Last update: 17 September 2013
A ${TL}_{k}\text{diagram}$ is a Brauer diagram on $k$ dots which can be drawn with no crossings of edges. $$\begin{array}{cc}\n\n\n\n\n\n& \n\n\n\n\n\n\end{array}$$ The TemperleyLieb algebra $T{L}_{k}\left(x\right)$ is the subalgebra of the Brauer algebra ${B}_{k}\left(x\right)$ which is the span of the $T{L}_{k}\text{diagrams.}$
Theorem B8.1. The TemperleyLieb algebra $T{L}_{k}\left(x\right)$ is the algebra over $\u2102$ given by generators ${E}_{1},{E}_{2},\dots ,{E}_{k1}$ and relations $$\begin{array}{c}{E}_{i}{E}_{j}={E}_{j}{E}_{i},\phantom{\rule{1em}{0ex}}\text{if}\hspace{0.17em}ij>1,\\ {E}_{i}{E}_{i\pm 1}{E}_{i}={E}_{i},\phantom{\rule{1em}{0ex}}\text{and}\\ {E}_{i}^{2}=x{E}_{i}\text{.}\end{array}$$
Theorem B8.2. Let $q\in {\u2102}^{*}$ be such that $q+{q}^{1}+2=1/{x}^{2}$ and let ${H}_{k}\left(q\right)$ be the IwahoriHecke algebra of type ${A}_{k1}\text{.}$ Then the map $$\begin{array}{ccc}{H}_{k}\left(q\right)& \u27f6& T{L}_{k}\left(x\right)\\ {T}_{i}& \u27fc& \frac{q+1}{x}{E}_{i}1\end{array}$$ is a surjective homomorphism and the kernel of this homomorphism is the ideal generated by the elements $${T}_{i}{T}_{i+1}{T}_{i}+{T}_{i}{T}_{i+1}+{T}_{i+1}{T}_{i}+{T}_{i}+{T}_{i+1}+1,\phantom{\rule{1em}{0ex}}\text{for}\hspace{0.17em}1\le i\le n2\text{.}$$
The Schur Weyl duality theorem for ${s}_{n}$ has the following analogue for the TemperleyLieb algebras. Let ${U}_{q}\U0001d530{\U0001d529}_{2}$ be the DrinfeldJimbo quantum group corresponding to the Lie algebra ${\U0001d530\U0001d529}_{2}$ and let $V$ be the $2\text{dimensional}$ representation of ${U}_{q}{\U0001d530\U0001d529}_{2}\text{.}$ There is an action, see [CPr1994], of the TemperleyLieb algebra ${TL}_{k}(q+{q}^{1})$ on ${V}^{\otimes k}$ which commutes with the action of ${U}_{q}{\U0001d530\U0001d529}_{2}$ on ${V}^{\otimes k}\text{.}$
Theorem B8.3.
(a)  The action of ${TL}_{k}(q+{q}^{1})$ on ${V}^{\otimes k}$ generates ${\text{End}}_{{U}_{q}{\U0001d530\U0001d529}_{2}}\left({V}^{\otimes k}\right)\text{.}$ 
(b)  The action of ${U}_{q}{\U0001d530\U0001d529}_{2}$ on ${V}^{\otimes k}$ generates ${\text{End}}_{T{L}_{k}(q+{q}^{1})}\left({V}^{\otimes k}\right)\text{.}$ 
Theorem B8.4. The TemperleyLieb algebra is semisimple if and only if $1/{x}^{2}\ne 4{\text{cos}}^{2}(\pi /\ell ),$ for any $2\le \ell \le k\text{.}$
Partial results for ${TL}_{k}\left(x\right)$
The following results giving answers to the main questions (Iac) for the TemperleyLieb algebras hold when $x$ is such that ${TL}_{k}\left(x\right)$ is semisimple.
(a)  How do we index/count them? 
How do we index/count them? $$\text{Partitions of}\hspace{0.17em}k\hspace{0.17em}\text{with at most two rows}\phantom{\rule{2em}{0ex}}\stackrel{11}{\u27f7}\phantom{\rule{2em}{0ex}}\text{Irreducible representations}\hspace{0.17em}{T}^{\lambda}\text{.}$$  
(b)  What are their dimensions? 
The dimension of the irreducible representation ${T}^{(k\ell ,\ell )}$ is given by $$\begin{array}{ccc}\text{dim}\left({T}^{(k\ell ,\ell )}\right)& =& \text{\# of standard tableaux of shape}\hspace{0.17em}(k\ell ,\ell )\\ & =& \left(\genfrac{}{}{0ex}{}{k}{\ell}\right)\left(\genfrac{}{}{0ex}{}{k}{\ell 1}\right)\text{.}\end{array}$$  
(c)  What are their characters? 
The character of the irreducible representation
${T}^{(k\ell ,\ell )}$ evaluated at the element
$${d}_{2h}=\underset{\underset{k2h}{\u23df}}{\begin{array}{c}\n\n\n\n\n\n\n\n\cdots \n\n\end{array}}\hspace{0.17em}\underset{\underset{2h}{\u23df}}{\begin{array}{c}\n\n\n\n\n\n\cdots \n\n\n\n\n\n\n\end{array}}$$

References
The book [GHJ1989] contains a comprehensive treatment of the basic results on the TemperleyLieb algebra. The SchurWeyl 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 DMS9510655.
Ram was supported in part by National Science Foundation grant DMS9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..