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Friday, September 21, 2007 |
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Tableaux, Puzzles and Mosaics |
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The Littlewood-Richardson numbers show up in a number of different areas of mathematics. They are structure constants of the ring of symmetric functions, which connects them to representation theory and cohomology of Grassmannians. There are now several well known combinatorial rules for computing Littlewood-Richardson numbers. I will talk about two of the main ones: the original rule of Littlewood and Richardson, which is phrased in terms of tableaux, and the Knutson-Tao puzzle rule, which looks very different. Most every other known rule is just a variant on one or the other. Yet it is not immediately obvious why these two are rules are the same, or why they are correct. I will give a new construction, mosaics, which interpolate between puzzles and tableaux. Then a miracle will occur: just using the fact that one can interpolate between them, a new and pleasant proof of correctness (for both rules) will appear out of thin air. |