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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f0417801.png" /> be independent variables. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f0417802.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f0417803.png" /> of non-negative integers satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f0417804.png" />, let
+
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 }  &\dots  &x _ {n} ^ {\lambda _ {1} + n - 1 }  \\
 +
\dots  &\dots  &\dots  \\
 +
x _ {1} ^ {\lambda _ {n} }  &\dots  &x _ {n} ^ {\lambda _ {n} }  \\
 +
\end{array}
 +
\
 +
\right | ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f0417805.png" /></td> </tr></table>
+
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
  
so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f0417806.png" /> is the ordinary Vandermonde determinant. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f0417807.png" />; then after discarding zeros the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f0417808.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f0417809.png" /> can be regarded as a partition of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178010.png" />. Consider the corresponding irreducible representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178011.png" /> of the symmetric group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178012.png" />. For any partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178014.png" /> one denotes by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178015.png" /> the value of the character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178016.png" /> on the [[conjugacy class]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178017.png" /> determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178018.png" />, and by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178019.png" /> the order of the centralizer of any permutation in this class. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178021.png" />. Then
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178022.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178023.png" />. Here, if the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178024.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178025.png" /> ones, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178026.png" /> twos, etc., then
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178027.png" /></td> </tr></table>
+
$$
 +
c _  \mu  =
 +
k _ {1} ! k _ {2} ! \dots
 +
1 ^ {k _ {1} } 2 ^ {k _ {2} } \dots .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178028.png" />, then Frobenius' formula can be put in the form
+
If $  n \geq  m $,  
 +
then Frobenius' formula can be put in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178029.png" /></td> </tr></table>
+
$$
 +
\sum _  \lambda
 +
a _ {\lambda \mu }
 +
W _  \lambda  = \
 +
s _  \mu  W _ {0} ,
 +
$$
  
where the sum is taken over all partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178030.png" /> (adding the appropriate number of zeros). The last formula can be used to compute the characters of the symmetric group. Namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178031.png" /> is the coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178032.png" /> in the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041780/f04178033.png" />.
+
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====
 
====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>
 
<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====
 
====Comments====

Revision as of 19:40, 5 June 2020


A formula expressing a relation between the generalized Vandermonde determinant and the ordinary one (see Vandermonde determinant) in terms of sums of powers. The characters of representations of a symmetric group (cf. 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

$$ W _ \lambda = \left | \begin{array}{lll} x _ {1} ^ {\lambda _ {1} + n - 1 } &\dots &x _ {n} ^ {\lambda _ {1} + n - 1 } \\ \dots &\dots &\dots \\ x _ {1} ^ {\lambda _ {n} } &\dots &x _ {n} ^ {\lambda _ {n} } \\ \end{array} \ \right | , $$

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

[1] F.D. Murnagan, "The theory of group representations" , Johns Hopkins Univ. Press (1938)

Comments

See also Character of a representation of a group.

References

[a1] H. Boerner, "Representations of groups" , North-Holland (1970) (Translated from German)
[a2] D.E. Littlewood, "The theory of group characters and matrix representations of groups" , Clarendon Press (1950)
[a3] I.G. Macdonald, "Symmetric functions and Hall polynomials" , Clarendon Press (1979)
[a4] B.G. Wybourne, "Symmetry principles and atomic spectroscopy" , Wiley (Interscience) (1970)
How to Cite This Entry:
Frobenius formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_formula&oldid=35124
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article