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 S_n. Among the other applications, the formula can be used to derive the hook length formula.

Statement

Let ${\chi }_{\lambda }$ be the character of an irreducible representation of the symmetric group $S_n$ corresponding to a partition $\lambda$ of n: $n={\lambda }_1+\cdots +{\lambda }_k$ and ${\ell }_j={\lambda }_j+k-j$. For each partition $\mu$ of n, let $C(\mu )$ denote the conjugacy class in $S_n$ corresponding to it (cf. the example below), and let $i_j$ denote the number of times j appears in $\mu$ (so $\sum _j{i}_{j}j=n$). Then the Frobenius formula states that the constant value of ${\chi }_{\lambda }$ on $C(\mu ),$

${\chi }_{\lambda }(C(\mu )),$

is the coefficient of the monomial $x_1^{{\ell }_1}\dots x_k^{{\ell }_k}$ in the homogeneous polynomial in $k$ variables

$\prod _{i<j}^k(x_i-x_j)\prod _j{P}_j(x_1,\dots ,x_k{)}^{i_j},$

where $P_j(x_1,\dots ,x_k)=x_1^j+\dots +x_k^j$ is the $j$-th power sum. Example: Take $n=4$. Let $\lambda :4=2+2={\lambda }_1+{\lambda }_2$ and hence $k=2$, ${\ell }_1=3$, ${\ell }_2=2$. If $\mu :4=1+1+1+1$ ($i_1=4$), which corresponds to the class of the identity element, then ${\chi }_{\lambda }(C(\mu ))$ is the coefficient of $x_1^3{x}_2^2$ in

$(x_1-x_2)P_1(x_1,x_2{)}^4=(x_1-x_2)(x_1+x_2{)}^4$

which is 2. Similarly, if $\mu :4=3+1$ (the class of a 3-cycle times an 1-cycle) and $i_1=i_3=1$, then ${\chi }_{\lambda }(C(\mu ))$, given by

$(x_1-x_2)P_1(x_1,x_2)P_3(x_1,x_2)=(x_1-x_2)(x_1+x_2)(x_1^3+x_2^3),$

is −1. For the identity representation, $k=1$ and ${\lambda }_1=n={\ell }_1$. The character ${\chi }_{\lambda }(C(\mu ))$ will be equal to the coefficient of $x_1^n$ in $\prod _j{P}_j(x_1{)}^{i_j}=\prod _j{x}_1^{i_{j}j}=x_1^{\sum _j{i}_{j}j}=x_1^n$, which is 1 for any $\mu$ as expected.

Analogues

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. Bibcode:1991InMat.106..461R. doi:10.1007/BF01243921.
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• 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