Partial fraction decompositions, and semilinear representations of infinite symmetric groups

M. Rovinsky

paper

Partial fraction decompositions, and semilinear representations of infinite symmetric groups

arXiv:2205.15144

Abstract

Let $F ∣ k$ be a non-trivial regular field extension, $S$ be an infinite (discrete) set, $G$ be the group of all permutations of $S$ endowed with the compact-open topology, $L$ be the fraction field of the tensor product over $k$ of the copies of $F$ labeled by $S$ . The field $L$ is endowed with the natural $G$ -action. For each $G$ -invariant subfield $K$ of $L$ , let $S m_{K}$ denote the category of smooth (i.e. with open stabilizers) $K$ -semilinear representations of $G$ . The categories $S m_{K}$ (especially, their simple and injective objects) are the principal object of the present study, though only in some particular cases. It is known that the indecomposable injective objects of the category $S m_{L}$ are the $L$ -exterior powers $L ⟨ (s S) ⟩$ ( $s \geq 0$ ) of the $L$ -vector space with the basis $S$ , while $L$ is the only simple object. It turns out that the objects $K ⟨ (s S) ⟩$ are injective quite generally. Let $K = L^{H} \subset L$ be the fixed field of an algebraic automorphism $k$ -group $H$ of $F ∣ k$ acting on $L$ diagonally. The question is: what could be a relation (a kind of the Schur--Weyl duality) between representations of $H$ and the indecomposable injectives or simple objects of $S m_{K}$ ? In this paper we consider several examples, where $H$ is either a subgroup of $P G L_{2, k}$ or a torus. In these examples: a) a natural bijection between the finite-dimensional simple objects of $S m_{K}$ and the irreducible rational representations of $H$ is constructed; b) for $H \neq = P G L_{2, k}$ , the indecomposable injectives and the simple objects of $S m_{K}$ are described completely. For $H = P G L_{2, k}$ , an infinite list of infinite-dimensional simple objects is produced, which is shown to be complete if $F \neq = k$ ; a system of indecomposable injective generators is described.

arXiv commentSupersedes arXiv:1405.3265 & 1508.02267; v2: many modifications, all irreducible finite-dimensional representations over `cross-ratio' fields $K$ are described; v3: more revisions, all irreducible level 1 representations over $K$ are described, the injectivity of objects induced from trivial representaions of sufficiently small subgroups is shown, a part of v2 moved to arXiv:2408.04028