Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 17 September 2013

Appendix B

B8. The Temperley-Lieb algebras ${TL}_k\left(x\right)$

A ${TL}_k\text{-diagram}$ is a Brauer diagram on $k$ dots which can be drawn with no crossings of edges. $$\begin{array}{cc} & \end{array}$$ The Temperley-Lieb 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 Temperley-Lieb algebra $T{L}_k\left(x\right)$ is the algebra over $ℂ$ given by generators $E_1,E_2,\dots ,E_{k-1}$ and relations $$\begin{array}{c}{E}_i{E}_j=E_j{E}_i,\text{if}\hspace{0.17em}|i-j|>1,\\ E_i{E}_{i\pm 1}{E}_i=E_i,\text{and}\\ E_i^2=x{E}_i\text{.}\end{array}$$

Theorem B8.2. Let $q\in {ℂ}^{*}$ be such that $q+q^{-1}+2=1/x^2$ and let $H_k\left(q\right)$ be the Iwahori-Hecke algebra of type $A_{k-1}\text{.}$ Then the map $$\begin{array}{ccc}{H}_k\left(q\right)& ⟶& T{L}_k\left(x\right)\\ T_i& ⟼& \frac{q+1}{x}{E}_i-1\end{array}$$ is a surjective homomorphism and the kernel of this homomorphism is the ideal generated by the elements $$T_i{T}_{i+1}{T}_i+T_i{T}_{i+1}+T_{i+1}{T}_i+T_i+T_{i+1}+1,\text{for}\hspace{0.17em}1\le i\le n-2\text{.}$$

The Schur Weyl duality theorem for $s_n$ has the following analogue for the Temperley-Lieb algebras. Let $U_q𝔰{𝔩}_2$ be the Drinfeld-Jimbo quantum group corresponding to the Lie algebra ${𝔰𝔩}_2$ and let $V$ be the $2\text{-dimensional}$ representation of $U_q{𝔰𝔩}_2\text{.}$ There is an action, see [CPr1994], of the Temperley-Lieb algebra ${TL}_k(q+q^{-1})$ on $V^{\otimes k}$ which commutes with the action of $U_q{𝔰𝔩}_2$ on $V^{\otimes k}\text{.}$

Theorem B8.3.

(a)	The action of ${TL}_k(q+q^{-1})$ on $V^{\otimes k}$ generates ${\text{End}}_{U_q{𝔰𝔩}_2}\left(V^{\otimes k}\right)\text{.}$
(b)	The action of $U_q{𝔰𝔩}_2$ on $V^{\otimes k}$ generates ${\text{End}}_{T{L}_k(q+q^{-1})}\left(V^{\otimes k}\right)\text{.}$

Theorem B8.4. The Temperley-Lieb 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 (Ia-c) for the Temperley-Lieb algebras hold when $x$ is such that ${TL}_k\left(x\right)$ is semisimple.

I. What are the irreducible ${TL}_k\left(x\right)\text{-modules?}$

(a)	How do we index/count them?
	How do we index/count them? $$\text{Partitions of}\hspace{0.17em}k\hspace{0.17em}\text{with at most two rows}\stackrel{1-1}{⟷}\text{Irreducible representations}\hspace{0.17em}{T}^{\lambda }\text{.}$$
(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 $$d_{2h}=\underset{\underset{k-2h}{⏟}}{\begin{array}{c} ⋯ \end{array}}\hspace{0.17em}\underset{\underset{2h}{⏟}}{\begin{array}{c} ⋯ \end{array}}$$ is $${\chi }^{(k-\ell ,\ell )}\left(d_{2h}\right)=\{\begin{array}{cc}\left(\genfrac{}{}{0ex}{}{k-2h}{\ell -h}\right)-\left(\genfrac{}{}{0ex}{}{k-2h}{\ell -h-1}\right),& \text{if}\hspace{0.17em}\ell \ge h,\\ 0,& \text{if}\hspace{0.17em}\ell <h\text{.}\end{array}$$ There is an algorithm for writing the character ${\chi }^{(k-\ell ,\ell )}\left(a\right)$ of a general element $a\in {TL}_k\left(x\right)$ as a linear combination of the characters ${\chi }^{(k-\ell ,\ell )}\left(d_{2h}\right)\text{.}$

References

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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