Young Straightening in a Quotient Sn-Module
Hélène Barcelo
DOI: 10.1023/A:1022416129423
Abstract
We describe a straightening algorithm for the action of S n on a certain graded ring R m R_μ . The ring R m R_μ appears in the work of C. de Concini and C. Procesi [2] and T. Tanisaki [8], and more recently in the work of A. Garsia and C. Procesi [4]. This ring is a graded version of the permutation representation resulting from the action of S n on the left cosets of a Young subgroup. As a corollary of our straightening algorithm we obtain a combinatorial proof of the fact that the top degree component of R m R_μ affords the irreducible representation of S n indexed by 
.
.Pages: 5–23
Keywords: graded permutation representation; straightening algorithm; Young's natural representation; symmetric group
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References
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2. C. de Concini and C. Procesi, "Symmetric functions, conjugacy classes and the flag variety," Inv, Math. 64 (1981), 203-279.
3. A.M. Garsia and N. Bergeron, "On certain spaces of harmonic polynomials," Contemp. Math. 138 (1992), 51-86.
4. A.M. Garsia and C. Procesi, "On certain graded Sn-modules and the q-Kostka polynomials," Advances in Math. 94 (1992), 82-138.
5. A.M. Garsia and M. Wachs, "Combinatorial aspects of skew representations of the symmetric group," J. Comb. Theory A, 50 (1989), 47-81.
6. H. Kraft, "Conjugacy classes and Weyl group representations," Proc. 1980 Torun Conf. Poland, Asterisque, 87-88 (1981), 191-205.
7. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979.
8. T. Tanisaki, "Defining ideals of the closures of the conjugacy classes and representations of the Weyl groups," Tohoku Math J. 34 (1982), 575-585.
9. A. Young, "On quantitative substitutional analysis (sixth paper)," The Collected Papers of A. Young, University of Toronto Press, Toronto, 1977.