In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Molien series is a generating function

Generating function

In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...

 attached to a linear representation ρ of a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 G on a finite-dimensional vector space V. It counts the homogeneous polynomial

Homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...

s of a given total degree d that are invariants

Invariant (mathematics)

In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...

 for G. It is named for Theodor Molien

Theodor Molien

Theodor Molien or Fedor Eduardovich Molin was a Baltic-German mathematician. He was born in Riga, Latvia, which at that time was a part of Russian Empire. Molien studied associative algebras and polynomial invariants of finite groups.-Youth in Riga:Theodor Molien's father Eduard Molien was a...

.

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

Formal power series

In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...

$M(t)\; =\; \backslash sum\_d\; n\_d\; t^d.$ This can be looked at another way, by considering the representation of G on the symmetric algebra

Symmetric algebra

In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....

 of V, and then the whole subalgebra

Subalgebra

In mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...

 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 function. 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

Rational function

In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

. The case of finite group

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

s is most often studied.

Formula

Molien

Theodor Molien

Theodor Molien or Fedor Eduardovich Molin was a Baltic-German mathematician. He was born in Riga, Latvia, which at that time was a part of Russian Empire. Molien studied associative algebras and polynomial invariants of finite groups.-Youth in Riga:Theodor Molien's father Eduard Molien was a...

showed that$M(t)\; =\; \backslash frac\{1\}\{|G|\}\; \backslash sum\_\{g\backslash in\; G\}\; \backslash frac\{1\}\{\backslash det(I-tg)\}$ 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 $\backslash left|G\backslash right|\; \backslash cdot\; M\backslash left(t\backslash right)\; =\; \backslash sum\_\{g\backslash in\; G\}\; \backslash frac\{1\}\{\backslash det(I-tg)\}$ is valid, although it does not help with computing M(t)).

Example

Consider $S\_3$ acting on R³ by permuting the coordinates. Note that $\backslash det(I-tg)$ is constant on conjugacy classes, so it is enough to take one from each of the three classes in $S\_3$; so $\backslash det(I-te)\; =\; (1-t)^3,\backslash det(I-t\backslash sigma\_2)\; =\; (1-t)(1-t^2)$ and $\backslash det(1-t\backslash sigma\_3)\; =\; (1-t^3)$ where $\backslash sigma\_2=(1,2)$ and $\backslash sigma\_3\; =\; (1,2,3)$. Then$M(t)\; =\; \backslash frac16\backslash left(\backslash frac\{1\}\{(1-t)^3\}\; +\; \backslash frac3\{(1-t)(1-t^2)\}\; +\; \backslash frac\{2\}\{1-t^3\}\backslash right)$