Beschreibung:
<jats:title>Abstract</jats:title>
<jats:p>The counting of the dimension of the space of <jats:inline-formula>
<jats:tex-math><?CDATA $U(N) \times U(N) \times U(N)$?></jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mml:mi>U</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>N</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>×</mml:mo>
<mml:mi>U</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>N</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>×</mml:mo>
<mml:mi>U</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>N</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="aac9b3bieqn1.gif" xlink:type="simple" />
</jats:inline-formula> polynomial invariants of a complex 3-index tensor as a function of degree <jats:italic>n</jats:italic> is known in terms of a sum of squares of Kronecker coefficients. For <jats:inline-formula>
<jats:tex-math><?CDATA $n \leqslant N$?></jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mml:mi>n</mml:mi>
<mml:mo>⩽</mml:mo>
<mml:mi>N</mml:mi>
</mml:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="aac9b3bieqn2.gif" xlink:type="simple" />
</jats:inline-formula>, the formula can be expressed in terms of a sum of symmetry factors of partitions of <jats:italic>n</jats:italic> denoted <jats:inline-formula>
<jats:tex-math><?CDATA $Z_3(n)$?></jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mml:msub>
<mml:mi>Z</mml:mi>
<mml:mn>3</mml:mn>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="aac9b3bieqn3.gif" xlink:type="simple" />
</jats:inline-formula>, which also counts the number of bipartite ribbon graphs with <jats:italic>n</jats:italic> edges. We derive the large <jats:italic>n</jats:italic> all-orders asymptotic formula for <jats:inline-formula>
<jats:tex-math><?CDATA $ Z_3(n)$?></jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mml:msub>
<mml:mi>Z</mml:mi>
<mml:mn>3</mml:mn>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>n</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="aac9b3bieqn4.gif" xlink:type="simple" />
</jats:inline-formula> making contact with high order results previously obtained numerically. We explain how the different terms in the asymptotics are associated with probability distributions over ribbon graphs. The derivation of the asymptotics relies on the dominance in the sum, of partitions with many parts of length 1. This large <jats:italic>n</jats:italic> dominance of small parts also leads to conjectured formulae for the asymptotics of invariants for general <jats:italic>d</jats:italic>-index tensors. The coefficients of powers of <jats:inline-formula>
<jats:tex-math><?CDATA $ 1/n$?></jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mml:mn>1</mml:mn>
<mml:mrow>
<mml:mo>/</mml:mo>
</mml:mrow>
<mml:mi>n</mml:mi>
</mml:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="aac9b3bieqn5.gif" xlink:type="simple" />
</jats:inline-formula> in these expansions involve Stirling numbers of the second kind along with restricted partition sums.</jats:p>