Counterexamples to the 0-1 conjecture
Timothy J. McLarnan, Gregory S. Warrington
Abstract
For permutations x and w, let mu(x,w) be the coefficient of highest possible degree in the Kazhdan-Lusztig polynomial Px,w. It is well-known that the coefficients mu(x,w) arise as the edge labels of certain graphs encoding the representations of Sn. 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 S16, for which x and w are in the same left cell, and the second in S10. The proof of the counterexample in S16 relies on computer calculations.
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