A Representation Stability Theorem for VI-modules

Abstract

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.

Stable Decompositions of Certain Representations of the Finite General Linear Groups

“…As we shall explain in Section 3, Theorem 4 generalizes the main results of [5]. However, the proof of Theorem 4 depends crucially on the key propositions in [5] and also uses Theorem 2.…”

“…However, the proof of Theorem 4 depends crucially on the key propositions in [5] and also uses Theorem 2. For each m 0, the sequence k[G n /G n−m ] where n m can be assembled into a free VIC-module P (m), and any finitely generated VIC-module over k is a quotient of a finite direct sum of VIC-modules of the form P (m).…”

Stability for Representations of Hecke Algebras of type $A$

“…In [3], Gan and Watterlond researched V I -modules, where V I is the category of finite dimensional vector spaces and injective linear maps. They proved a result about the representation stability for the family of finite general linear groups GL n (F q ).…”