The sheaf homology of the Cayley module for G2(q)

Butler, Mark William (2022). The sheaf homology of the Cayley module for G2(q). University of Birmingham. Ph.D.

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Abstract

In the 1980s, Ronan and Smith developed a homology theory to describe modules for groups of Lie type, using sheaves constructed on geometries associated with such groups. For a finite field $k$ of characteristic $\pi$ , they establish a one-to-one correspondence between certain ‘fixed-point’ sheaves $F_V$ and the irreducible $kG$ -modules for $G$ a Chevalley group defined over $k$ . This correspondence is given by the irreducible $kG$ -module $V$ being a unique irreducible quotient of the zero-homology module $H_0(F_V)$ . In certain cases, this homology module $H_0(F_V)$ is in fact isomorphic to the irreducible module $V$ . The question explored in this thesis is whether or not the Cayley module $C$ for the group $G_2(k)$ is isomorphic to $H_0(F_C)$ , as was speculated by Segev and Smith in 1986.