Last update: 13 March 2014
We shall use the notations for partitions given in [Mac1979]. In particular, a partition of the positive integer denoted is a decreasing sequence of non-negative integers such that The length is the largest such that The Ferrers diagram of is the left-justified array of boxes with boxes in the row. For example, is a partition of length 4. Given two partitions we write if for all We have that if the Ferrers diagram of is a subset of the Ferrers diagram of
The Bratteli diagram given in Fig. 1 is called the Young lattice. 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 by adding a box to The Young lattice is a multiplicity free Bratteli diagram.
Classically, a standard tableau of shape is a filling of the boxes in the Ferrers diagram of with the numbers such that the numbers are increasing left to right in the rows and increasing down the columns. Each tableau in the Bratteli diagram can be identified in a natural way with a standard tableau of shape Let be a standard tableau of shape and let be the tableau such that is the partition given by the set of boxes of which contain the numbers One easily shows that this identification is a bijection between the standard tableaux of shape and the tableaux
The Young lattice is the Bratteli diagram which is given by the sets A partition is connected by an edge to a partition if or equivalently, if can be obtained by adding a box to The Young lattice can be obtained by removing all the partitions with more than rows (and the edges connected to them) from the full Young lattice It is easy to see that tableaux in the Young lattice correspond to standard tableaux of shapes in exactly the same way as tableaux in correspond to standard tableaux. Note also that the full Young lattice can be viewed as the limit of the Young lattices 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 Suppose that is obtained by adding a box to the row of and that is obtained by adding a box to the row of Now suppose that is such that If then we must have that If then either or is the shape obtained by adding a box to the row of Thus,
The Centralizer Algebras
For the remainder of this section fix Let be an orthonormal basis of Then the simple roots, the fundamental weights, and the element are given by The finite dimensional irreducible modules of are indexed by the dominant integral weights, where It is sometimes helpful to identify each dominant integral weight with the partition Note that all partitions in have at most rows. It will also be helpful to note that, if then
Let the irreducible of highest weight The decomposition rule for tensoring by is given by where the sum is over all partitions that are gotten by adding a box to the partition It follows that the Bratteli diagram for tensor powers of is the Young lattice
Let be the irreducible indexed by the fundamental weight The matrices satisfy the relations
The first two relations follow from Proposition (2.18). From (4.3), we have Use (4.2) to show that It follows that The result now follows from Proposition (2.22) part (3) and the standard fact that is antisymmetric part of and is the symmetric part of
A Path Algebra Formula for
Let be the Bratteli diagram for tensor powers of 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 a tableau in the Bratteli diagram Then where is obtained by adding a box to the row of and is obtained by adding a box to the row of
Let be a tableau in the Bratteli diagram Then, since differs from by adding a box in the row we have and Using (4.2) to compute we get The formula for follows since
The formula for now follows easily since
Remark. In terms of standard tableaux, the value of is the "axial distance" between the box containing and the box containing
One can choose the identification (Section 1) of the centralizer algebras with the path algebras corresponding to the Bratteli diagram so that the matrices are given by the formula where for each we have and for each pair such that we have
Since commutes with all elements of it follows from 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 The relation from Proposition (4.4) can be written in the form or, equivalently, in the form Let and view (4.9) as an equation in the path algebra. Since the matrices and are diagonal, taking the of this equation yields or, equivalently, Since the right hand side of this equation is nonzero, the left hand side is also nonzero and we may write Plugging in the following gives the first formula in Theorem (4.8).
Now let us prove the second formula in Theorem (4.8). Let and suppose that is such that and By the remark in (4.1), is unique. It follows that On the other hand, the relation from Proposition (4.4) can be written in the form giving that Equating (4.11) and (4.12) and using the formula for gives
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.
Remark. If such that then Thus, the formula for given in Theorem (4.8) is actually symmetric in and
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 (4.8) in the case that and let otherwise.
Let and let be the set of extensions of Then the values are all different as ranges over all elements of
Let By the previous lemma, if is obtained by adding a box to the row of Since it follows that
[RWe1992]. 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 are of the form M and|
(1) Let It follows from the formula for that The identity (1) follows if we show that the values are all different as runs over all tableaux in Since it follows that the values are all different as runs over all tableaux in if and only if the values are all different as runs over all tableaux in Statement (1) now follows from Lemma (4.13).
(2) follows from the definition (1.3) of the embedding of path algebras.
(3) We must show two things: (a) For each possible choice of and the formula determines (b) There exist tableaux and in of the form and
Suppose that and are given. Since it follows from (4.1) that and are the unique extensions of and respectively, such that By Theorem (4.8) we know that the values are nonzero. It follows that proving (a). To see that (b) is true we reason as follows. Suppose that is a partition that is the same as except that there is a box missing from the row. Suppose that is a partition that is the same as except that there is a box missing from the row. Since we know that Then there is a unique partition that is the same as except that there is a box missing from the row and a box missing from the row. The partition is uniquely determined by and and and can be determined by fixing some tableau of shape
The centralizer is generated by the matrices
It follows from the identification of the centralizer algebras with the path algebras that the matrix units span the centralizer algebras In view of Theorem (4.14), the matrix units can be written in terms of the matrices. The statement follows.
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.