Last update: 17 September 2013
A is a Brauer diagram on dots which can be drawn with no crossings of edges. The Temperley-Lieb algebra is the subalgebra of the Brauer algebra which is the span of the
Theorem B8.1. The Temperley-Lieb algebra is the algebra over given by generators and relations
Theorem B8.2. Let be such that and let be the Iwahori-Hecke algebra of type Then the map is a surjective homomorphism and the kernel of this homomorphism is the ideal generated by the elements
The Schur Weyl duality theorem for has the following analogue for the Temperley-Lieb algebras. Let be the Drinfeld-Jimbo quantum group corresponding to the Lie algebra and let be the representation of There is an action, see [CPr1994], of the Temperley-Lieb algebra on which commutes with the action of on
|(a)||The action of on generates|
|(b)||The action of on generates|
Theorem B8.4. The Temperley-Lieb algebra is semisimple if and only if for any
Partial results for
The following results giving answers to the main questions (Ia-c) for the Temperley-Lieb algebras hold when is such that is semisimple.
|(a)||How do we index/count them?|
|How do we index/count them?|
|(b)||What are their dimensions?|
|The dimension of the irreducible representation is given by|
|(c)||What are their characters?|
The character of the irreducible representation
evaluated at the element
The book [GHJ1989] contains a comprehensive treatment of the basic results on the Temperley-Lieb algebra. The Schur-Weyl duality theorem is treated in the book [CPr1994], see also the references there. The character formula given above is derived in [HRa1995].
This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.
Key words and phrases. Algebraic combinatorics, representations.
Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..