On the Action of the Symmetric Group on the Cohomology of Groups Related to (Virtual) Braids

In this paper we consider the cohomology of four groups related to the virtual braids of [Kauffman] and [Goussarov-Polyak-Viro], namely the pure and non-pure virtual braid groups (PvB_n and vB_n, respectively), and the pure and non-pure flat braid groups (PfB_n and fB_n, respectively). The cohomologies of PvB_n and PfB_n admit an action of the symmetric group S_n. We give a description of the cohomology modules H^i(PvB_n,Q) and H^i(PfB_n,Q) as sums of S_n-modules induced from certain one-dimensional representations of specific subgroups of S_n. This in particular allows us to conclude that H^i(PvB_n,Q) and H^i(PfB_n,Q) are uniformly representation stable, in the sense of [Church-Farb]. We also give plethystic formulas for the Frobenius characteristics of these S_n-modules. We then derive a number of constraints on which S_n irreducibles may appear in H^i(PvB_n,Q) and H^i(PfB_n,Q). In particular, we show that the multiplicity of the alternating representation in H^i(PvB_n,Q) and H^i(PfB_n,Q) is identical, and moreover is nil for sufficiently large $n$. We use this to recover the (previously known) fact that the multiplicity of the alternating representation in H^i(PB_n,Q) is nil (here PB_n is the ordinary pure braid group). We also give an explicit formula for H^i(vB_n,Q) and show that H^i(fB_n,Q)=0. Finally, we give Hilbert series for the character of the action of S_n on H^i(PvB_n,Q) and H^i(PfB_n,Q). An extension of the standard `Koszul formula' for the graded dimension of Koszul algebras to graded characters of Koszul algebras then gives Hilbert series for the graded characters of the respective quadratic dual algebras.Comment: This version substantially extends version 1, with minimal changes to the original material. 49 pages, several figure

Twenty-five years of two-dimensional rational conformal field theory

In this article we try to give a condensed panoramic view of the development of two-dimensional rational conformal field theory in the last twenty-five years.Comment: A review for the 50th anniversary of the Journal of Mathematical Physics. Some references added, typos correcte

Representation stability for the cohomology of arrangements associated to root systems

From a root system, one may consider the arrangement of reflecting hyperplanes, as well as its toric and elliptic analogues. The corresponding Weyl group acts on the complement of the arrangement and hence on its cohomology. We consider a sequence of linear, toric, or elliptic arrangements which arise from a family of root systems of type A, B, C, or D, and we show that the rational cohomology stabilizes as a sequence of Weyl group representations. Our techniques combine a Leray spectral sequence argument similar to that of Church in the type A case along with FI$_W$-module theory which Wilson developed and used in the linear case. A key to the proof relies on a combinatorial description, using labelled partitions, of the poset of connected components of intersections of subvarieties in the arrangement.Comment: 20 pages; improved exposition and minor correction