## Puzzles, Tableaux, and Mosaics

- Date: 11/28/2006

Kevin Purbhoo (University of British Columbia)

University of British Columbia

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 interpolates 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.

Discrete Math Seminar 2006