View source for Frobenius formula

<!--
f0417801.png
$#A+1 = 33 n = 0
$#C+1 = 33 : ~/encyclopedia/old_files/data/F041/F.0401780 Frobenius formula
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A formula expressing a relation between the generalized Vandermonde determinant and the ordinary one (see [[Vandermonde determinant|Vandermonde determinant]]) in terms of sums of powers. The characters of representations of a symmetric group (cf. [[Representation of the symmetric groups|Representation of the symmetric groups]]) appear as coefficients in the Frobenius formula.

Let  $  x _ {1} \dots x _ {n} $
be independent variables. For any  $  n $-tuple  $  \lambda = ( \lambda _ {1} \dots \lambda _ {n} ) $
of non-negative integers satisfying the condition  $  \lambda _ {1} \geq  \dots \geq  \lambda _ {n} $, 
let

$$ 
W _  \lambda   =  \left |

\begin{array}{lll}
x _ {1} ^ {\lambda _ {1} + n - 1 }  &\cdots  &x _ {n} ^ {\lambda _ {1} + n - 1 }  \\
\vdots  &\ddots  &\vdots  \\
x _ {1} ^ {\lambda _ {n} }  &\cdots  &x _ {n} ^ {\lambda _ {n} }  \\
\end{array}
 \ 
\right | ,
$$

so that  $  W _ {0} $
is the ordinary Vandermonde determinant. Let  $  \sum \lambda _ {i} = m $; 
then after discarding zeros the  $  n $-tuple  $  \lambda $
can be regarded as a partition of the number  $  m $. 
Consider the corresponding irreducible representation  $  T _  \lambda  $
of the symmetric group  $  S _ {m} $. 
For any partition  $  \mu = ( \mu _ {1} \dots \mu _ {r} ) $
of  $  m $
one denotes by  $  a _ {\lambda \mu }  $
the value of the character of  $  T _  \lambda  $
on the [[conjugacy class]] of  $  S _ {m} $
determined by  $  \mu $, 
and by  $  c _  \mu  $
the order of the centralizer of any permutation in this class. Let  $  s _  \mu  = s _ {\mu _ {1}  } \dots s _ {\mu _ {r}  } $, 
where  $  s _ {k} = x _ {1}  ^ {k} + \dots + x _ {n}  ^ {k} $. 
Then

$$ 

\frac{W _  \lambda  }{W _ {0} }
  = \ 
\sum _  \mu  a _ {\lambda \mu } 
c _  \mu   ^ {- 1} s _  \mu  ,
$$

where the sum is taken over all (unordered) partitions of  $  m $. 
Here, if the partition  $  \mu $
contains  $  k _ {1} $
ones,  $  k _ {2} $
twos, etc., then

$$ 
c _  \mu  =
k _ {1} ! k _ {2} ! \dots
1 ^ {k _ {1} } 2 ^ {k _ {2} } \dots .
$$

If  $  n \geq  m $, 
then Frobenius' formula can be put in the form

$$ 
\sum _  \lambda 
a _ {\lambda \mu } 
W _  \lambda   = \ 
s _  \mu  W _ {0} ,
$$

where the sum is taken over all partitions of  $  m $(
adding the appropriate number of zeros). The last formula can be used to compute the characters of the symmetric group. Namely,  $  a _ {\lambda \mu }  $
is the coefficient of  $  x _ {1} ^ {\lambda _ {1} + n - 1 } {} \dots x _ {n} ^ {\lambda _ {n} } $
in the polynomial  $  s _  \mu  W _ {0} $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F.D. Murnagan,   "The theory of group representations" , Johns Hopkins Univ. Press  (1938)</TD></TR></table>

====Comments====
See also [[Character of a representation of a group|Character of a representation of a group]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Boerner,   "Representations of groups" , North-Holland  (1970)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.E. Littlewood,   "The theory of group characters and matrix representations of groups" , Clarendon Press  (1950)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.G. Macdonald,   "Symmetric functions and Hall polynomials" , Clarendon Press  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B.G. Wybourne,   "Symmetry principles and atomic spectroscopy" , Wiley (Interscience)  (1970)</TD></TR></table>