VI-modules in nondescribing characteristic, part I

Abstract

We classify all irreducible generic VI-modules in non-describing characteristic. Our result degenerates to yield a classification of irreducible generic FI-modules in arbitrary characteristic. Our result can also be viewed as a classification theorem for a natural class of representations of GL∞(Fq).

Linear and quadratic ranges in representation stability

“…We say that an FI-module is semi-induced 1 if it admits a finite length filtration where the quotients are induced modules. The following is a useful property of semi-induced modules which holds not only for FI-modules but also many other similar functor categories; see for example [Ram1,Remark 2.33] and [Nag2,Corollary 4.23].…”

Categorifications of rational Hilbert series and characters of $FS^{op}$ modules

“…• Let VI q be the category of finite dimensional vector spaces over F q and linear injections between them. If V is a finitely generated VI q module and char k = char F q , then h V (t) is rational with denominator d−1 j=0 (1 − q j t) for some d ∈ N. Equivalently, (dim V n ) n∈N eventually agrees with a polynomial in q n , proved by Nagpal [14].…”

“…Thus to prove the statement for M it suffices to prove that B F d q (Σ V M) is exact for dim V ≫ 0 and d ≫ 0. Theorem 1.2 of Nagpal [14] states that if q is invertible in k there is V ∈ VI q such that Σ V M admits a finite filtration whose associated graded modules are induced. The spectral sequence for this filtration reduces us to the case where M is an induced module.…”

“…Despite these applications, the representation theory of FS op is not well understood. For finitely generated FI modules and VI q modules in non-describing characteristic, Nagpal has proved structure theorems [13,14], known as shift theorems, which we use to establish our results. But FS op modules do not satisfy a shift theorem, and are qualitatively different: the category of finitely generated FS op modules has infinite Krull-Gabriel dimension, as opposed to dimension 1.…”