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Representation Theory

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Counterexamples to the 0-1 Conjecture

Authors: Timothy J. McLarnan and Gregory S. Warrington
Journal: Represent. Theory 7 (2003), 181-195
MSC (2000): Primary 05E15; Secondary 20F55
Published electronically: May 7, 2003
MathSciNet review: 1973372
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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 $\mu(x,w)$ 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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Additional Information

Timothy J. McLarnan
Affiliation: Department of Mathematics, Earlham College, Richmond, Indiana 47374

Gregory S. Warrington
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003

Received by editor(s): October 1, 2002
Received by editor(s) in revised form: March 24, 2003
Published electronically: May 7, 2003
Article copyright: © Copyright 2003 American Mathematical Society