Counterexamples to the 0-1 conjecture

الأصل · EN

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.