Cohomology of $SL_2$ and related structures

TL;DR

This paper introduces a new method to compute the dimensions of cohomology spaces for $SL_2$ modules, providing explicit formulas for low degrees and bounds for higher degrees, with applications to related algebraic structures.

Contribution

It develops a novel approach for calculating cohomology dimensions of $SL_2$ modules, including explicit formulas and bounds, and extends results to extension spaces and related algebraic structures.

Findings

Closed formula for $ ext{dim} ext{H}^n(SL_2,V(m))$ when $n \\le 2p-3$

Dimension bounded by Fibonacci numbers for low degrees

Explicit computation of degree three extensions between Weyl modules

Abstract

Let $SL_2$ be the rank one simple algebraic group defined over an algebraically closed field $k$ of characteristic $p>0$. The paper presents a new method for computing the dimension of the cohomology spaces $\text{H}^n(SL_2,V(m))$ for Weyl $SL_2$-modules $V(m)$. We provide a closed formula for $\text{dim}\text{H}^n(SL_2,V(m))$ when $n\le 2p-3$ and show that this dimension is bounded by the $(n+1)$-th Fibonacci number. This formula is then used to compute $\text{dim}\text{H}^n(SL_2, V(m))$ for $n=1, 2,$ or $3$. For $n>2p-3$, an exponential bound, only depending on $n$, is obtained for $\text{dim}\text{H}^n(SL_2,V(m))$. Analogous results are also established for the extension spaces $\text{Ext}^n_{SL_2}(V(m_2),V(m_1))$ between Weyl modules $V(m_1)$ and $V(m_2)$. In particular, we determine the degree three extensions for all Weyl modules of $SL_2$. As a byproduct, our results and techniques give explicit upper bounds for the dimensions of the cohomology of the Specht modules of symmetric groups, the cohomology of simple modules of $SL_2$, and the finite group of Lie type $SL_2(p^s)$.