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تسجيل مستخدم جديدHomology representations arising from the half cube, II
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تأليف
R.M. Green
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In a previous work (arXiv:0806.1503v2), we defined a family of subcomplexes of the $n$-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least $k$, and we proved that the homology of such a subcomplex is concentrated in degree $k-1$. This homology group supports a natural action of the Coxeter group $W(D_n)$ of type $D$. In this paper, we explicitly determine the characters (over ${Bbb C}$) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group $S_n$ by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of $sym_n$ agree (over ${Bbb C}$) with the representations of $sym_n$ on the $(k-2)$-nd homology of the complement of the $k$-equal real hyperplane arrangement.
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