Abstract
We describe a straightening algorithm for the action of S n on a certain graded ring \(R_\mu\). The ring \(R_\mu\) 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 \(R_\mu\) affords the irreducible representation of S n indexed by μ.
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Barcelo, H. Young Straightening in a Quotient Sn-Module. Journal of Algebraic Combinatorics 2, 5–23 (1993). https://doi.org/10.1023/A:1022416129423
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DOI: https://doi.org/10.1023/A:1022416129423