Counterexamples to the 0-1 Conjecture

McLarnan, Timothy; Warrington, Gregory

doi:10.1090/S1088-4165-03-00178-X

Counterexamples to the 0-1 Conjecture
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by Timothy J. McLarnan and Gregory S. Warrington PDF

Represent. Theory 7 (2003), 181-195 Request permission

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 $\mu (x,w)$ arise as the edge labels of certain graphs encoding the representations of $S_n$. The 0-1 Conjecture states that the $\mu (x,w) \in \{0,1\}$. 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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