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.

In (Ram 1991) harv error: no target: CITEREFRam1991 (help), Arun Ram gives a q-analog of the Frobenius formula.

Statement

Let ${\displaystyle \chi _{\lambda }}$ be the character of an irreducible representation of the symmetric group ${\displaystyle S_{n}}$ corresponding to a partition ${\displaystyle \lambda }$ of n: ${\displaystyle n=\lambda _{1}+\cdots +\lambda _{k}}$ and ${\displaystyle \ell _{j}=\lambda _{j}+k-j}$. For each partition ${\displaystyle \mu }$ of n, let ${\displaystyle C(\mu )}$ denote the conjugacy class in ${\displaystyle S_{n}}$ corresponding to it (cf. the example below), and let ${\displaystyle i_{j}}$ denote the number of times j appears in ${\displaystyle \mu }$ (so ${\displaystyle \Sigma \,i_{j}j=n}$). Then the Frobenius formula states that the constant value of ${\displaystyle \chi _{\lambda }}$ on ${\displaystyle C(\mu ),}$

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

is the coefficient of the monomial ${\displaystyle x_{1}^{\ell _{1}}\dots x_{k}^{\ell _{k}}}$ in the homogeneous polynomial

${\displaystyle \prod _{i<j}(x_{i}-x_{j})\prod _{j}P_{j}(x_{1},\dots ,x_{k})^{i_{j}},}$

where ${\displaystyle P_{j}(x_{1},\dots ,x_{k})=x_{1}^{j}+\dots +x_{k}^{j}}$ is the ${\displaystyle j}$-th power sum.

Example: Take ${\displaystyle n=4}$ and ${\displaystyle \lambda :4=2+2}$. If ${\displaystyle \mu :4=1+1+1+1}$, which corresponds to the class of the identity element, then ${\displaystyle \chi _{\lambda }(C(\mu ))}$ is the coefficient of ${\displaystyle x_{1}^{3}x_{2}^{2}}$ in

${\displaystyle (x_{1}-x_{2})(x_{1}+x_{2})^{4}}$

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

${\displaystyle (x_{1}-x_{2})(x_{1}+x_{2})(x_{1}^{3}+x_{2}^{3}),}$

is −1.

See also

• Representation theory of symmetric groups

References

• A. Ram, A Frobenius formula for the characters of the Hecke algebras, Inventiones mathematicae, vol 106, no 1, pp 461–488, 1991.
• Fulton, William; Harris, Joe (1991). 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. 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