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 Incoming links: Past exam of the groups and representations course of the University of Oxford / 2019 / 2 / b / Solution

Past exam of the groups and representations course of the University of Oxford / 2019 / 2 / c / Solution 2026-10-25

The irreducible representations can all be specified on the generators:

1 ε R_{2} R (r) 11 (α 0 0 α^{2}) R (s) 1 - 1 (0110)

The first is the trivial representation and the second is the sign representation. To see directly that these exhaust the one-dimensional representations, their scalar images commute, so the relation

srs = r^{- 1}

r = r^{- 1}

. Combining

r^{2} = 1

with

r^{3} = 1

gives

r = 1

, and then

s

may be either sign.

Taking traces gives the character table, in the conjugacy class order of solution:

class size 1 ε R_{2} I_{2} 1112 r 211 - 1 s 31 - 1 0

Here

α + α^{2} = - 1

gives the middle entry in the last row. The weighted character inner products of distinct rows vanish, and the norm of each row is one. Together with solution, this proves completeness of the table.

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