Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 17 September 2013

Appendix B

B8. The Temperley-Lieb algebras TLk(x)

A TLk-diagram is a Brauer diagram on k dots which can be drawn with no crossings of edges. The Temperley-Lieb algebra TLk(x) is the subalgebra of the Brauer algebra Bk(x) which is the span of the TLk-diagrams.

Theorem B8.1. The Temperley-Lieb algebra TLk(x) is the algebra over given by generators E1,E2,,Ek-1 and relations EiEj= EjEi,if |i-j|>1, EiEi±1 Ei=Ei, and Ei2=xEi.

Theorem B8.2. Let q* be such that q+q-1+2=1/x2 and let Hk(q) be the Iwahori-Hecke algebra of type Ak-1. Then the map Hk(q) TLk(x) Ti q+1xEi-1 is a surjective homomorphism and the kernel of this homomorphism is the ideal generated by the elements TiTi+1Ti+ TiTi+1+ Ti+1Ti+ Ti+ Ti+1+1, for1in-2.

The Schur Weyl duality theorem for sn has the following analogue for the Temperley-Lieb algebras. Let Uq𝔰𝔩2 be the Drinfeld-Jimbo quantum group corresponding to the Lie algebra 𝔰𝔩2 and let V be the 2-dimensional representation of Uq𝔰𝔩2. There is an action, see [CPr1994], of the Temperley-Lieb algebra TLk(q+q-1) on Vk which commutes with the action of Uq𝔰𝔩2 on Vk.

Theorem B8.3.

(a) The action of TLk(q+q-1) on Vk generates EndUq𝔰𝔩2(Vk).
(b) The action of Uq𝔰𝔩2 on Vk generates EndTLk(q+q-1)(Vk).

Theorem B8.4. The Temperley-Lieb algebra is semisimple if and only if 1/x24cos2(π/), for any 2k.

Partial results for TLk(x)

The following results giving answers to the main questions (Ia-c) for the Temperley-Lieb algebras hold when x is such that TLk(x) is semisimple.

I. What are the irreducible TLk(x)-modules?

(a) How do we index/count them?
How do we index/count them? Partitions ofkwith at most two rows 1-1 Irreducible representationsTλ.
(b) What are their dimensions?
The dimension of the irreducible representation T(k-,) is given by dim(T(k-,)) = # of standard tableaux of shape (k-,) = (k)- (k-1).
(c) What are their characters?
The character of the irreducible representation T(k-,) evaluated at the element d2h= k-2h 2h is χ(k-,) (d2h)= { (k-2h-h)- (k-2h-h-1), ifh, 0, if<h. There is an algorithm for writing the character χ(k-,)(a) of a general element aTLk(x) as a linear combination of the characters χ(k-,)(d2h).


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].

Notes and references

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..

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