[Paper Review] Coinduction functor in representation stability theory

Wee Liang Gan, Liping Li|arXiv (Cornell University)|Feb 24, 2015

Algebraic structures and combinatorial models23 references8 citations

TL;DR

This paper introduces and analyzes the coinduction functor in the context of representation stability theory for FI₉ and VI categories over fields of characteristic 0. By leveraging the coinduction functor and its adjunction with restriction, the authors provide new homological proofs of stability results and establish that all finitely generated projective modules over these categories are injective, a key structural result in representation stability.

ABSTRACT

We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new and simpler proofs of (generalizations of) various results on homological properties of FI-modules. We also prove that any finitely generated projective VI-module over a field of characteristic 0 is injective.

Motivation & Objective

• To develop and apply the coinduction functor in the category of FI₉- and VI-modules to study homological properties.
• To provide new, functorial proofs of representation stability results using coinduction and restriction functors.
• To establish that finitely generated projective modules over FI₉ and VI categories are injective when the base field has characteristic 0.
• To clarify the structure of coinduced modules, particularly Q(kC eₘ), in FI₉ and VI settings.
• To demonstrate the utility of coinduction in proving condition (RS3) for finitely generated projective modules.

Proposed method

• The coinduction functor Q is defined as the right adjoint to the restriction functor S induced by the embedding ι: C → C, n ↦ 1 ⊕ n.
• The authors compute Q(kC eₘ) for FI₉ and VI categories, showing it contains kC eₘ₊₁ as a direct summand under suitable conditions.
• They use the Eckman-Shapiro lemma to relate Ext groups via the adjunction (S, Q), enabling transfer of injectivity properties.
• For VI categories, a nontrivial surjection from Q(kC eₘ) to kC eₘ₊₁ is constructed when q (the size of the finite field) is invertible in k.
• The proof of injectivity relies on induction, starting from the known injectivity of kC e₀ in characteristic 0.
• The structure of induced modules is analyzed using wreath product group representations and Pieri’s formula for plethysm.

Experimental results

Research questions

• RQ1 How does the coinduction functor behave on standard modules kC eₘ in FI₉ and VI categories?
• RQ2 Can the coinduction functor be used to give new homological proofs of representation stability results?
• RQ3 Why are finitely generated projective modules over FI₉ and VI categories injective over fields of characteristic 0?
• RQ4 What is the precise structure of Q(kC eₘ) for FI₉ and VI categories, and how does it relate to kC eₘ₊₁?
• RQ5 Does the coinduction functor preserve or reflect injective modules in these categories?

Key findings

• For the FI₉ category, Q(kC eₘ) is isomorphic to kC eₘ ⊕ kC eₘ₊₁ when k is a commutative ring.
• For the VI category, if q is a unit in k, then Q(kC eₘ) contains kC eₘ₊₁ as a direct summand.
• Any finitely generated projective VI-module over a field of characteristic 0 is injective.
• Any finitely generated projective FI₉-module over a field of characteristic 0 is injective.
• The coinduction functor allows a new proof that finite generation implies condition (RS3) in representation stability.
• The structure of induced modules over wreath products is analyzed using Pieri’s formula, showing that the map ν ↦ ħ(ν) is bijective when n ≥ 2m.