In mathematics, specifically in representation theory, the Frobenius formula, introduced by G. Frobenius, computes the characters of irreducible representations of the symmetric group Sn.
Among the other applications, the formula can be used to derive the hook length formula.
be the character of an irreducible representation of the symmetric group
corresponding to a partition
ℓ
μ
( μ )
denote the conjugacy class in
corresponding to it (cf.
denote the number of times j appears in
Then the Frobenius formula states that the constant value of
χ
is the coefficient of the monomial
ℓ
ℓ
in the homogeneous polynomial in
-th power sum.
ℓ
ℓ
), which corresponds to the class of the identity element, then
χ
(the class of a 3-cycle times an 1-cycle) and
χ
For the identity representation,
ℓ
χ
will be equal to the coefficient of
Arun Ram gives a q-analog of the Frobenius formula.
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