Evaluation of divisor functions of matrices

Gautami Bhowmik

Acta Arithmetica (1996)

  • Volume: 74, Issue: 2, page 155-159
  • ISSN: 0065-1036

Abstract

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1. Introduction. The study of divisor functions of matrices arose legitimately in the context of arithmetic of matrices, and the question of the number of (possibly weighted) inequivalent factorizations of an integer matrix was asked. However, till now only partial answers were available. Nanda [6] evaluated the case of prime matrices and Narang [7] gave an evaluation for 2×2 matrices. We obtained a recursion in the size of the matrices and the weights of the divisors [1,2] which helped us obtain a result for 3×3 matrices but no closed formula for the general case. In this paper we obtain the complete evaluation of the divisor functions by a combinatorial consideration (see Theorem 1). Because of the existence of a bijection (detailed in a forthcoming paper [3]) between the set of divisors of an r×r integer matrix and the set of subgroups of an abelian group of rank at most r, we have here a rather simple proof to obtain the number of subgroups of a finite abelian group.

How to cite

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Gautami Bhowmik. "Evaluation of divisor functions of matrices." Acta Arithmetica 74.2 (1996): 155-159. <http://eudml.org/doc/206843>.

@article{GautamiBhowmik1996,
abstract = {1. Introduction. The study of divisor functions of matrices arose legitimately in the context of arithmetic of matrices, and the question of the number of (possibly weighted) inequivalent factorizations of an integer matrix was asked. However, till now only partial answers were available. Nanda [6] evaluated the case of prime matrices and Narang [7] gave an evaluation for 2×2 matrices. We obtained a recursion in the size of the matrices and the weights of the divisors [1,2] which helped us obtain a result for 3×3 matrices but no closed formula for the general case. In this paper we obtain the complete evaluation of the divisor functions by a combinatorial consideration (see Theorem 1). Because of the existence of a bijection (detailed in a forthcoming paper [3]) between the set of divisors of an r×r integer matrix and the set of subgroups of an abelian group of rank at most r, we have here a rather simple proof to obtain the number of subgroups of a finite abelian group.},
author = {Gautami Bhowmik},
journal = {Acta Arithmetica},
keywords = {Smith normal form; Hermite normal form; multiplicative arithmetical function; integral matrices; divisor function; Smith invariants},
language = {eng},
number = {2},
pages = {155-159},
title = {Evaluation of divisor functions of matrices},
url = {http://eudml.org/doc/206843},
volume = {74},
year = {1996},
}

TY - JOUR
AU - Gautami Bhowmik
TI - Evaluation of divisor functions of matrices
JO - Acta Arithmetica
PY - 1996
VL - 74
IS - 2
SP - 155
EP - 159
AB - 1. Introduction. The study of divisor functions of matrices arose legitimately in the context of arithmetic of matrices, and the question of the number of (possibly weighted) inequivalent factorizations of an integer matrix was asked. However, till now only partial answers were available. Nanda [6] evaluated the case of prime matrices and Narang [7] gave an evaluation for 2×2 matrices. We obtained a recursion in the size of the matrices and the weights of the divisors [1,2] which helped us obtain a result for 3×3 matrices but no closed formula for the general case. In this paper we obtain the complete evaluation of the divisor functions by a combinatorial consideration (see Theorem 1). Because of the existence of a bijection (detailed in a forthcoming paper [3]) between the set of divisors of an r×r integer matrix and the set of subgroups of an abelian group of rank at most r, we have here a rather simple proof to obtain the number of subgroups of a finite abelian group.
LA - eng
KW - Smith normal form; Hermite normal form; multiplicative arithmetical function; integral matrices; divisor function; Smith invariants
UR - http://eudml.org/doc/206843
ER -

References

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  1. [1] G. Bhowmik, Divisor functions of integer matrices: evaluations, average orders and applications, Astérisque 209 (1992), 169-177. Zbl0787.11007
  2. [2] G. Bhowmik and O. Ramaré, Average orders of multiplicative arithmetical functions of integer matrices, Acta Arith. 66 (1994), 45-62. Zbl0795.11047
  3. [3] G. Bhowmik and O. Ramaré, Factorisation of matrices, partitions and Hecke algebra, to appear. 
  4. [4] L. M. Butler, A unimodality result in the enumeration of subgroups of a finite abelian group, Proc. Amer. Math. Soc. 101 (1987), 771-775. Zbl0647.20053
  5. [5] A. Krieg, Hecke Algebras, Mem. Amer. Math. Soc. 435 (1990). 
  6. [6] V. C. Nanda, Arithmetic functions of matrices and polynomial identities, in: Colloq. Math. Soc. János Bolyai 34, North-Holland, 1984, 1107-1126. 
  7. [7] A. Narang, Ph. D. Thesis, Panjab University, India, 1979. 

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