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Tensoring Polytopes

By trdunlap2

Fix a Kac-Moody Algebra \mathfrak{g}.

For every dominant (resp. anti-dominant) weight \lambda There is a collection of MV polytopes that forms a basis for the irreducible representation, L_\lambda, of highest (resp. lowest) weight \lambda. Furthermore this basis respects the weight decomposition of the L_\lambda. ( I’d really like to get a firm grasp on how \mathfrak{g} acts on this basis, but for now I only have a vague idea. )

For the tensor product of two representations, then, we can take as a basis, ordered pairs of MV-cycles — one part of the basis for the first and one part of the basis for the second. Since polytopes have a highest and lowest vertex, thinking of these like “head” and “tail” we draw these pairs in a manner analogous to summing vectors. This process basically gives the Minkowski sum.

But lately I’m beginning to think that a better method is to turn the second polytope “upside down” and draw them head to head. ( I hoped to have some pictures justifying this earlier this week, but at best I may have them up by Monday.) In words, if we associate to each polytope a path in the crystal (understanding that some paths are equivelant) then putting two polytopes head to tail is like concatenating these paths, head to head concatenating one path with the reverse of the other. The new path also corresponds to a polytope (two if you consider its reverse). I have no abstract justification for doing this other than it can be done — but the results for the handful of calculations I’ve done so far are very interesting, by which I mean indicative of symmetries.

The description by Anderson of can also be thought of as adding two polytopes (head to tail or head to head) and in terms of concatenating paths through the crystal may have some parallels (it certainly does for what might be called “balanced” paths) but I’m not to excited about those.

Let me summarize these two methods, let MV_\lambda be the polytopes forming a basis for L_\lambda then a basis for L_\lambda\otimes L_\mu can be given either by
\{ (P,Q) | P\in MV_\lambda,Q\in MV_\mu\}
or by
\{ (P,Q) | P\in MV_\lambda\cap (MV^-_\mu+\gamma),Q\in MV_{\lambda+\mu-\gamma} \}.
The second one is a theorem due to Anderson — I’m recalling it off the top of my head so I may revise it later.

This entry was posted on January 30, 2009 at 1:27 pm and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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