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

Table of contents
- Solution iv

Solution

The eight-dimensional simple module is projective, so its tensor product of group representations with the natural three-dimensional module is projective. Its Brauer character is

ϕ_{8} ϕ_{3} = (24, 0, α, \overline{α}) .

From part (iii),

Φ_{3} + Φ_{8} = (24, 0, α, \overline{α}) .

The duality of simple and projective Brauer characters makes the decomposition multiplicities unique. Hence, writing

P_{3}

and

P_{8}

for the corresponding projective covers,

8 \otimes 3 ≅ P_{3} \oplus P_{8} .

This completes the explicit 2-modular representation theory of GL3 of F2 calculation.

 Ancestors (11)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version