an:06839337
Zbl 06839337
Gan, Wee Liang; Watterlond, John
A representation stability theorem for VI-modules
EN
Algebr. Represent. Theory 21, No. 1, 47-60 (2018).
00373735
2018
j
20C33
representation stability; multiplicity stability; finite general linear groups; VI-modules
Summary: 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.