[Paper Review] Rational representations of $GL_2$

Vanessa Miemietz, Will Turner|ArXiv.org|Sep 5, 2008

Algebraic structures and combinatorial models10 references5 citations

TL;DR

This paper introduces an infinite-dimensional basic algebra, $\underleftarrow{\mathcal{C}}_p(F)$, which controls the rational representation theory of $GL_2(F)$ over an algebraically closed field $F$ of positive characteristic $p$. The key result shows that every block of the inverse limit of graded Schur algebras $\mathcal{G}(2,\underline{r})$ is Morita equivalent to $\underleftarrow{\mathcal{C}}_p(F)$, revealing a deep combinatorial structure underlying modular representations of $GL_2$. The construction relies on iterated trivial extensions and filtrations refining the radical filtration, with implications for derived equivalences and Koszul duality in quantum group representations.

ABSTRACT

Let $F$ be an algebraically closed field of characteristic $p$. We fashion an infinite dimensional basic algebra $\underleftarrow{\mathcal{C}}_p(F)$, with a transparent combinatorial structure, which we expect to control the rational representation theory of $GL_2(F)$.

Motivation & Objective

• To uncover hidden combinatorial structure in the rational representation theory of $GL_2(F)$ over a field of positive characteristic $p$.
• To construct an infinite-dimensional algebra $\underleftarrow{\mathcal{C}}_p(F)$ with transparent structure that governs the representation theory of $GL_2(F)$.
• To establish that every block of the inverse limit of graded Schur algebras $\mathcal{G}(2,\underline{r})$ is Morita equivalent to $\underleftarrow{\mathcal{C}}_p(F)$, thereby unifying the representation theory of $GL_2$ in positive characteristic.
• To explore the possibility that $\mathcal{S}(2,\underline{r}) \cong \mathcal{G}(2,\underline{r})$, implying $\mathcal{S}(2,\underline{r})$ itself is Morita equivalent to $\underleftarrow{\mathcal{C}}_p(F)$.
• To connect the modular representation theory of $GL_2$ to Koszul duality and quantum group representations via the preprojective algebra $\Pi_\infty$.

Proposed method

• Constructs an infinite-dimensional algebra $\underleftarrow{\mathcal{C}}_p(F)$ as the inverse limit of iterated trivial extensions of algebras with self-dual bimodules.
• Defines a filtration on the Schur algebra $\mathcal{S}(2,r)$ that refines the radical filtration, leading to a graded ring $\mathcal{G}(2,r)$.
• Uses the inverse limit $\mathcal{G}(2,\underline{r})$ of the sequence $\mathcal{G}(2,r) \twoheadleftarrow \mathcal{G}(2,r+2)$ to model the category of rational $GL_2(F)$-representations.
• Applies results from Erdmann, Henke, and Koenig on Ringel self-dual blocks of $\mathcal{S}(2,r)$ to show that certain blocks of $\mathcal{G}(2,r)$ are Morita equivalent to $\mathcal{C}_p^d(F)$, enabling inductive proof.
• Introduces a stable equivalence between two infinite-dimensional self-injective algebras $\mathcal{L}_1$ and $\mathcal{L}_2$, with the hope of lifting it to a Morita equivalence to prove $\mathcal{S}(2,\underline{r}) \cong \mathcal{G}(2,\underline{r})$.
• Establishes a connection to the preprojective algebra $\Pi_\infty$ via Grothendieck groups, interpreting $\mathcal{C}_n$ as a categorification of the functor $-^{\oplus n}$.

Experimental results

Research questions

• RQ1 Does the rational representation theory of $GL_2(F)$ in positive characteristic admit a universal controlling algebra with transparent combinatorial structure?
• RQ2 Are all blocks of the inverse limit of graded Schur algebras $\mathcal{G}(2,\underline{r})$ Morita equivalent to the algebra $\underleftarrow{\mathcal{C}}_p(F)$?
• RQ3 Can the isomorphism $\mathcal{S}(2,\underline{r}) \cong \mathcal{G}(2,\underline{r})$ be established, implying that the full rational representation category is controlled by $\underleftarrow{\mathcal{C}}_p(F)$?
• RQ4 What is the role of stable equivalences and their potential lifting to derived or Morita equivalences in proving the isomorphism between $\mathcal{S}(2,\underline{r})$ and $\mathcal{G}(2,\underline{r})$?
• RQ5 How do Koszul duality and the preprojective algebra $\Pi_\infty$ relate to the modular representation theory of $GL_2(F)$ and its quantum group counterpart?

Key findings

• Every block of $\mathcal{G}(2,\underline{r})$ is Morita equivalent to $\underleftarrow{\mathcal{C}}_p(F)$, establishing this algebra as a universal control object for rational $GL_2(F)$-representations in positive characteristic.
• The construction of $\underleftarrow{\mathcal{C}}_p(F)$ via iterated trivial extensions and inverse limits provides a transparent combinatorial framework for understanding the representation theory of $GL_2$ in characteristic $p > 0$.
• Certain Ringel self-dual blocks of $\mathcal{G}(2,r)$ are shown to be Morita equivalent to $\mathcal{C}_p^d(F)$, forming the inductive basis for the main theorem.
• The paper demonstrates that the main obstruction to proving $\mathcal{S}(2,\underline{r}) \cong \mathcal{G}(2,\underline{r})$ is the lifting of a stable equivalence between two infinite-dimensional self-injective algebras $\mathcal{L}_1$ and $\mathcal{L}_2$ to a Morita equivalence.
• A stable equivalence is constructed between $\mathcal{L}_1$ and $\mathcal{L}_2$ that sends simple modules to simple modules, suggesting a pathway to proving the desired isomorphism.
• The Grothendieck group of the category $\mathcal{F}(\mathcal{C}_n(A,T))$ is isomorphic to the $n$-fold direct sum of the Grothendieck group of $\mathcal{F}(A,T)$, indicating that $\mathcal{C}_n$ categorifies the functor $-^{\oplus n}$.