Abstract
Let VI be the category whose objects are the finite dimensional vector spaces over a finite field of order q and whose morphisms are the injective linear maps. A VI-module over a ring is a functor from the category VI to the category of modules over the ring. A VI-module gives rise to a sequence of representations of the finite general linear groups. We prove that the sequence obtained from any finitely generated VI-module over an algebraically closed field of characteristic zero is representation stable - in particular, the multiplicities which appear in the irreducible decompositions eventually stabilize. We deduce as a consequence that the dimension of the representations in the sequence {V n } obtained from a finitely generated VI-module V over a field of characteristic zero is eventually a polynomial in q n. Our results are analogs of corresponding results on representation stability and polynomial growth of dimension for FI-modules (which give rise to sequences of representations of the symmetric groups) proved by Church, Ellenberg, and Farb.
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Presented by Henning Krause.
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Gan, W.L., Watterlond, J. A Representation Stability Theorem for VI-modules. Algebr Represent Theor 21, 47–60 (2018). https://doi.org/10.1007/s10468-017-9703-2
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DOI: https://doi.org/10.1007/s10468-017-9703-2