Posted: Aug. 10, 2022

Abstract: This Maple package implements Allan Berele's constant-term expressions for certain Poincaré series called Cbar(n,k), described in his paper

Denominators for One Variable Poincaré series of generic matrices .

Added Aug. 3, 2024: The paper appeared in J. Algebra 618 (2023) , 195-213.

It arose in the theory of Polynomial Identity Rings and Invariant Theory, that thanks to Wilf-Zeilberger algorithmic proof theory can be used to derive linear recurrences for n=2 and n=3. This enables very fast computations of these series for many values of k. Alas, while the analogous recurrences exist for n=4 and above, finding them is currently beyond our meager computational powers.

Note: There is no accompanying article. Everything is contained in the Maple package and the output files.

Maple package

• Berele.txt, a Maple package for fast computation of Poincaré series for Cbar(n,k) for n=2 and n=3 and very slow one for higher n.

Sample Input and Output for Berele.txt

• If you want to see a computer-generated article with the recurrence, in k, for Cbar(2,k)(t) and the first 100 terms

  The input file generates the output file  

• If you want to see a computer-generated article with the recurrence, in k, for Cbar(3,k)(t) and the first 30 terms

  The input file generates the output file  

• If you want to see a computer-generated article with the recurrence, in k, for Cbar(3,k)(t) and the first 100 terms (WARNING: BIG FILE!)

  The input file generates the output file  

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