Geometry and the compression of combinatorial formulas for Macdonald polynomals

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The aim of the project is to give a better understanding of the connection between the work of Gaussent, Littelmann, Schwer, Ram and Yip and the work of Haglund, Haiman and Loehr [3,4,5]. Both methods lead to rather different combinatorial formulas for Macdonald polynomials. It is expected that the connection between these formulas has a geometric background. The project has a geometric and a combinatorial part. The aim of the geometric part is to generalize the methods developed in [2] by Gaussent and Littelmann to a different class of Bott-Samelson varieties. The combinatorial part has as a goal to translate "geometry = counting points in a certain variety over a finite field with q elements" into a statistic on (generalized) Young diagrams producing a polynomial in q describing exactly the desired number.