Young-diagrammatic Methods for the Representation Theory of $G_2$

We study the G2 fixed-point spaces ∇_{GL7} (k) (λ)^{G2} for an algebraically closed field k of
characteristic p > 2, and dominant GL7 (k)-weights λ ∈ X^{+} (7). Our primary focus
is to develop a formula for the calculation of the dimension dim ∇_{GL7 (k)} (λ)^{G2} . We
obtain this formula by studying the fixed-point spaces ∇_{SO7 (k)} (µ)^{G2} for dominant
SO7 (k)-weights µ ∈ X^{+} (T_{SO7 (k)} ), and then obtaining a good SO7 (k)-filtration of
∇_{GL7 (k)} (λ) when viewed as an SO7 (k)-module.