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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  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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