<p>An <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F upper I">
<mml:semantics>
<mml:mi>FI</mml:mi>
<mml:annotation encoding="application/x-tex">\operatorname {FI}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>- or an <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O upper I">
<mml:semantics>
<mml:mi>OI</mml:mi>
<mml:annotation encoding="application/x-tex">\operatorname {OI}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-module <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper M">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">M</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathbf {M}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> over a corresponding noetherian polynomial algebra <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper P">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">P</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathbf {P}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> may be thought of as a sequence of compatible modules <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper M Subscript n">
<mml:semantics>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">M</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">\mathbf {M}_n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> over a polynomial ring <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper P Subscript n">
<mml:semantics>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">P</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">\mathbf {P}_n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> whose number of variables depends linearly on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n">
<mml:semantics>
<mml:mi>n</mml:mi>
<mml:annotation encoding="application/x-tex">n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. In order to study invariants of the modules <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper M Subscript n">
<mml:semantics>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">M</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">\mathbf {M}_n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> in dependence of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n">
<mml:semantics>
<mml:mi>n</mml:mi>
<mml:annotation encoding="application/x-tex">n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, an equivariant Hilbert series is introduced if <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper M">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">M</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathbf {M}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is graded. If <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper M">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">M</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathbf {M}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is also finitely generated, it is shown that this series is a rational function. Moreover, if this function is written in reduced form rather precise information about the irreducible factors of the denominator is obtained. This is key for applications. It follows that the Krull dimension of the modules <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper M Subscript n">
<mml:semantics>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">M</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">\mathbf {M}_n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> grows eventually linearly in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n">
<mml:semantics>
<mml:mi>n</mml:mi>
<mml:annotation encoding="application/x-tex">n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, whereas the multiplicity of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper M Subscript n">
<mml:semantics>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">M</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">\mathbf {M}_n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> grows eventually exponentially in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n">
<mml:semantics>
<mml:mi>n</mml:mi>
<mml:annotation encoding="application/x-tex">n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. Moreover, for any fixed degree <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j">
<mml:semantics>
<mml:mi>j</mml:mi>
<mml:annotation encoding="application/x-tex">j</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, the vector space dimensions of the degree <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j">
<mml:semantics>
<mml:mi>j</mml:mi>
<mml:annotation encoding="application/x-tex">j</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> components of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper M Subscript n">
<mml:semantics>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">M</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">\mathbf {M}_n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> grow eventually polynomially in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n">
<mml:semantics>
<mml:mi>n</mml:mi>
<mml:annotation encoding="application/x-tex">n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. As a consequence, any graded Betti number of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper M Subscript n">
<mml:semantics>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">M</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">\mathbf {M}_n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> in a fixed homological degree and a fixed internal degree grows eventually polynomially in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n">
<mml:semantics>
<mml:mi>n</mml:mi>
<mml:annotation encoding="application/x-tex">n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. Furthermore, evidence is obtained to support a conjecture that the Castelnuovo-Mumford regularity and the projective dimension of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper M Subscript n">
<mml:semantics>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">M</mml:mi>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:msub>
<mml:annotation encoding="application/x-tex">\mathbf {M}_n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> both grow eventually linearly in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n">
<mml:semantics>
<mml:mi>n</mml:mi>
<mml:annotation encoding="application/x-tex">n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. It is also shown that modules <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper M">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="bold">M</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\mathbf {M}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> whose width <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n">
<mml:semantics>
<mml:mi>n</mml:mi>
<mml:annotation encoding="application/x-tex">n</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> components <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.
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