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The Temperley-Lieb algebra has the same generators as the $S_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen in the Wikipedia article.

Is there a standard analogue for other Weyl groups? Perhaps some specialization of the Hecke algebra?

There is a standard basis for the Temperley-Lieb algebra, which can be indexed by 321-avoiding permutations: pick a reduced word for $v$ in $S_n$'s generators, and interpret the generators inside T-L instead. This association is well-defined, because the 321-avoiders are also the "fully commutative" elements, meaning that each has only one reduced word up to commutation relations.

This subset of $S_n$ is also the set of "lambda-cominuscule" elements (meaning, the $T$-action on the Bruhat cell $BvB/B$ includes the dilation action), and is an ideal in left/right weak Bruhat order. In Weyl groups other than $S_n$ the lambda-cominuscule condition is strictly stronger than "fully commutative".

Assuming the first question, do the lambda-cominuscules, or some other subset of W, similarly give a basis?

Of course references would be most appreciated.

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First I don't think you have stated the defining relations correctly. [AK: you were quite right, and I've edited the question.]

It sounds as though you should start with the work of Richard Green. The following papers seem relevant.

http://arxiv.org/abs/q-alg/9712018

http://arxiv.org/abs/math/0102003

http://arxiv.org/abs/math/0108076

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  • $\begingroup$ This seems to be exactly what I want. (Except that the basis ends up indexed by fully commutative elements, rather than lambda-cominuscule, but apparently that's life.) $\endgroup$
    – Allen Knutson
    Commented May 28, 2013 at 19:38
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There are some ad-hoc definitions for some types. Type B can be defined using diagrams that have a left-right symmetry. Tammo tom Dieck has proposed a definition for type D here: (http://www.uni-math.gwdg.de/tammo/preprints/1dk.pdf).

But there is no satisfactory general answer, as far as I know. In particular it is not clear if the proposed type B and D cases are good ideas or not.

One could also wonder whether this has something to do with cluster algebras or noncrossing partitions, given that Catalan numbers also appear there.

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  • $\begingroup$ Catalan numbers appear everywhere! :-) More seriously, I've thought a little in the first direction you propose in the last paragraph but haven't seen a cluster algebra connection so far. The TL relations are not obviously cluster algebra-like, but part of the thing about cluster algebras is that often one needs to look at more elements than in the defining presentation (usually the game is to find small generating sets for one's favourite algebra but this is absolutely not the case for cluster algebras, where one wants "simple" relations and have to live with needing large generating sets). $\endgroup$
    – Jan Grabowski
    Commented May 28, 2013 at 21:02
  • $\begingroup$ Indeed. But the work of Genauer and Stoltzfus (arxiv.org/abs/math/0511003) has at least a small amount of cluster flavour. And one can also note that the cluster theory for surfaces now involves skein relations. $\endgroup$
    – F. C.
    Commented May 29, 2013 at 9:35

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