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- W1941848763 abstract "This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group $S_n$ on the cohomology of the configuration space of $n$ ordered points in $mathbf{R}^d$. This cohomology is known to vanish outside of dimensions divisible by $d-1$; it is shown here that the $S_n$-representation on the $i(d-1)^{st}$ cohomology stabilizes sharply at $n=3i$ (resp. $n=3i+1$) when $d$ is odd (resp. even). The result comes from analyzing $S_n$-representations known to control the cohomology: the Whitney homology of set partition lattices for $d$ even, and the higher Lie representations for $d$ odd. A similar analysis shows that the homology of any rank-selected subposet in the partition lattice stabilizes by $ngeq 4i$, where $i$ is the maximum rank selected. Further properties of the Whitney homology and more refined stability statements for $S_n$-isotypic components are also proven, including conjectures of J. Wiltshire-Gordon." @default.
- W1941848763 created "2016-06-24" @default.
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- W1941848763 date "2015-05-15" @default.
- W1941848763 modified "2023-09-27" @default.
- W1941848763 title "Representation stability for cohomology of configuration spaces in $mathbf{R}^d$" @default.
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