreducible representations of the affine Hecke algebra of type A is due to Zelevinsky [Zel1980] and is an analogue of the Langlands construction of admissible representations of real reductive Lie groups. Zelevinsky used the combinatorics of multisegments which is easily seen to be equivalent to the combinatorics of unipotent-semisimple pairs used later in [KLu0862716] (see [Ari1996]). Here we show how the construction of affine Hecke algebra representations via the functors naturally matches up with Zelevinsky’s indexings by multisegments. Using the multisegment indexing of representations, Theorem 6.6 below explicitly matches up the decomposition numbers for affine Hecke algebras with Kazhdan-Lusztig polynomials. Recall that the functor gives representations of the affine Hecke algebra in the setting of Theorem 6.1a when is of type and is the fundamental representation.
Consider an (infinite) sheet of graph paper which has its diagonals labeled consecutively by The content of a box on this sheet of graph paper is (a natural generalization of the definition of in (6.15)). A multisegment is a collection of rows of boxes (segments) placed on graph paper. We can label this multisegment by a pair of weights and by setting For example (the numbers in the boxes in the picture are the contents of the boxes). The construction forces the condition
|(a')||is a weight of where is the number of boxes in|
|(b')||is integrally dominant,|
|(c')||with integrally dominant and maximal length in the coset|
Let be a multisegment with boxes and number the boxes of from left to right (like a book). Define to be the subalgebra of generated by so that is the “parabolic” subalgebra of corresponding to the multisegment Define a one-dimensional module by setting for and such that is not at the end of its row.
Let be of type and let be the functor from the setting of Theorem 6.1a, where The standard module for the affine Hecke algebra is as defined in (4.1). It follows from the above discussion that these modules are naturally indexed by multisegments The following proposition shows that this standard module coincides with the usual standard module for the affine Hecke algebra as considered by Zelevinsky [Zel1980] (see also [Ari1996], [CGi1997] and [KLu0862716]).
Let be a multisegment determined by a pair weights with integrally dominant. Let be the one dimensional representation of the parabolic subalgebra of the affine Hecke algebra defined in (6.21). Then
To remove the constants that come from the difference between and the affine braid group action in Theorem 6.1a should be normalized so that and
By Proposition 4.2a, as a vector space. Let be the standard basis of with If we let the symmetric group act on by permuting the tensor factors then where and is the parabolic subgroup of which stabilizes the vector This shows that, as vector spaces, are isomorphic.
For notational purposes let and let be the image of in Since is integrally dominant and has weight it must be a highest weight vector. We will show that acts on by the constant where is the content of the box of the multisegment (read left to right and top to bottom like a book).
Consider the projections where and acts as the identity on the last factors of Then and for each (the first components of) form a highest weight vector of weight in It is the “highest” highest weight vector of with respect to the ordering in Lemma 4.1 and thus it is deepest in the filtration constructed there. Note that the quantum Casimir element acts on the space in (6.24) as the constant times a unipotent transformation, and the unipotent transformation must preserve the filtration coming from Lemma 4.1. Since is the highest weight vector of the smallest submodule of this filtration (which is isomorphic to a Verma module by Lemma 4.1b) it is an eigenvector for the action of the quantum Casimir. Thus, by (2.11) and (2.14), acts on by the constant (see [LRa1977]). Since commutes with for this also specifies the action of on
The explicit for this case of type and is well known (see, for example, the proof of [LRa1977, Prop. 4.4]) and given by Since acts by on the and tensor factors of and commutes with the projection it follows that if is not a box at the end of a row of This analysis of the action of on shows that there is an This map is surjective since is generated by (the action on generates all of Finally, (6.23) guarantees that it is an isomorphism.
In the same way that each weight has a normal form every multisegment has a normal form The element in the normal form of can be constructed combinatorially by the following scheme. We number (order) the boxes of in two different ways.
First ordering: To each box of associate the following triple where, if a box is the leftmost box in a row “the box to its left” is the rightmost box in the same row. The lexicographic ordering on these triples induces an ordering on the boxes of
Second ordering: To each box of associate the following pair The lexicographic ordering of these pairs induces a second ordering on the boxes of
If is the permutation defined by these two numberings of the boxes then For example, for the multisegment displayed in (6.20) the numberings of the boxes are given by and the normal form of is and where Let be of type and and let as defined in (4.1). It is known (a consequence of Proposition 6.5 and Proposition 4.2c) that is always a simple or Furthermore, all simple modules are obtained by this construction. See [Suz1998] for proofs of these statements. The following theorem is a reformulation of Proposition 4.5 in terms of the combinatorics of our present setting.
Let and be multisegments with boxes (with and assumed to be integral) and let be their normal forms. Then the multiplicities of in a Jantzen filtration of are given by where is the Kazhdan-Lusztig polynomial for the symmetric group
Theorem 6.6 says that every decomposition number for affine Hecke algebra representations is a Kazhdan-Lusztig polynomial. The following is a converse statement which says that every Kazhdan-Lusztig polynomial for the symmetric group is a decomposition number for affine Hecke algebra representations. This statement is interesting in that Polo [Pol1999] has shown that every polynomial in is a Kazhdan-Lusztig polynomial for some choice of and permutations Thus, the following proposition also shows that every polynomial arises as a generalized decomposition number for an appropriate pair of affine Hecke algebra modules.
Let and Then, for each pair of permutations the Kazhdan-Lusztig polynomial for the symmetric group is equal to
Since and are both regular, and the standard and irreducible modules and range over all Thus, this statement is a corollary of Proposition 4.5.
This is a typed version of the paper Affine Braids, Markov Traces and the Category by Rosa Orellana and Arun Ram*.
*Research supported in part by National Science Foundation grant DMS-9971099, the National Security Agency and EPSRC grant GR K99015.
This paper is a slightly revised version of a preprint of 2001. We thank F. Goodman, A. Henderson and an anonymous referee for their very helpful comments on the original preprint.