## Counterexamples to the 0-1 Conjecture

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- by Timothy J. McLarnan and Gregory S. Warrington
- Represent. Theory
**7**(2003), 181-195 - DOI: https://doi.org/10.1090/S1088-4165-03-00178-X
- Published electronically: May 7, 2003
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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.## References

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## Bibliographic Information

**Timothy J. McLarnan**- Affiliation: Department of Mathematics, Earlham College, Richmond, Indiana 47374
- Email: timm@earlham.edu
**Gregory S. Warrington**- Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
- MR Author ID: 677560
- Email: warrington@math.umass.edu
- Received by editor(s): October 1, 2002
- Received by editor(s) in revised form: March 24, 2003
- Published electronically: May 7, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Represent. Theory
**7**(2003), 181-195 - MSC (2000): Primary 05E15; Secondary 20F55
- DOI: https://doi.org/10.1090/S1088-4165-03-00178-X
- MathSciNet review: 1973372