Number Theory: An Introduction to Mathematics

420 X A Character Study

With respect to a given basis ofVletA(t)now denote the matrix representing
t∈Hand putA(s)=Oifs∈G\H. If one adopts corresponding bases for each of
the subspacesVi, then the matrixA ̃(s)representings∈Gin the induced representa-
tion is the block matrix

A ̃(s)=

⎛

⎜

⎜

⎝

A(s 1 −^1 ss 1 ) A(s− 11 ss 2 ) ··· A(s− 11 ssn)

A(s 2 −^1 ss 1 ) A(s− 21 ss 2 ) ··· A(s− 21 ssn)

··· ··· ··· ···

A(sn−^1 ss 1 ) A(s−n^1 ss 2 ) ··· A(s−n^1 ssn)

⎞

⎟

⎟

⎠.

Evidently each row and each column contains exactly one nonzero block. It
should be noted also that a different choice of coset representativessi′=siti,where
ti∈H(i= 1 ,...,n), yields an equivalent representation, since

⎛

⎝

A(t 1 )−^1 ··· 0

··· ··· ···

0 ··· A(tn)−^1

⎞

⎠A ̃(s)

⎛

⎝

A(t 1 ) ··· 0

··· ··· ···

0 ··· A(tn)

⎞

⎠

=

⎛

⎝

A(s 1 ′−^1 ss′ 1 ) ··· A(s 1 ′−^1 ss′n)

··· ··· ···

A(sn′−^1 ss′ 1 ) ··· A(sn′−^1 ss′n)

⎞

⎠.

Furthermore, changing the order of the cosets corresponds to performing the same
permutation on the rows and columns ofA ̃(s), and thus also yields an equivalent rep-
resentation.
It follows that ifψis the character of the original representationσofH, then the
characterψ ̃of the induced representationσ ̃ofGis given by

ψ( ̃ s)=

∑n

i= 1

ψ(s−i^1 ssi),

where we setψ(s)=0ifs∈/H.IfHis of finite orderh, this can be rewritten in the
form

ψ( ̃ s)=h−^1

∑

u∈G

ψ(u−^1 su), (12)

sinceψ(t−^1 s−i^1 ssit)=ψ(si−^1 ssi)ift∈H.
From any representation of a groupGwe can also obtain a representation of a
subgroupH simply by restricting the given representation toH. We will say that
the representation ofHisdeducedfrom that ofG. There is a remarkable reciprocity
between induced and deduced representations, discovered by Frobenius (1898):

Proposition 13Letρ:s→A(s)be an irreducible representation of the finite group
G andσ:t→B(t)an irreducible representation of the subgroup H. Then the number
of times thatσoccurs in the representation of H deduced from the representationρof
G is equal to the number of times thatρoccurs in the representation of G induced by
the representationσof H.