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Counterexamples to the 0-1 conjecture

Timothy J. McLarnan·Gregory S. Warrington
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Abstract

For permutations x and w, let mu(x,w) be the coefficient of highest possible degree in the Kazhdan-Lusztig polynomial P_{x,w}. It is well-known that the coefficients mu(x,w) arise as the edge labels of certain graphs encoding the representations of S_n. The 0-1 Conjecture states that the mu(x,w) are either 0 or 1. We present two counterexamples to this conjecture, the first in S_16, for which x and w are in the same left cell, and the second in S_10. The proof of the counterexample in S_16 relies on computer calculations.

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Timothy J. McLarnan, Gregory S. Warrington. 2002-09-18. Counterexamples to the 0-1 conjecture. https://arxiv.org/abs/math/0209221

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