2003/01/31 by Benoit Fresse, Benoît Fresse, Fresse, Benoit · 5 citations
Mathematics · #05C05 #05E25 #17B01 #18C15 #18D50 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:05C05 #msc:05E25 #msc:17B01 #msc:18C15 #msc:18D50
paper · pdf · doi:10.48550/arxiv.math/0301365
144 pages. Includes a glossary and notation index
arxiv created 2003/01/31 · openalex publication_date 2003/01/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider partitions of a set with r elements ordered by refinement. We consider the simplicial complex K(r) formed by chains of partitions which starts at the smallest element and ends at the largest element of the partition poset. A classical theorem asserts that K(r) is equivalent to a wedge of r-1-dimensional spheres. In addition, the poset of partitions is equipped with a natural action of the symmetric group in r letters. Consequently, the associated homology modules are representations of the symmetric groups. One observes that the r-1th homology modules of K(r), where r = 1,2,..., are dual to the Lie representation of the symmetric groups. In this article, we would like to point out that this theorem occurs a by-product of the theory of Koszul operads. For that purpose, we improve results of V. Ginzburg and M. Kapranov in several directions. More particularly, we extend the Koszul duality of operads to operads defined over a field of positive characteristic (or over a ring). In addition, we obtain more conceptual proofs of theorems of V. Ginzburg and M. Kapranov.