[Paper Review] On the Action of the Symmetric Group on the Cohomology of Groups Related to (Virtual) Braids

Peter Lee|arXiv (Cornell University)|Apr 16, 2013

Geometric and Algebraic Topology3 citations

TL;DR

This paper investigates the action of the symmetric group $S_n$ on the rational cohomology of pure and non-pure virtual and flat braid groups. Using representation theory and Koszul duality, it decomposes the cohomology modules $H^i(P\negthinspace v\negthinspace B_n,\mathbb{Q})$ and $H^i(P\negthinspace f\negthinspace B_n,\mathbb{Q})$ as sums of induced $S_n$-modules from one-dimensional representations, proving uniform representation stability and deriving plethystic formulas and Hilbert series for the graded characters. It further shows that the alternating representation appears with multiplicity zero for large $n$, recovering a known result for ordinary pure braid groups.

ABSTRACT

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.

Motivation & Objective

• To understand the action of the symmetric group $S_n$ on the rational cohomology of pure and non-pure virtual and flat braid groups.
• To decompose the cohomology modules $H^i(P\negthinspace v\negthinspace B_n,\mathbb{Q})$ and $H^i(P\negthinspace f\negthinspace B_n,\mathbb{Q})$ as representations of $S_n$.
• To establish uniform representation stability for these cohomology modules.
• To derive plethystic formulas and Hilbert series for the graded characters of the $S_n$-action.
• To determine constraints on which irreducible $S_n$-representations appear, particularly the alternating representation.

Proposed method

• The cohomology modules are analyzed via the associated graded algebras of the group algebras of the pure braid groups.
• The action of $S_n$ on the top-degree component of the quadratic dual algebras $\mathfrak{pvb}_n^!$ and $\mathfrak{pfb}_n^!$ is explicitly described.
• Plethystic formulas are used to compute the Frobenius characteristics of the $S_n$-modules $H^i(P\negthinspace v\negthinspace B_n,\mathbb{Q})$ and $H^i(P\negthinspace f\negthinspace B_n,\mathbb{Q})$.
• The Koszul complex is constructed for the relevant algebras, and the generalized Koszul formula $A_\sigma(z)A_\sigma^!(-z) = 1$ is applied to derive character relations.
• The structure of induced representations from specific subgroups of $S_n$ is used to describe the cohomology modules as direct sums of induced $S_n$-modules.
• The vanishing of $H^i(f\negthinspace B_n,\mathbb{Q})$ is established using the Koszul complex and character analysis.

Experimental results

Research questions

• RQ1 How does the symmetric group $S_n$ act on the rational cohomology of the pure virtual braid group $P\negthinspace v\negthinspace B_n$?
• RQ2 Can the cohomology modules $H^i(P\negthinspace v\negthinspace B_n,\mathbb{Q})$ and $H^i(P\negthinspace f\negthinspace B_n,\mathbb{Q})$ be expressed as sums of induced $S_n$-modules from one-dimensional representations of subgroups?
• RQ3 Is the decomposition of $H^i(P\negthinspace v\negthinspace B_n,\mathbb{Q})$ and $H^i(P\negthinspace f\negthinspace B_n,\mathbb{Q})$ as $S_n$-modules uniformly representation stable?
• RQ4 What constraints exist on the appearance of irreducible $S_n$-representations, especially the alternating representation, in these cohomology modules?
• RQ5 What are the Hilbert series and graded characters for the $S_n$-action on $H^i(P\negthinspace v\negthinspace B_n,\mathbb{Q})$ and $H^i(P\negthinspace f\negthinspace B_n,\mathbb{Q})$?

Key findings

• The cohomology modules $H^i(P\negthinspace v\negthinspace B_n,\mathbb{Q})$ and $H^i(P\negthinspace f\negthinspace B_n,\mathbb{Q})$ are shown to decompose as direct sums of $S_n$-modules induced from one-dimensional representations of specific subgroups of $S_n$.
• The cohomology modules are uniformly representation stable in the sense of Church and Farb, meaning the decomposition stabilizes for large $n$.
• The multiplicity of the alternating representation in both $H^i(P\negthinspace v\negthinspace B_n,\mathbb{Q})$ and $H^i(P\negthinspace f\negthinspace B_n,\mathbb{Q})$ is identical and vanishes for sufficiently large $n$.
• The cohomology of the non-pure flat braid group satisfies $H^i(f\negthinspace B_n,\mathbb{Q}) = 0$ for all $i$.
• The cohomology of the non-pure virtual braid group is explicitly computed as $H^i(v\negthinspace B_n,\mathbb{Q})$ via the generalized Koszul formula.
• Hilbert series for the graded characters of the $S_n$-action on $H^i(P\negthinspace v\negthinspace B_n,\mathbb{Q})$ and $H^i(P\negthinspace f\negthinspace B_n,\mathbb{Q})$ are derived, and the same is extended to the quadratic dual algebras using the Koszul formula for graded characters.