arXiv subjects
Publications and source records attributed to Cihan Bahran.
We establish bounds on the local degrees of FI-modules (in the sense of Church-Miller-Nagpal-Reinhold [CMNR18]) through a spectral sequence which only depend on the initial page and is independent from how many pages have been flipped.
These notes include introductory material on the notion of splitting fields for modules over a k-algebra where k is a field.
We characterize the regularity of an FI-module using the derivative functors.
We improve the central stability ranges for H2(Torelli subgroup of Aut(Fn)'s) as GL_n(Z)-representations, H2(Torelli subgroup of mapping class groups) as Sp_2g(Z)-representations, Hk(congruence subgroups of GL_n(R)'s) as SL_n^U(R/I)-representations.
In terms of local cohomology, we give an explicit range as to when the FI-homology of an FI-module attains its regularity.
We identify two recursively defined polynomial conditions for FI-modules in the literature. We characterize these conditions using homological invariants of FI-modules (namely the local degree and regularity, together with the stable degree) and clarify their relationship. For one of these conditions, we give improved twisted homological stability ranges for the symmetric groups. As another application, we improve the representation stability ranges for congruence subgroups with respect to the action of an appropriate linear group by a factor of 2 in its slope.
We give refined bounds for the regularity of FI-modules and the stable ranges of FI-modules for various forms of their stabilization studied in the representation stability literature. We show that our bounds are sharp in several cases. We apply these to get explicit stable ranges for diagonal coinvariant algebras, and improve those for ordered configuration spaces of manifolds and congruence subgroups of general linear groups.
For a fixed prime $p$, we consider a filtration of the commuting complex of elements of order $p$ in the symmetric group $\mathfrak{S}_n$. The filtration is obtained by imposing successively relaxed bounds on the number of disjoint $p$-cycles in the cycle decomposition of the elements. We show that each term in the filtration becomes highly acyclic as $n$ increases. We use $\mathbf{FI}$-modules in the proof.
We show how to get explicit induction formulae for finite group representations, and more generally for rational Green functors, by summing a divergent series over Dwyer's subgroup and centralizer decomposition spaces. This results in formulae with rational coefficients. The former space yields a well-known induction formula, the latter yields a new one. As essentially immediate corollaries of the existing literature, we get similar formulae in group cohomology and stable splittings of classifying spaces.
Recently, Church-Miller-Nagpal-Reinhold [arXiv:1706.03845] obtained linear stable ranges for the integral cohomology of ordered configuration spaces, in the sense of representation stability. We note that the constants in these linear stable ranges can be further improved for an orientable manifold of dimension at least 3. The idea is to simply exploit the sparsity of Totaro's spectral sequence.