Frobenius formula
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.
Statement
Let [math]\displaystyle{ \chi_\lambda }[/math] be the character of an irreducible representation of the symmetric group [math]\displaystyle{ S_n }[/math] corresponding to a partition [math]\displaystyle{ \lambda }[/math] of n: [math]\displaystyle{ n = \lambda_1 + \cdots + \lambda_k }[/math] and [math]\displaystyle{ \ell_j = \lambda_j + k - j }[/math]. For each partition [math]\displaystyle{ \mu }[/math] of n, let [math]\displaystyle{ C(\mu) }[/math] denote the conjugacy class in [math]\displaystyle{ S_n }[/math] corresponding to it (cf. the example below), and let [math]\displaystyle{ i_j }[/math] denote the number of times j appears in [math]\displaystyle{ \mu }[/math] (so [math]\displaystyle{ \sum_j i_j j = n }[/math]). Then the Frobenius formula states that the constant value of [math]\displaystyle{ \chi_\lambda }[/math] on [math]\displaystyle{ C(\mu), }[/math]
- [math]\displaystyle{ \chi_\lambda(C(\mu)), }[/math]
is the coefficient of the monomial [math]\displaystyle{ x_1^{\ell_1} \dots x_k^{\ell_k} }[/math] in the homogeneous polynomial in [math]\displaystyle{ k }[/math] variables
- [math]\displaystyle{ \prod_{i \lt j}^k (x_i - x_j) \; \prod_j P_j(x_1, \dots, x_k)^{i_j}, }[/math]
where [math]\displaystyle{ P_j(x_1, \dots, x_k) = x_1^j + \dots + x_k^j }[/math] is the [math]\displaystyle{ j }[/math]-th power sum.
Example: Take [math]\displaystyle{ n = 4 }[/math]. Let [math]\displaystyle{ \lambda: 4 = 2 + 2 = \lambda_1 + \lambda_2 }[/math] and hence [math]\displaystyle{ k=2 }[/math], [math]\displaystyle{ \ell_1=3 }[/math], [math]\displaystyle{ \ell_2=2 }[/math]. If [math]\displaystyle{ \mu: 4 = 1 + 1 + 1 + 1 }[/math] ([math]\displaystyle{ i_1=4 }[/math]), which corresponds to the class of the identity element, then [math]\displaystyle{ \chi_\lambda(C(\mu)) }[/math] is the coefficient of [math]\displaystyle{ x_1^3 x_2^2 }[/math] in
- [math]\displaystyle{ (x_1 - x_2)P_1(x_1,x_2)^4=(x_1 - x_2)(x_1 + x_2)^4 }[/math]
which is 2. Similarly, if [math]\displaystyle{ \mu: 4 = 3 + 1 }[/math] (the class of a 3-cycle times an 1-cycle) and [math]\displaystyle{ i_1=i_3=1 }[/math], then [math]\displaystyle{ \chi_{\lambda}(C(\mu)) }[/math], given by
- [math]\displaystyle{ (x_1 - x_2)P_1(x_1,x_2)P_3(x_1,x_2)=(x_1 - x_2)(x_1 + x_2)(x_1^3 + x_2^3), }[/math]
is −1.
For the identity representation, [math]\displaystyle{ k=1 }[/math] and [math]\displaystyle{ \lambda_1=n=\ell_1 }[/math]. The character [math]\displaystyle{ \chi_\lambda(C(\mu)) }[/math] will be equal to the coefficient of [math]\displaystyle{ x_1^n }[/math] in [math]\displaystyle{ \prod_j P_j(x_1)^{i_j}=\prod_j x_1^{i_j j}= x_1^{\sum_j i_j j}=x_1^n }[/math], which is 1 for any [math]\displaystyle{ \mu }[/math] as expected.
Analogues
In (Ram 1991), Arun Ram gives a q-analog of the Frobenius formula.
See also
- Representation theory of symmetric groups
References
- Ram, Arun (1991). "A Frobenius formula for the characters of the Hecke algebras". Inventiones mathematicae 106 (1): 461–488.
- Fulton, William; Harris, Joe (1991) (in en-gb). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. OCLC 246650103. https://link.springer.com/10.1007/978-1-4612-0979-9.
- Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN 0-19-853489-2 MR1354144
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