We present a class of subposets of the partition lattice n with the following property: The order complex is homotopy equivalent to the order complex of n?1 ; and the Sn-module structure of the homology coincides with a recently discovered lifting of the S n?1-action on the homology of n?1 : This is the Whitehouse representation on Robinson's space of fully-grown trees, and has also appeared in...

We show that the order complex of any finite lattice with a chain 0̂ < m1 < · · · < mr < 1̂ of modular elements is at least (r − 2)-connected. In [St], Stanley shows that a finite lattice L with a maximal chain consisting entirely of modular elements is supersolvable and therefore graded and EL-shellable in the sense of [Bj1]. Hence, the order complex ∆(L \ {0̂, 1̂}) has the homotopy type of a wedg...

Given a permutation w ∈ Sn, we consider a determinantal ideal Iw whose generators are certain minors in the generic n × n matrix (filled with independent variables). Using ‘multidegrees’ as simple algebraic substitutes for torus-equivariant cohomology classes on vector spaces, our main theorems describe, for each ideal Iw: • variously graded multidegrees and Hilbert series in terms of ordinary ...

The amplituhedron \(\mathcal{A}_{n,k,m}\) was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes planar \(\mathcal{N}=4\) supersymmetric Yang-Mills theory. It is projection inside the Grassmannian \(\text{Gr}_{k,k+m}\) of totally nonnegative part \(\text{Gr}_{k,n}\). Karp Williams (2019) studied \(m=1\) \(\mathcal{A}_{n,k,1}\), g...