The common basis complex and the partial decomposition poset
arxiv(2024)
摘要
For a finite-dimensional vector space V, the common basis complex of V is
the simplicial complex whose vertices are the proper non-zero subspaces of V,
and σ is a simplex if and only if there exists a basis B of V that
contains a basis of S for all S∈σ. This complex was introduced by
Rognes in 1992 in connection with stable buildings.
In this article, we prove that the common basis complex is homotopy
equivalent to the proper part of the poset of partial direct sum decompositions
of V. Moreover, we establish this result in a more general combinatorial
context, including the case of free groups, matroids, vector spaces with
non-degenerate sesquilinear forms, and free modules over commutative Hermite
rings, such as local rings or Dedekind domains.
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