Kazhdan-Lusztig polynomials and counterexamples to the 0-1 conjecture

Greg Warrington

UMass Amherst

November 13,
refreshments at 3:45pm


The Kazhdan-Lusztig polynomials P_{x,w} associated to permutations x and w arise in the representations of the symmetric group S_n, in the geometry of Schubert varieties and in the context of Verma modules. There are many questions about these areas whose answers require a better combinatorial understanding of these polynomials. We look at one such question.

We present several counterexamples to the 0-1 conjecture: "The coefficient of highest possible degree in P_{x,w} is either 0 or 1." These counterexamples imply, in particular, that the representations of S_n constructed by Kazhdan and Lusztig are encoded by weighted graphs rather than simply by graphs. We will also mention how the combinatorial techniques used in the discovery and proof of the counterexamples may shed light on the coefficients of lower order terms of these polynomials.

This is joint work with Tim McLarnan.
Joint meeting with Lie Theory Seminar.

Speaker's Contact Info: warrington(at-sign)math.umass.edu

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Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)math.mit.edu

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