Annals of Combinatorics 9 (2005) 281-291


Minimal Bar Tableaux

Peter Clifford

Hamilton Institute, NUI Maynooth, Ireland
peterc@alum.mit.edu

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 Sn 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.

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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.


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