Characters of finite quasigroups VII: permutation characters

Kenneth Walter Johnson; Jonathan D. H. Smith

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 2, page 265-273
  • ISSN: 0010-2628

Abstract

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Each homogeneous space of a quasigroup affords a representation of the Bose-Mesner algebra of the association scheme given by the action of the multiplication group. The homogeneous space is said to be faithful if the corresponding representation of the Bose-Mesner algebra is faithful. In the group case, this definition agrees with the usual concept of faithfulness for transitive permutation representations. A permutation character is associated with each quasigroup permutation representation, and specialises appropriately for groups. However, in the quasigroup case the character of the homogeneous space determined by a subquasigroup need not be obtained by induction from the trivial character on the subquasigroup. The number of orbits in a quasigroup permutation representation is shown to be equal to the multiplicity with which its character includes the trivial character.

How to cite

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Johnson, Kenneth Walter, and Smith, Jonathan D. H.. "Characters of finite quasigroups VII: permutation characters." Commentationes Mathematicae Universitatis Carolinae 45.2 (2004): 265-273. <http://eudml.org/doc/249324>.

@article{Johnson2004,
abstract = {Each homogeneous space of a quasigroup affords a representation of the Bose-Mesner algebra of the association scheme given by the action of the multiplication group. The homogeneous space is said to be faithful if the corresponding representation of the Bose-Mesner algebra is faithful. In the group case, this definition agrees with the usual concept of faithfulness for transitive permutation representations. A permutation character is associated with each quasigroup permutation representation, and specialises appropriately for groups. However, in the quasigroup case the character of the homogeneous space determined by a subquasigroup need not be obtained by induction from the trivial character on the subquasigroup. The number of orbits in a quasigroup permutation representation is shown to be equal to the multiplicity with which its character includes the trivial character.},
author = {Johnson, Kenneth Walter, Smith, Jonathan D. H.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quasigroup; association scheme; permutation character; finite quasigroups; association schemes; permutation characters; multiplication groups; homogeneous spaces},
language = {eng},
number = {2},
pages = {265-273},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Characters of finite quasigroups VII: permutation characters},
url = {http://eudml.org/doc/249324},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Johnson, Kenneth Walter
AU - Smith, Jonathan D. H.
TI - Characters of finite quasigroups VII: permutation characters
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 2
SP - 265
EP - 273
AB - Each homogeneous space of a quasigroup affords a representation of the Bose-Mesner algebra of the association scheme given by the action of the multiplication group. The homogeneous space is said to be faithful if the corresponding representation of the Bose-Mesner algebra is faithful. In the group case, this definition agrees with the usual concept of faithfulness for transitive permutation representations. A permutation character is associated with each quasigroup permutation representation, and specialises appropriately for groups. However, in the quasigroup case the character of the homogeneous space determined by a subquasigroup need not be obtained by induction from the trivial character on the subquasigroup. The number of orbits in a quasigroup permutation representation is shown to be equal to the multiplicity with which its character includes the trivial character.
LA - eng
KW - quasigroup; association scheme; permutation character; finite quasigroups; association schemes; permutation characters; multiplication groups; homogeneous spaces
UR - http://eudml.org/doc/249324
ER -

References

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  1. Barnsley M.F., Fractals Everywhere, Academic Press, San Diego, CA, 1988. Zbl0784.58002MR0977274
  2. Chari V., Pressley A.N., A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994. Zbl0839.17010MR1300632
  3. Johnson K.W., Smith J.D.H., Characters of finite quasigroups, Eur. J. Comb. 5 (1984), 43-50. (1984) Zbl0537.20042MR0746044
  4. Johnson K.W., Smith J.D.H., Characters of finite quasigroups II: induced characters, Eur. J. Comb. 7 (1986), 131-137. (1986) Zbl0599.20110MR0856325
  5. Johnson K.W., Smith J.D.H., Characters of finite quasigroups III: quotients and fusion, Eur. J. Comb. 10 (1989), 47-56. (1989) Zbl0667.20053MR0977179
  6. Johnson K.W., Smith J.D.H., Characters of finite quasigroups IV: products and superschemes, Eur. J. Comb. 10 (1989), 257-263. (1989) Zbl0669.20053MR1029172
  7. Johnson K.W., Smith J.D.H., Characters of finite quasigroups V: linear characters, Eur. J. Comb. 10 (1989), 449-456. (1989) Zbl0679.20059MR1014553
  8. Johnson K.W., Smith J.D.H., Song S.Y., Characters of finite quasigroups VI: critical examples and doubletons, Eur. J. Comb. 11 (1990), 267-275. (1990) Zbl0704.20056MR1059557
  9. Mack G., Schomerus V., Conformal field algebras with quantum symmetry from the theory of superselection sectors, Comm. Math. Phys. 134 (1990), 139-196. (1990) Zbl0715.17028MR1079804
  10. Penrose P., A generalised inverse for matrices, Proc. Cambridge. Phil. Soc. 51 (1955), 406-413. (1955) MR0069793
  11. Smith J.D.H., Centraliser rings of multiplication groups on quasigroups, Math. Proc. Cambridge Phil. Soc. 79 (1976), 427-431. (1976) Zbl0335.20035MR0399333
  12. Smith J.D.H., Quasigroup actions: Markov chains, pseudoinverses, and linear representations, Southeast Asia Bull. Math. 23 (1999), 719-729. (1999) Zbl0944.20059MR1810836
  13. Smith J.D.H., Quasigroup homogeneous spaces and linear representations, J. Algebra 241 (2001), 193-203. (2001) Zbl0994.20054MR1838850
  14. Smith J.D.H., A coalgebraic approach to quasigroup permutation representations, Algebra Universalis 48 (2002), 427-438. (2002) Zbl1068.20070MR1967091

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