Representation stability in the level 4 braid group (Q2163311)

Let \(k\) be the field of either \(\mathbb{Q}\) or \(\mathbb{C}\). Let \(\mathbf{B}_n\) and \(\mathbf{PB}_n\) be the braid and pure braid groups of \(n\)-strands, respectively. Then there is an exact sequence \[ 1 \to \mathbf{PB}_n \to \mathbf{B}_n \to \mathcal{S}_n \to 1, \] where \(\mathcal{S}_n\) is the symmetric group. There is a natural (conjugation) \(\mathcal{S}_n\)-action on \(\mathbf{PB}_n\) permuting the strands, giving rise to \(\mathcal{S}_n\)-representations \(H_q(\mathbf{PB}_n,k)\). There is a natural inclusion morphism \(\iota : \mathbf{PB}_n \to \mathbf{PB}_{n+1}\). From this, \textit{T. Church} and \textit{B. Farb} [Adv. Math. 245, 250--314 (2013; Zbl 1300.20051); Duke Math. J. 164, No. 9, 1833--1910 (2015; Zbl 1339.55004)] showed a remarkable relation between the \(H_q(\mathbf{PB}_n,\mathbb{Q})\) as one varies \(n\) for large \(n\). This is called ``representation stability''. Consider the sequence \[ \mathbf{B}_n \to \mathbf{GL}_n(\mathbb{Z}[t, t^{-1}]) \to \mathbf{GL}_n(\mathbb{Z}) \to \mathbf{GL}_n(\mathbb{Z}/m\mathbb{Z}), \] where the first map is the Burau representation, the second one is by setting \(t=-1\) and the last one being \(mod \ m\) and \(m\) here is known as the level in the tradition of modular forms. This induces the exact sequence \[ 1 \to \mathbf{B}_n[m] \to \mathbf{B}_n \to \mathcal{Z}_n \to 1, \] where \(\mathcal{Z}_n\) is the image of the composition in \(\mathbf{GL}_n(\mathbb{Z}/m\mathbb{Z})\). The group \(\mathcal{Z}_n\) has a quotient (compatible with first sequence above): \[ 1 \to \mathcal{PZ}_n \to \mathcal{Z}_n \to \mathcal{S}_n \to 1. \] Again, there is an analogous natural map \(\iota_n : \mathbf{B}_n[m] \to \mathbf{B}_{n+1}[m]\). Set \(m = 4\) and \(q = 1\). This paper studies the natural \(\mathcal{Z}_n\)-representations \(H_1(\mathbf{B}_n[4];k)\) in the context of representation stability. The first result (Thm 2.1) is that the dimension of \(H_1(\mathbf{B}_n[4];\mathbb{Q})\) is a degree 4 polynomial in \(n\). Along the line, the authors produce a basis in terms of the natural (Artin) generators of \(\mathbf{B}_n\). The main result (Thm 2.5) shows that \(H_1(\mathbf{B}_n[4];\mathbb{C})\) is a direct sum of four irreducible representations for \(n \ge 4\), inductively defined on \(n\). For a closed orientable surface \(\Sigma_g\) of genus \(g \ge 1\), denote by \(\mathbf{Mod}_g\) its mapping class group. There is a double cover \(\psi : \Sigma_g \to S^2\) to the 2-sphere, ramified at the \(2g+2\) points on \(S^2\) together with an involution \(\tau\) which induces an element in \(\mathbf{Mod}_g\), preserving the fibres of \(\psi\). Denote by \(\mathbf{SMod}_g\) the mapping class group of \(\Sigma_g\) and the subgroup centralizing \(\tau\), respectively. Denote by \(\mathbf{Mod}_g[m]\) the subgroup of \(\mathbf{Mod}_g\) fixing \(H_1(\Sigma_g, \mathbb{Z}/m\mathbb{Z})\) (in the spirit of Torelli) and \[ \mathbf{SMod}_g[m] = \mathbf{SMod}_g \cap \mathbf{Mod}_g[m]. \] From \(\psi, \tau\), we obtain morphisms \[ \mathbf{SMod}_g[m] \to \mathbf{B}_{2g+1}[m] \to \mathbf{B}_{2g+3}[m] \to \mathbf{SMod}_{g+1}[m]. \] From this, the paper shows a similar representation stability, namely that the \(\mathbf{SMod}_g[m]\)-representation \(H_1(\mathbf{SMog}_g[4]; \mathbb{C})\) decomposes into four irreducible representations defined inductively on \(g \ge 2\) (Cor 2.6). There is a quartic polynomial lower bound on the dimension of \(H_1(\mathbf{B}_{2g+1}[0]; \mathbb{Q})\) in terms of genus \(g\). There are a few more negative results (corollaries) concerning the level 4 Albanese cohomology of \(\mathbf{SMod}_g[4]\) and the \(d\)-characteristic variety of the complement of the braid arrangement moduli space.