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.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

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