Annals of Combinatorics 9 (2005) 281-291Minimal Bar Tableaux Peter Clifford
Hamilton Institute, NUI Maynooth, Ireland Received November 20, 2003 AMS Subject Classification: 05E05, 05E10, 20C25, 20C30 Abstract. Motivated by Stanley's results in [7], we generalize the rank of a partition to the rank of a shifted partition . We show that the number of bars required in a minimal bar tableau of is , where o and e are the number of odd and even rows of . As a consequence we show that the irreducible projective characters of S_{n} vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur's symmetric functions in terms of the power sum symmetric functions. Keywords: bar tableau, strip tableau, rank, shifted partition, shifted shape, projective character, negative character, Schur Q-functions References 1. P. Clifford, Algebraic and combinatorial properties of minimal border strip tableaux, Ph. D. Thesis, M.I.T., 2003. 2. P.N. Hoffman and J.F. Humphreys, Projective Representations of the Symmetric Groups, Oxford Science Publications, 1992. 3. T. Józefiak, Characters of projective representations of symmetric groups, Exposition. Math. 7 (1989) 193?–247. 4. I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd Edition, Oxford University Press, Oxford, 1995. 5. A.O. Morris, The spin representation of the symmetric group, Proc. London Math. Soc. 12 (3) (1962) 55–76. 6. I. Schur, Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 127 (1911) 20–50. 7. R.P. Stanley, The rank and minimal border strip decompositions of a skew partition, J. Combin. Theory Ser. A 100 (2002) 349–375. 8. J.R. Stembridge, On symmetric functions and the spin characters of Sn, In: Topics in Algebra, Vol. 26, Part 2, Banach Centre Publications, PWN Polish Scientific Publishers,Warsaw. 9. J.R. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989) 87–134. 10. J.R. Stembridge, The SF Package for Maple, Version 2.3, July 22, 2001, http://www.math .lsa.umich.edu/~jrs/maple.html#SF. |