Last update: 29 January 2014
The affine braid group is the group of affine braids with strands (braids with a flagpole). The group is presented by generators and with relations
For define By drawing pictures of the corresponding affine braids it is easy to check that so that the elements are a commuting family for Thus is an abelian subgroup of The free abelian group generated by is and for in
An alternate presentation of can be given using the generators and where
The affine braid group is the affine braid group of type The affine braid groups of type and are the subgroup Then is a central element of (where the indices are taken mod and and In we have and is defined to be the image of under the homomorphism so that
A Temperley-Lieb diagram on dots is a graph with dots in the top row, dots in the bottom row, and edges pairing the dots such that the graph is planar (without edge crossings). For example, are Temperley-Lieb diagrams on 7 dots. The composition of two diagrams is the diagram obtained by placing above and identifying the bottom vertices of with the top dots of removing any connected components that live entirely in the middle row. If is the set of Temperley-Lieb diagrams on dots then the Temperley-Lieb algebra is the associative algebra with basis where is the number of blocks removed from the middle row when constructing the composition and is a fixed element of the base ring. For example, using the diagrams and above, we have The algebra is presented by generators (see [GHJ1989, Lemma 2.8.4]).
In the definition of the Temperley-Lieb algebra, and for other algebras defined in this paper, the base ring could be any one of several useful rings (e.g. or localizations of these at special primes). The most useful approach is to view the results of computations as valid over any ring with such that the formulas make sense.
The affine Hecke algebra is the quotient of the group algebra of the affine braid group by the relations is a surjective homomorphism is a fixed element of the base ring). The affine Hecke algebra is the affine Hecke algebra of type The affine Hecke algebras of types and are, respectively, the quotients and of the group algebras of and (see Remark 2.2) by the relations (2.8).
The Iwahori-Hecke algebra is the subalgebra of generated by In the Iwahori-Hecke algebra define Direct calculations show that and that and if and only if Thus, setting there are surjective algebra homomorphisms given by The kernel of is generated by the element on the left hand side of equation (2.10). In the notation of Theorem 4.1, the representations of correspond to the case when Writing as the element from (2.10) acts as on the irreducible Iwahori-Hecke algebra modules and and (up to a scalar multiple) it is a projection onto
There is an alternative surjective homomorphism that instead sends This alternative surjection has kernel generated by This element is on and and (up to a scalar multiple) it is a projection onto
A priori, there are two different kinds of integrality for the Temperley-Lieb algebra: coefficients in or coefficients in (in terms of the basis of Temperley-Lieb diagrams). The relation between these is as follows. If since Then so that is a polynomial in The polynomials all form bases of the ring The transition matrix between the and the is triangular (with 1s on the diagonal) and the transition matrix between the and the is also triangular (the non zero entries are binomial coefficients). Hence, the transition matrix between and has integer entries and so is, in fact, a polynomial in with integer coefficients.
The affine Temperley-Lieb algebra is the diagram algebra generated by The generators of satisfy (where the indices are taken mod and (see [GLe2004, 4.15(iv)]). In we let (see Remark 2.1).
Graham and Lehrer [GLe2004, §4.3] define four slightly different affine Temperley-Lieb algebras, the diagram algebra and the algebras defined as follows: For each invertible element in the base ring there is a surjective homomorphism and every irreducible representation of factors through one of these homomorphisms (see [GLe2004, Prop. 4.14(v)]). In Proposition 3.2 we shall see that these homomorphisms arise naturally in the Schur-Weyl duality setting.
View the elements in the affine Temperley-Lieb algebra via the surjective algebra homomorphism of (2.13). Define for Since for all and the are linear combinations of the
Rewrite (2.14) as and use induction, to obtain the formula for in (a). Summing the formula in (a) over gives and, thus, formula (c) follows from
The following Lemma is a transfer of the recursion to the The following are the base cases of Lemma 2.7.
Let be the constant defined by the equation For
From (2.3) and (2.9) we have Substituting this into the definition of gives Use Proposition 2.6 (a) to substitute for which gives Using induction, substitute for the first in this equation to get
Label the vertices from left to right in the top row of a diagram with and label the corresponding vertices in the bottom row with The cycle type of a diagram is the set partition of obtained from by setting If is a set partition of the form where is a composition of then we simplify notation by writing For example There are diagrams whose cycle type cannot be written as a composition (for example has cycle type but all of the diagrams needed here have cycle types that are compositions.
If is a composition of define as the sum of the Temperley-Lieb diagrams on dots with cycle type Define be the sum of diagrams obtained from the summands of by wrapping the first edge in each row around the pole, with the orientation coming from as shown in the examples below. When the first edge in the top row connects to the first vertex in the bottom row only one new diagram is produced, otherwise there are two. For example, in View and as elements of by setting With this notation, expanding the first few in terms of diagrams gives where, as in Lemma 2.7, is the constant defined by the equation
Let be the constant defined by the equation Then and, for where the sum is over all compositions of with and with
From our computations above, and where Let For the recursion in Lemma 2.7 gives So if has cycle type with then
To view in the (nonaffine) Temperley-Lieb algebra (via (2.11)) let so that If then and if then In both cases the coefficients in Theorem 2.8 specialize to and where the sum is over compositions of with The first few examples are
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.