Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



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
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.

References [Enhancements On Off] (What's this?)

  • Sara Billey and V. Lakshmibai, Singular loci of Schubert varieties, Progress in Mathematics, vol. 182, Birkhäuser Boston, Inc., Boston, MA, 2000. MR 1782635
  • gwsb-msl S. Billey and G. Warrington, Maximal singular loci of Schubert varieties in $SL(n)/B$, Trans. Amer. Math. Soc. (to appear). ducloux F. du Cloux, Personal communication, 2002.
  • William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
  • A. M. Garsia and T. J. McLarnan, Relations between Young’s natural and the Kazhdan-Lusztig representations of $S_n$, Adv. in Math. 69 (1988), no. 1, 32–92. MR 937317, DOI
  • James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460
  • David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI
  • Donald E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 (1970), 709–727. MR 272654
  • Donald E. Knuth, The art of computer programming. Volume 3, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973. Sorting and searching; Addison-Wesley Series in Computer Science and Information Processing. MR 0445948
  • Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Kazhdan & Lusztig pour les grassmanniennes, Young tableaux and Schur functors in algebra and geometry (Toruń, 1980), Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266 (French). MR 646823
  • mclarnan T. J. McLarnan and G. Warrington, Counterexamples to the 0-1 conjecture, arXiv:math.CO/0209221 (2002).
  • Mitsuyuki Ochiai and Fujio Kako, Computational construction of $W$-graphs of Hecke algebras $H(q,n)$ for $n$ up to $15$, Experiment. Math. 4 (1995), no. 1, 61–67. MR 1359418

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 05E15, 20F55

Retrieve articles in all journals with MSC (2000): 05E15, 20F55

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
MR Author ID: 677560

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