Last update: 30 January 2014
Let be the Drinfeld-Jimbo quantum group corresponding to a finite dimensional complex semisimple Lie algebra We shall use the notations and conventions for as in [LRa1977] and [ORa0401317]. There is an invertible element in (a suitable completion of) such that, for two modules and the map is a module isomorphism. In order to be consistent with the graphical calculus these operators should be written on the right. The element satisfies “quasitriangularity relations” (see [LRa1977, (2.1-2.3)]) which imply that, for modules and a module isomorphism which, together, imply the braid relation
Let be such that for all simple roots As explained in [LRa1977, (2.14)] and [Dri1990], there is a quantum Casimir element in the center of and, for a module we define a module isomorphism and the elements satisfy if is a module generated by a highest weight vector of weight (see [LRa1977, Prop. 2.14] or [Dri1990, Prop. 3.2]). Note that are the eigenvalues of the classical Casimir operator [Dix1996, 7.8.5]. From the relation (3.1) it follows that if are finite dimensional irreducible modules then acts on the isotypic component of the decomposition
Let be a quantum group and let and be such that the operators and are well defined. Define and in by Then the braid relations and imply that there is a well defined map which makes into a right module. By (3.1) and the fact that the eigenvalues of are related to the eigenvalues of the Casimir. The Schur functors are the functors where is the vector space of highest weight vectors of weight in
Although the Lie algebra is reductive, not semisimple, all of the general setup of Sections 3.1 and 3.2 can be applied without change. The simple roots are and The dominant integral weights of are and these index the simple finite dimensional A partition with rows is a dominant integral weight with If and denotes the “determinant” representation of then (see [FHa1991, §15.5])
Identify each partition with the configuration of boxes which has boxes in row For example, If and are partitions with (as collections of boxes) then the skew shape is the collection of boxes of that are not in For example, if is as in (3.8) and If is a box in position of the content of is are the contents of the boxes for the partition in (3.8).
If is a partition and where the sum is over all partitions with rows that are obtained by adding a box to [Mac1995, I App. A (8.4) and I (5.16)], Hence, the decompositions of are encoded by the graph with For example if then the first rows of are
The following result is well known (see [Jim1986] or [LRa1977, (4.4)]).
If and is the “standard” representation of then the map of (3.3) factors through the surjective homomorphism (2.8) to give a representation of the affine Hecke algebra.
For a skew shape with boxes identify paths from to in with standard tableaux of shape by filling the boxes, successively, with as they appear. In the example graph above
In the case when and the partitions which appear in (3.11) and in the graph all have rows. For example if then the first few rows of are and this is the graph which describes the decompositions in (3.11).
If where is a partition of with rows, and is the 2-dimensional "standard" representation of then the map of (3.3) factors through the surjective homomorphism of (2.13) with to give a representation of the affine Temperley-Lieb algebra
The proof that the kernel of contains the element (2.12) is exactly as in the proof of [ORa0401317, Thm. 6.1(c)]: The element in acts on as where pr is the unique projection onto in Using and the pictorial equalities it follows that acts as By (3.1), this is equal to and the coefficient simplifies to where
The restriction of an irreducible representation of to is irreducible and all irreducible representations of are obtained in this fashion. Since the “determinant” representation is trivial as an module it follows from (3.7) that the irreducible representations of are indexed by partitions with Hence, the graph which describes the decompositions of is exactly the same as the graph for except with all columns of length 2 removed from the partitions. More precisely, the decompositions are encoded by the graph with For example if and then the first few rows of are Paths in (3.17) correspond to paths in (3.14) which correspond to standard tableaux of shape
This is a typed version of Commuting families in Hecke and Temperley-Lieb Algebras by Tom Halverson, Manuela Mazzocco and Arun Ram.
AMS Subject Classifications: Primary 20G05; Secondary 16G99, 81R50, 82B20.