Last update: 13 March 2014
The Bratteli diagram is given in Fig. 2. The shapes of which are on level are the partitions of A partition is connected by an edge to a partition if can be obtained from by adding a box to or by removing a box from The diagram is a multiplicity free Bratteli diagram. The tableaux in the Bratteli diagram are called up-down tableaux since they are sequences of partitions in which each partition differs from the previous one by either adding or removing a box.
The Bratteli diagram is given by the sets A partition is connected by an edge to a partition if can be obtained by adding or removing a box from The Bratteli diagram is a multiplicity free Bratteli diagram. It can be obtained from the Bratteli diagram by removing all the partitions with more than rows (and the edges connected to them). It is easy to see that tableaux in the Bratteli diagram are up-down tableaux that never pass through a partition of length greater than The Bratteli diagram can be viewed as the limit of the Bratteli diagrams as goes to infinity.
For the remainder of this section let us fix and, unless otherwise specified, all paths and tableaux shall be from the Bratteli diagram
Fix and assume that as partitions. Then there is at most one such that
Given a partition let us write (resp. to denote that is obtained by adding (resp. removing) a box to (resp. from) the row of Fix and assume that as partitions. Suppose that and that where and are either If exists then is given by and The path exists when is a partition and not equal to
The Centralizer Algebras
For the remainder of this section fix to be a complex simple Lie algebra of type or and let be the corresponding quantum group. We shall use the standard notations ([Bou1981], pp. 252-258) for the root systems of Types and so that are an orthonormal basis of and the element is given by where The finite dimensional irreducible representations of which appear as irreducible summands in the tensor powers of of are indexed by the dominant integral weights in the set We shall identify each dominant integral weight with the partition It will be helpful to note that, if then
Let the irreducible of highest weight In type the decomposition rule for tensoring by is given by where the sum is over all partitions that are gotten by adding or removing a box from the partition It follows that in Type the Bratteli diagram for tensor powers of is given by In Types and the tensor product rule given in (5.5) holds whenever but must be modified slightly when In order to avoid this complication
(5.6) For the remainder of this section, we shall assume that in Types and we have that in particular, and whenever the constants and are used,
The weights of the Markov traces on are given by where the quantum dimension of is given by Since all automorphisms of the Dynkin diagram corresponding to fix the node corresponding to the fundamental weight it follows that As in (3.2), define to be the projection onto the invariants, Define where the factor appears as a transformation on the and the tensor factors and By Theorem (3.12), there is a natural identification of the centralizer algebras with the path algebras corresponding to the Bratteli diagram so that where, if and then where we have replaced the weights of the Markov trace by
Let be the irreducible indexed by the fundamental weight The matrices and in satisfy the relations
(a) and (b) follow from Proposition (2.18). From (5.5), we have Use (5.4) to show that It follows that Relation (c) now follows from Corollary (2.22); the signs of the eigenvalues of are determined by which summands are in (d) follows from Proposition (3.11) part (2) and the fact that
Let us prove (e). By Corollary (2.22), acts by the eigenvalue on the irreducible summand in Thus, it follows from relation (c) that Using this formula and the relation it can be easily checked that relation (c) is equivalent to relation (e).
The relation follows by noting that, except for the constant is the projection onto the invariants and that acts by constant on All of the relations in (f) follow silimlarly.
A Path Algebra Formula for
Let be the Bratteli diagram for tensor powers of (with the assumptions in (5.6)). Identify the centralizer algebras with the path algebras corresponding to the Bratteli diagram Recall that the path algebras have a natural basis of matrix units.
For each tableau define Let be a pair of tableaux and in such that and are the same except possibly at the shape at level In other words the pair Define These constants are defined so that, if then and The first of these formulas is a consequence of Corollary (2.25).
Let be as given in (5.3).
|(a)||Let be a tableau in the Bratteli diagram Then|
|(b)||Let and be such that Then|
Let be a tableau in the Bratteli diagram Then, since differs from by either adding or removing a box in the row, Using (5.4) to compute we get The formula for follows since and
(b) The formulas for now follow from the definition of and the formula for in (a).
One can choose the identification (Section 1) of the centralizer algebras with the path algebras corresponding to the Bratteli diagram (with the assumption in (5.6)) so that the matrices are given by the formula where, for each and for each pair such that where and are given by (5.11) and (5.12) respectively.
Since commutes with all elements of it follows from Proposition Corollary (1.5) that for some constants In view of the imbeddings it is sufficient to show that the formulas for hold for
By definition and it follows that Thus, we may rewrite the relation (5.10e) in the form Let and view (5.15) as an equation in the path algebra. Since the matrices and are diagonal, taking the of this equation yields or, equivalently, Hence, Plugging in the following we get All except the last of the formulas in Theorem (5.14) now follow immediately from (5.9) and the following lemma.
Let If or if then
Consider the equation (5.16).
Case 1. If and then and since the weights are all nonzero. Thus the right hand side of (5.16) is nonzero. This implies that is nonzero.
Case 2. If and then and Thus the right hand side of (5.16) is nonzero. This implies that is nonzero.
Case 3. Suppose and Clearly is nonzero if and only if Then, by Proposition (5.13), there is some such that This value is nonzero in Types and since is odd, and is nonzero in type since, by the assumption in (5.6), and
Now let us prove the last formula in Theorem (5.14). Let and suppose that is such that and By Lemma (5.1), is unique. It follows that On the other hand since Since it follows that and thus that Equating (5.18) and (5.19) and using the formula for gives exactly as in the proof of Theorem (4.8). It follows from the remarks at the end of Section 1 that we can choose the normalization of the elements so that and are as given in the theorem.
Given a tableau let denote the tableau Let denote the set of all extensions of Given tableaux and in let be the constant given by Theorem (5.14) in the case that and let otherwise.
Let and let be the set of extensions of Then the values are different as ranges over all elements of
Let By Proposition (5.13), for some positive integer Let be the largest value of such that By the assumption in (5.6), Since it follows that Since and it follows that The result follows.
The proofs of the following results are essentially the same as the proofs of Theorem (4.14) and Corollary (4.15).
The matrix units are given in terms of the inductively, by the following formulas.
|(2)||Let If then where is given by (1).|
|(3)||Let If then where and are tableaux in of the form and|
The centralizer is generated by the matrices
This is a typed version of the paper A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras by Robert and Arun
The paper was received June 24, 1994; accepted September 12, 1994.
Supported in part by National Science Foundation Grant DMS-9300523 to the University of Wisconsin.
Supported in part by National Science Foundation Postdoctoral Fellowship DMS-9107863.