THE YONEDA EXTENSION ALGEBRA OF GL2(Fp)

We compute the Yoneda extension algebra of GL2 over an algebraically closed field of characteristic p > 0. 1. Intro Let F be an algebraically closed field of characteristic p > 0. Let G = GL2(F ) denote the group of 2 × 2 invertible matrices over F . Let L denote a complete set of irreducible objects in the category G -mod of rational representations of G. The object of this paper is to give an explicit description of the Yoneda extension algebra Y = ⊕ L,L∈L ExtG -mod(L,L ) of G. The best previous work in this direction was done by A. Parker, who outlined an intricate algorithm to compute the dimension of ExtnG -mod(L,L ) for L,L ∈ L and n ≥ 0 [6]. Our approach is quite different. We develop a theory of homological duality for certain algebraic operators O which we introduced previously in our study of the category of rational representations of G, and use this to give a combinatorial description of Y as an algebra. The categoryG -mod has countably many blocks, all of which are equivalent. Therefore, the algebra Y is isomorphic to a direct sum of countably many copies of y, where y is the Yoneda extension algebra of the principal block of G. Our problem is to compute y. Suppose Γ = ⊕ i,j∈Z Γ ij is a bigraded algebra. We have a combinatorial operator OΓ which acts on the collection of Z-graded algebras ∆ after the formula OΓ(∆) i = ⊕