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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The center of $ \mathbb{Z}[S^{n+1}]$ is the set of symmetric polynomials in $ n$ commuting transposition-sums


Author: Gadi Moran
Journal: Trans. Amer. Math. Soc. 332 (1992), 167-180
MSC: Primary 20C30; Secondary 05E05, 05E10
DOI: https://doi.org/10.1090/S0002-9947-1992-1062873-1
MathSciNet review: 1062873
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Abstract: Let $ {S_{n + 1}}$ be the symmetric group on the $ n + 1$ symbols $ 0,1,2, \ldots ,n$. We show that the center of the group-ring $ \mathbb{Z}[{S_{n + 1}}]$ coincides with the set of symmetric polynomials with integral coefficients in the $ n$ elements $ {s_1}, \ldots ,{s_n} \in \mathbb{Z}[{S_{n + 1}}]$, where $ {s_k} = \sum\nolimits_{0 \leq i < k} {(i,k)} $ is a sum of $ k$ transpositions, $ k = 1, \ldots ,n$. In particular, every conjugacy-class-sum of $ {S_{n + 1}}$ is a symmetric polynomial in $ {s_1}, \ldots ,{s_n}$.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1062873-1
Article copyright: © Copyright 1992 American Mathematical Society