Representation stability for the cohomology of the pure string motion groups

By Jenny Wilson. 2011

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The cohomology of the pure string motion group PSigma_n admits a natural action by the hyperoctahedral group W_n. Church and Farb conjectured that for each k > 0, the sequence of degree k rational cohomology groups of PSigma_n is uniformly representation stable with respect to the induced action by W_n, that is, the description of the groups' decompositions into irreducible W_n representations stabilizes for n >> k. We use a characterization of the cohomology groups given by Jensen, McCammond, and Meier to prove this conjecture. Using a transfer argument, we further deduce that the rational cohomology groups of the string motion group vanish in positive degree. We also prove that the subgroup of orientation-preserving string motions, also known as the braid-permutation group, is rationally cohomologically stable in the classical sense.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4eb72ea5f38794aaff85a0b55bf68a6c3107c5a1

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Citations in the corpus

FI_W-modules and stability criteria for representations of the classical Weyl groups (2013, methodology): “We recover (with considerably less effort) the main result of our previous paper [Wil12], which stated that for eachm, the sequence ofBn–representations {H(PΣn,Q)}n”
Representation theory and homological stability (2010, methodology): “ n-pointed Riemann surfaces [JR1, JR2], the cohomology of certain hyperplane arrangements [Mo], syzygies of line bundles on Segre– Veronese varieties [Ra], the cohomology of pure string motion groups [Wi1] (verifying Conjecture 4.8 below), and the cohomology of conﬁguration spaces of nordered points”
Representation Stability (2014, background): “Wilson applies this theory to a number of examples, including the cohomology of the pure string motion groups (see also [33]), the cohomology of various hyperplane arrangements, and diagonal co-invariant algebras for Weyl groups.”
A journey through loop braid groups (2016, result): “However, as Wilson recalls in [51], it follows from two results of Wattenberg [50][Lemma 1.”