Cohomology of $S{L}_2$ and related structures

Abstract

Let $S{L}_2$ 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 ${\text{H}}^n(S{L}_2,V(m))$ for Weyl $S{L}_2$-modules $V(m)$. We provide a closed formula for $\text{dim}{\text{H}}^n(S{L}_2,V(m))$ 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 $\text{dim}{\text{H}}^n(S{L}_2,V(m))$ for $n=1,2,$ or $3$. For $n>2p-3$, an exponential bound, only depending on $n$, is obtained for $\text{dim}{\text{H}}^n(S{L}_2,V(m))$. Analogous results are also established for the extension spaces ${\text{Ext}}_{S{L}_2}^n(V(m_2),V(m_1))$ between Weyl modules $V(m_1)$ and $V(m_2)$. In particular, we determine the degree three extensions for all Weyl modules of $S{L}_2$. 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 $S{L}_2$, and the finite group of Lie type $S{L}_2(p^s)$.

References in corpus (1)

• Cohomology for Frobenius kernels of $S{L}_2$