Quotients of the order complex \(\Delta (\overline \Pi _n)\) by subgroups of the Young subgroup \(S_{1}\times S_{n-1}\) (Q712199)

The author studies the action of subgroups of the symmetric group \(S_n\) on the order complex \(\Delta(\overline{\Pi}_n)\) of the poset \(\overline{\Pi}_n\) obtained from the partition lattice \(\Pi_n\) on \([n]=\{1,2,\dots,n\}\) by removing the smallest and the greatest elements. It was known from [\textit{D. N. Kozlov}, Proc. Am. Math. Soc. 128, No.~8, 2253--2259 (2000; Zbl 0939.05086)] that \(\Delta(\overline{\Pi}_n)/S_n\) is contractible. The main result of this paper is that \(\Delta(\overline{\Pi}_n)/G\) is homotopy equivalent to a wedge of \((n-3)\)-dimensional spheres when \(G\) is a subgroup of \(S_1\times S_{n-1}:=\{\sigma \in S_n: \sigma(1)=1\}\) with \(n\geq 3\). The technique of the proof uses discrete Morse theory.