A Rational-Function Identity Related to the Murnaghan-Nakayama Formula for the Characters of Sn
Curtis Greene
DOI: 10.1023/A:1022435901373
Abstract
The Murnaghan-Nakayama formula for the characters of S n is derived from Young's seminormal representation, by a direct combinatorial argument. The main idea is a rational function identity which when stated in a more general form involves Möbius functions of posets whose Hasse diagrams have a planar embedding. These ideas are also used to give an elementary exposition of the main properties of Young's seminormal representations.
Pages: 235–255
Keywords: symmetric group; representation; character; Young tableau; Möbius function
Full Text: PDF
References
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3. P. Diaconis and C. Greene, "Applications of Murphy's elements," unpublished manuscript, 1989.
4. A.M. Garsia, M.L. Wachs, "Combinatorial aspects of skew representations of the symmetric group," J. Combin. Theory A 50 (1989), 47-81.
5. C. Greene, On the Mobius algebra of a partially ordered set, Adv. Math. 10 (1973), 177-187.
6. C. Greene, "Proof of a conjecture of Goulden and Jackson on immanants of the Jacobi-Trudi matrix," Linear Algebra Appl., to appear.
7. G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, MA, 1981.
8. D.E. Littlewood, The Theory of Croup Characters, Oxford University Press, Oxford, 1950.
9. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.
10. G.-C. Rota, "On the foundations of combinatorial theory I: Theory of Mobius functions," Z. Wahrsch. 2 (1964), 340-368.
11. D.E. Rutherford, Substitutional Analysis, University Press, Edinburgh, 1948.
12. R.P. Stanley, Enumerative Combinatorics, Vol. I. Brooks-Cole, Belmont, MA, 1986.
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14. A. Young, The Collected Papers of Alfred Young, University of Toronto Press, Toronto, 1977.