Congruences between word length statistics for the finitary alternating and symmetric groups

Congruences between word length statistics for the finitary alternating and symmetric groups Bacher and de la Harpe ( arxiv:1603.07943 , 2016) study conjugacy growth series of infinite permutation groups and their relationships with p(n), the partition function, and $$p(n)_\mathbf{e }$$ p ( n ) e , a generalized partition function. They prove identities for the conjugacy growth series of the finitary symmetric group and the finitary alternating group. The group theory due to Bacher and de la Harpe ( arxiv:1603.07943 , 2016) also motivates an investigation into congruence relationships between the finitary symmetric group and the finitary alternating group. Using the Ramanujan congruences for the partition function p(n) and Atkin’s generalization to the k-colored partition function $$p_{k}(n)$$ p k ( n ) , we prove the existence of congruence relations between these two series modulo arbitrary powers of 5 and 7, which we systematically describe. Furthermore, we prove that such relationships exist modulo powers of all primes $$\ell \ge 5$$ ℓ ≥ 5 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archiv der Mathematik Springer Journals

Congruences between word length statistics for the finitary alternating and symmetric groups

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
0003-889X
eISSN
1420-8938
D.O.I.
10.1007/s00013-017-1058-8
Publisher site
See Article on Publisher Site

Abstract

Bacher and de la Harpe ( arxiv:1603.07943 , 2016) study conjugacy growth series of infinite permutation groups and their relationships with p(n), the partition function, and $$p(n)_\mathbf{e }$$ p ( n ) e , a generalized partition function. They prove identities for the conjugacy growth series of the finitary symmetric group and the finitary alternating group. The group theory due to Bacher and de la Harpe ( arxiv:1603.07943 , 2016) also motivates an investigation into congruence relationships between the finitary symmetric group and the finitary alternating group. Using the Ramanujan congruences for the partition function p(n) and Atkin’s generalization to the k-colored partition function $$p_{k}(n)$$ p k ( n ) , we prove the existence of congruence relations between these two series modulo arbitrary powers of 5 and 7, which we systematically describe. Furthermore, we prove that such relationships exist modulo powers of all primes $$\ell \ge 5$$ ℓ ≥ 5 .

Journal

Archiv der MathematikSpringer Journals

Published: Jun 22, 2017

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