Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 5 / c / iii

Table of contents
- Solution iii

Solution

The rows of the Cartan matrix of a group algebra express the projective characters in the simple Brauer-character basis. Thus

Φ_{1} Φ_{3} Φ_{\overline{3}} Φ_{8} = 2 ϕ_{1} + ϕ_{3} + ϕ_{\overline{3}}, = ϕ_{1} + 3 ϕ_{3} + 2 ϕ_{\overline{3}}, = ϕ_{1} + 2 ϕ_{3} + 3 ϕ_{\overline{3}}, = ϕ_{8} .

Evaluating gives

Φ_{1} Φ_{3} Φ_{\overline{3}} Φ_{8} 1816168 3211 - 1 7 A 1 α - 1 \overline{α} - 1 1 7 B 1 \overline{α} - 1 α - 1 1 .

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