Last update: 30 January 2014
Recall, from (2.8), that the affine Hecke algebra is the quotient of the group algebra of the affine braid group by the relations As observed in Proposition 3.2 the map in (3.3) makes the module in (3.11) into an module. Thus the vector spaces in (3.11) are the given by The following theorem is well known (see, for example, [Che1987]).
|(a)||The mutually commute in the affine Hecke algebra|
|(b)||The eigenvalues of are given by the graph of (3.13) in the sense that if for then and where is the content of box of|
|(c)||is a central element of and|
(a) is a restatement of (2.4). (b) Since the action and the action commute on it follows that the decomposition in (3.11) is a decomposition as bimodules, where the are some Comparing the components on each side of gives for any and skew shape with boxes. Iterate (4.2) (with to produce a decomposition where the summands are 1-dimensional vector spaces. This determines a basis (unique up to multiplication of the basis vectors by constants) of which respects the decompositions in (4.2) for
Combining (3.1), (3.2) and (3.4) gives that acts on the component of the decomposition (3.10) by the constant since if so that is the same as except with an additional box in row then and Hence, where is the box containing in
The remainder of the proof, including the simplicity of the is accomplished as in [Ram0401326, Thm. 4.1].
(c) The element is central in (it is a full twist) and hence its image is central in The constant describing its action on follows from the formula
Let be the commuting family in the affine Temperley-Lieb algebra as defined in (2.14). We will use the results of Theorem 4.1 to determine the eigenvalues of the in the (generically) irreducible representations.
|(a)||The elements mutually commute in|
|(b)||The eigenvalues of the elements are given by the graph of (3.17) in the sense that if the set of vertices on level is for then and with where is the partition (a single part in this case) on level of the path|
|(c)||is a central element of and acts on by the constant|
(a) The elements commute with one another in the affine Hecke algebra (see (2.4) and the are by definition linear combinations of the (see 2.14), so they commute.
(b) Let be a path to in and let be the corresponding standard tableau on 2 rows. If or if then and, from (2.14) and Theorem 4.1(b), If with then and and where If with then and and where
(c) Let The identity is best visible in an example: With and Then Proposition 2.6 says and so acts on by the constant since and
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.