\FI_G-modules, orbit configuration spaces, and complex\n reflection groups

2016/08/22 by Kevin Casto, Casto, Kevin
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1608.06317

openalex publication_date 2016/08/22 · openalex created_date 2022/09/02 · openalex updated_date 2026/07/28

Abstract

The category \FI_G was first defined and explored by Sam-Snowden.\nHere, we develop more of the machinery of \FI_G-modules and find\nnumerous examples to apply it to, extending the work of Church-Ellenberg-Farb\nand Wilson.\n In particular we develop a notion of character polynomials for\n\FI_G-modules with G finite, a notion of representation stability\nwhich we call K₀-stability even when G is infinite virtually polycyclic,\nand apply the notion of finite presentation degree when G is a general\ninfinite group. We use this to analyze numerous families of (Gⁿ rtimes\nS_n)-modules, such as:\n -the cohomology and homotopy groups of orbit configuration spaces\n -the diagonal coinvariant algebra of complex reflection groups\n -the homology of affine pure braid groups of type widetildeA_n and\n widetildeC_n\n -the cohomology of Fouxe-Rabinowitsch groups\n and many more examples.\n

Citations

• Coinduction functor in representation stability theory
• Two homological proofs of the Noetherianity of FI_G
• On the degree-wise coherence of FI_G-modules

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