Molien series

In mathematics, a Molien series is a generating function attached to a linear representation ρ of a group G on a finite-dimensional vector space V. It counts the homogeneous polynomials of a given total degree d that are invariants for G. It is named for Theodor Molien.

Formulation

More formally, there is a vector space of such polynomials, for each given value of d = 0, 1, 2, ..., and we write n_d for its vector space dimension, or in other words the number of linearly independent homogeneous invariants of a given degree. In more algebraic terms, take the d-th symmetric power of V, and the representation of G on it arising from ρ. The invariants form the subspace consisting of all vectors fixed by all elements of G, and n_d is its dimension.

The Molien series is then by definition the formal power series

[math]\displaystyle{ M(t) = \sum_d n_d t^d. }[/math]

This can be looked at another way, by considering the representation of G on the symmetric algebra of V, and then the whole subalgebra R of G-invariants. Then n_d is the dimension of the homogeneous part of R of dimension d, when we look at it as graded ring. In this way a Molien series is also a kind of Hilbert series. Without further hypotheses not a great deal can be said, but assuming some conditions of finiteness it is then possible to show that the Molien series is a rational function. The case of finite groups is most often studied.

Formula

Molien showed that

[math]\displaystyle{ M(t) = \frac{1}{|G|} \sum_{g\in G} \frac{1}{\det(I-t\rho(g))} }[/math]

This means that the coefficient of t^d in this series is the dimension n_d defined above. It assumes that the characteristic of the field does not divide |G| (but even without this assumption, Molien's formula in the form [math]\displaystyle{ |G| \cdot M(t) = \sum_{g\in G} \frac{1}{\det(I-t\rho(g))} }[/math] is valid, although it does not help with computing M(t)).

Example

Consider the symmetric group [math]\displaystyle{ S_3 }[/math] acting on R³ by permuting the coordinates. We add up the sum by group elements, as follows. Starting with the identity, we have

[math]\displaystyle{ \det \begin{pmatrix} 1-t & 0 & 0 \\ 0 & 1-t & 0 \\ 0 & 0 & 1-t \end{pmatrix} = (1-t)^3 }[/math].

There is a three-element conjugacy class of [math]\displaystyle{ S_3 }[/math], consisting of swaps of two coordinates. This gives three terms of the form

[math]\displaystyle{ \det \begin{pmatrix} 1 & -t & 0 \\ -t & 1 & 0 \\ 0 & 0 & 1-t \end{pmatrix} = (1-t)(1-t^2) }[/math].

There is a two-element conjugacy class of cyclic permutations, yielding two terms of the form

[math]\displaystyle{ \det \begin{pmatrix} 1 & -t & 0 \\ 0 & 1 & -t \\ -t & 0 & 1 \end{pmatrix} =(1-t^3) }[/math].

Notice that different elements of the same conjugacy class yield the same determinant. Thus

[math]\displaystyle{ M(t) = \frac16\left(\frac{1}{(1-t)^3} + \frac3{(1-t)(1-t^2)} + \frac{2}{1-t^3}\right) = \frac{1}{(1-t)(1-t^2)(1-t^3)}. }[/math]

On the other hand, we can expand the geometric series and multiply out to get

[math]\displaystyle{ M(t) = (1 + t + t^2 + t^3 + \ldots)(1+ t^2 + t^4 + \ldots)(1 + t^3 + t^6 + \ldots) = 1 + t + 2t^2 + 3t^3 + 4t^4 + 5t^5 + 6t^6 + 7t^7 + 9 t^8 + 11 x^9\ldots }[/math]

The coefficients of the series tell us the number of linearly independent homogeneous polynomials in three variables which are invariant under permutations of the three variables, i.e. the number of independent symmetric polynomials in three variables. In fact, if we consider the elementary symmetric polynomials

[math]\displaystyle{ \sigma_1 = x + y + z }[/math]

[math]\displaystyle{ \sigma_2 = xy + xz + yz }[/math]

[math]\displaystyle{ \sigma_3 = xyz }[/math]

we can see for example that in degree 5 there is a basis consisting of [math]\displaystyle{ \sigma_3 \sigma_2, \sigma_3 \sigma_1^2, \sigma_2^2 \sigma_1, \sigma_1^3 \sigma_2 }[/math] and [math]\displaystyle{ \sigma_1^5 }[/math].

(In fact, if you multiply the series out by hand, you can see that the [math]\displaystyle{ t^k }[/math] term comes from combinations of [math]\displaystyle{ t, t^2 }[/math] and [math]\displaystyle{ t^3 }[/math] exactly corresponding to combinations of [math]\displaystyle{ \sigma_1, \sigma_2 }[/math] and [math]\displaystyle{ \sigma_3 }[/math], also corresponding to partitions of [math]\displaystyle{ k }[/math] with [math]\displaystyle{ 1, 2, }[/math] and [math]\displaystyle{ 3 }[/math] as parts. See also Partition (number theory) and Representation theory of the symmetric group.)

References

• David A. Cox, John B. Little, Donal O'Shea (2005), Using Algebraic Geometry, pp. 295–8
• Molien, Th. (1897). "Uber die Invarianten der linearen Substitutionsgruppen.". Sitzungber. Konig. Preuss. Akad. Wiss. (J. Berl. Ber.) 52: 1152–1156. https://books.google.com/books?id=EIxK-opAmJYC&pg=PA1152. 
• Mukai, S. (2002). An introduction to invariants and moduli. Cambridge Studies in Advanced Mathematics. 81. ISBN 978-0-521-80906-1. http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521809061.