2015/08/22 by Klaus Lux, Lux, Klaus, Nham V. Ngo +3
Mathematics · #20C30 #20C33 #20G10 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1508.05534
openalex publication_date 2015/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let SL2 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 Hn(SL2,V(m)) for Weyl SL2-modules V(m). We provide a closed formula for dimHn(SL2,V(m)) when n≤ 2p-3 and show that this dimension is bounded by the (n+1)-th Fibonacci number. This formula is then used to compute dimHn(SL2, V(m)) for n=1, 2, or 3. For n>2p-3, an exponential bound, only depending on n, is obtained for dimHn(SL2,V(m)). Analogous results are also established for the extension spaces ExtnSL2(V(m2),V(m1)) between Weyl modules V(m1) and V(m2). In particular, we determine the degree three extensions for all Weyl modules of SL2. 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 SL2, and the finite group of Lie type SL2(ps).