VI-modules in non-describing characteristic. II
This paper establishes fundamental results about the category \(\mathrm{Mod}_{\mathrm{VI}}\) of \(\mathrm{VI}\)-modules and its localization away from torsion. Here, \(\mathrm{VI}\) is the category of finite-dimensional \(\mathbb{F}_q\)-vector spaces and injective linear maps; \(\mathrm{Mod}_{\mathrm{VI}}\) is the category of functors from \(\mathrm{VI}\) to \(\mathbf{k}\)-vector spaces, where \(\mathbf{k}\) is taken to be of non-describing characteristic for the main results. When \(\mathbf{k}\) is of characteristic zero, the results recover and strengthen those of [\textit{W. L. Gan} and \textit{J. Watterlond}, Algebr. Represent. Theory 21, No. 1, 47--60 (2018; Zbl 1485.20036)]. The category of generic \(\mathrm{VI}\)-modules is defined to be the quotient \[ \mathrm{Mod}_{\mathrm{VI}} / \mathrm{Mod}_{\mathrm{VI}}^{\mathrm{tors}} \] where \(\mathrm{Mod}_{\mathrm{VI}}^{\mathrm{tors}}\) is the full subcategory of torsion \(\mathrm{VI}\)-modules. The main results lead to the classification of the irreducible generic \(\mathrm{VI}\)-modules: their isomorphism classes are in bijection with \[ \amalg_{n\geq 0} \mathrm{Irr}(\mathrm{Mod}_{\mathbf{k}[\mathrm{GL}_n]}). \] One construction of the irreducibles is as follows. The induction functor \(\mathcal{J}\) is defined as the left adjoint to restriction to the maximal subgroupoid of \(\mathrm{VI}\) and a \(\mathrm{VI}\)-module is said to be semi-induced if it admits a finite filtration with associated graded that is induced. Thus, for \(\Theta\) an irreducible \(\mathrm{GL}_d\)-module, one has the induced \(\mathrm{VI}\)-module \(\mathcal{J}(\Theta)\). The associated irreducible generic \(\mathrm{VI}\)-module \(L(\Theta)\) is represented by the \(\mathrm{VI}\)-module \(\mathcal{L}(\Theta)\subset \mathcal{J}(\Theta)\) given as the intersection of the kernels of all maps from \(\mathcal{J}(\Theta)\) to semi-induced modules generated in degree \(<d\). The proofs rely on a different, more explicit approach, and are established by combinatorial and representation-theoretic arguments. Namely, the author constructs an explicit descending filtration \[ \mathcal{J}(\Theta) \supset M_{d,\Theta} \supset M_{d+1, \Theta} \supset \dots \] of non-zero subobjects which has the remarkable property that any subobject \(0\neq M \subset \mathcal{J}(\Theta)\) contains almost all the \(M_{n,\Theta}\). In non-describing characteristic, for any \(n \geq d\), the inclusion \(M_{n,\Theta} \subset M_{d,\Theta}\) induces an isomorphism of generic \(\mathrm{VI}\)-modules. Then \(L(\Theta)\) is represented by \(M_{d,\Theta}\) (which is not isomorphic to \(\mathcal{L}(\Theta)\) in general). The proof of the main results concludes by applying results from Part I [the author, Algebra Number Theory 13, No. 9, 2151--2189 (2019; Zbl 1461.20004)] where the \(\delta\)-invariant of finitely-generated \(\mathrm{VI}\)-modules is introduced. One key property is that, if \(0 \neq M \subset \mathcal{J}(\Theta)\), then \(\delta (M)=d\), so that \(\mathcal{J}(\Theta)\) contains no non-zero subobject with \(\delta\)-invariant \(<d\). The irreducible \(L(\Theta)\) is constructed by removing quotients that are of \(\delta\)-invariant \(<d\). The methods also apply to the study of (generic) \(\mathrm{FI}\)-modules, where \(\mathrm{FI}\) is the category of finite sets and injections, extending known results. The author also discusses the relationship with \(\mathrm{GL}_\infty\)-representations and Deligne categories.
scientific article; zbMATH DE number 7436430
scientific article