Counterexamples to the 0-1 conjecture
Combinatorics
2007-05-23 v1 Representation Theory
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 coefficients 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) are either 0 or 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.
Keywords
Cite
@article{arxiv.math/0209221,
title = {Counterexamples to the 0-1 conjecture},
author = {Timothy J. McLarnan and Gregory S. Warrington},
journal= {arXiv preprint arXiv:math/0209221},
year = {2007}
}
Comments
15 pages, 4 figures; code for computer calculations included in source package