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Relations between Young’s natural and the Kazhdan-Lusztig representations of \(S_ n\). (English) Zbl 0657.20014
The authors investigate the relation between two different constructions of all the irreducible representations (in characteristic 0) of the finite symmetric groups \(S_ n\). In section 1 Young’s “natural” representations \(A^{\lambda}\), \(\lambda\) \(\vdash n\), of \(S_ n\) are described in a combinatorial manner using standard tableaux. Then in section 2 a more general construction of Kazhdan and Lusztig is specialized to \(S_ n\) to give a set \(B^{\lambda}\), \(\lambda\) \(\vdash n\) of irreducible representations of \(S_ n\). A sketch of this complicated construction is presented, which involves the Bruhat order, the Robinson-Schensted correspondence and Kazhdan-Lusztig polynomials and graphs. The remainder of the paper is devoted to the description of integral transformation matrices \(W^{\lambda}\), such that \(A^{\lambda}(\sigma)W^{\lambda}=W^{\lambda}B^{\lambda}(\sigma)\) for \(\lambda\) \(\vdash n\), \(\sigma \in S_ n\). It is shown that the \(W^{\lambda}\) are upper triangular matrices and can be chosen with 1’s in the diagonal. The description of the \(W^{\lambda}\) is explicit for hook partitions.
Reviewer: J.B.Olsson

MSC:
20C30 Representations of finite symmetric groups
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