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 VImodule 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 {Vn} 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.
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 VImodule 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 {Vn} 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. 2010 Mathematics Subject Classification. 20C33.
Abstract. We prove that the irreducible decomposition of the permutation representation of GLn(Fq) on GLn(Fq)/GLn−m(Fq) stabilizes for large n. We deduce, as a consequence, a representation stability theorem for finitely generated VIC-modules.
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