Rocky Mountain Journal of Mathematics

The integer group determinants for the symmetric group of degree four

Christopher Pinner

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Abstract

For the symmetric group $S_4$ we determine all the integer values taken by its group determinant when the matrix entries are integers.

Article information

Source
Rocky Mountain J. Math., Volume 49, Number 4 (2019), 1293-1305.

Dates
First available in Project Euclid: 29 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1567044040

Digital Object Identifier
doi:10.1216/RMJ-2019-49-4-1293

Mathematical Reviews number (MathSciNet)
MR3998922

Zentralblatt MATH identifier
07104718

Subjects
Primary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure 15B36: Matrices of integers [See also 11C20]
Secondary: 11B83: Special sequences and polynomials 11C08: Polynomials [See also 13F20] 11C20: Matrices, determinants [See also 15B36] 11G50: Heights [See also 14G40, 37P30] 11R09: Polynomials (irreducibility, etc.) 11T22: Cyclotomy 43A40: Character groups and dual objects

Keywords
Group determinant symmetric group Lind-Lehmer constant Mahler measure

Citation

Pinner, Christopher. The integer group determinants for the symmetric group of degree four. Rocky Mountain J. Math. 49 (2019), no. 4, 1293--1305. doi:10.1216/RMJ-2019-49-4-1293. https://projecteuclid.org/euclid.rmjm/1567044040


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References

  • T. Boerkoel and C. Pinner, Minimal group determinants and the Lind-Lehmer problem for dihedral groups, Acta Arith. \bf186 (2018), 377–395.
  • K. Conrad, The origin of representation theory, Enseign. Math. 44 (1998), 361–392.
  • D. De Silva, M. Mossinghoff, V. Pigno and C. Pinner, The Lind-Lehmer constant for certain $p$-groups, Math. Comp. 88 (2018), 949–972.
  • D. De Silva and C. Pinner, The Lind-Lehmer constant for $\mathbb Z_p^n$, Proc. Amer. Math. Soc. 142 (2014), 1935–1941.
  • E. Formanek and D. Sibley, The group determinant determines the group, Proc. Amer. Math. Soc. 112 (1991), 649–656.
  • F.G. Frobenius, Über die Primfaktoren der Gruppendeterminante, Gesam. Abhandl. 3 (1968), 38–77.
  • T. Hawkins, The origins of the theory of group characters. Arch. Hist. Exact Sci. \bf7 (1971), 142–170.
  • N. Kaiblinger, On the Lehmer constant of finite cyclic groups, Acta Arith. 142 (2010), 79–84.
  • ––––, Progress on Olga Taussky-Todd's circulant problem, Ramanujan J. 28 (2012), 45–60.
  • H. Laquer, Values of circulants with integer entries, pp. 212–217 in A collection of manuscripts related to the Fibonacci sequence, Fibonacci Association, Santa Clara, CA (1980).
  • D. Lind, Lehmer's problem for compact abelian groups, Proc. Amer. Math. Soc. 133 (2005), 1411–1416.
  • M. Newman, On a problem suggested by Olga Taussky-Todd, Illinois J. Math. 24 (1980), 156–158.
  • ––––, Determinants of circulants of prime power order, Lin. Multilin. Alg. 9 (1980), 187–191.
  • V. Pigno and C. Pinner, The Lind-Lehmer constant for cyclic groups of order less than $892,371,480$, Ramanujan J. 33 (2014), 295–300.
  • C. Pinner and W. Vipismakul, The Lind-Lehmer constant for $\mathbb Z_{m} \times \mathbb Z^{n}_{p}$, Integers 16 (2016).
  • C. Pinner and C. Smyth, Integer group determinants for small groups, Ramanujan J., arXiv:1806.00199 [math.NT].
  • J.-P. Serre, Linear representations of finite groups, Grad. Texts Math. 42 (1977).
  • W. Vipismakul, The stabilizer of the group determinant and bounds for Lehmer's conjecture on finite abelian groups, Ph.D. dissertation, The University of Texas at Austin (2013).