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Past exam of the mathematics course of the University of Cambridge
/
2021
/
iii
/
Paper 160
/
1
/
c
/
ii
/
Solution
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2021
iii
Paper 160
1
c
ii
2026-09-28
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Because
C
(
t
′
)
=
R
(
t
)
,
e
(
t
′
)
=
∑
r
∈
R
(
t
)
sgn
(
r
)
{
r
t
′
}
.
(1)
Applying
θ
and using
re
(
u
)
=
sgn
(
r
)
e
(
u
)
gives
θ
(
e
(
t
′
))
=
(
∑
r
∈
R
(
t
)
re
(
t
)
)
⊗
e
(
u
)
.
(2)
Thus
m
=
∑
r
∈
R
(
t
)
re
(
t
)
. Every
r
fixes the
tabloid
{
t
}
, while
e
(
t
)
has
coefficient
one at
{
t
}
. Invariance of the
tabloid bilinear form
now yields
⟨
m
,
{
t
}⟩
=
∑
r
∈
R
(
t
)
⟨
re
(
t
)
,
{
t
}⟩
=
∑
r
∈
R
(
t
)
1
=
∣
R
(
t
)
∣
.
(3)
Ancestors
(12)
ii
c
1
Paper 160
iii
2021
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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