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

DOI:
https://doi.org/10.1090/S1088-4165-03-00178-X

Published electronically:
May 7, 2003

MathSciNet review:
1973372

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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

Article copyright:
© Copyright 2003
American Mathematical Society