| Title:
|
Characters of finite quasigroups VII: permutation characters (English) |
| Author:
|
Johnson, K. W. |
| Author:
|
Smith, J. D. H. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
45 |
| Issue:
|
2 |
| Year:
|
2004 |
| Pages:
|
265-273 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
quasigroup |
| Keyword:
|
association scheme |
| Keyword:
|
permutation character |
| MSC:
|
05E30 |
| MSC:
|
20C99 |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 1101.20037 |
| idMR:
|
MR2075274 |
| . |
| Date available:
|
2009-05-05T16:44:57Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119455 |
| . |
| Reference:
|
[1] Barnsley M.F.: Fractals Everywhere.Academic Press, San Diego, CA, 1988. Zbl 0784.58002, MR 0977274 |
| Reference:
|
[2] Chari V., Pressley A.N.: A Guide to Quantum Groups.Cambridge University Press, Cambridge, 1994. Zbl 0839.17010, MR 1300632 |
| Reference:
|
[3] Johnson K.W., Smith J.D.H.: Characters of finite quasigroups.Eur. J. Comb. 5 (1984), 43-50. Zbl 0537.20042, MR 0746044 |
| Reference:
|
[4] Johnson K.W., Smith J.D.H.: Characters of finite quasigroups II: induced characters.Eur. J. Comb. 7 (1986), 131-137. Zbl 0599.20110, MR 0856325 |
| Reference:
|
[5] Johnson K.W., Smith J.D.H.: Characters of finite quasigroups III: quotients and fusion.Eur. J. Comb. 10 (1989), 47-56. Zbl 0667.20053, MR 0977179 |
| Reference:
|
[6] Johnson K.W., Smith J.D.H.: Characters of finite quasigroups IV: products and superschemes.Eur. J. Comb. 10 (1989), 257-263. Zbl 0669.20053, MR 1029172 |
| Reference:
|
[7] Johnson K.W., Smith J.D.H.: Characters of finite quasigroups V: linear characters.Eur. J. Comb. 10 (1989), 449-456. Zbl 0679.20059, MR 1014553 |
| Reference:
|
[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. Zbl 0704.20056, MR 1059557 |
| Reference:
|
[9] Mack G., Schomerus V.: Conformal field algebras with quantum symmetry from the theory of superselection sectors.Comm. Math. Phys. 134 (1990), 139-196. Zbl 0715.17028, MR 1079804 |
| Reference:
|
[10] Penrose P.: A generalised inverse for matrices.Proc. Cambridge. Phil. Soc. 51 (1955), 406-413. MR 0069793 |
| Reference:
|
[11] Smith J.D.H.: Centraliser rings of multiplication groups on quasigroups.Math. Proc. Cambridge Phil. Soc. 79 (1976), 427-431. Zbl 0335.20035, MR 0399333 |
| Reference:
|
[12] Smith J.D.H.: Quasigroup actions: Markov chains, pseudoinverses, and linear representations.Southeast Asia Bull. Math. 23 (1999), 719-729. Zbl 0944.20059, MR 1810836 |
| Reference:
|
[13] Smith J.D.H.: Quasigroup homogeneous spaces and linear representations.J. Algebra 241 (2001), 193-203. Zbl 0994.20054, MR 1838850 |
| Reference:
|
[14] Smith J.D.H.: A coalgebraic approach to quasigroup permutation representations.Algebra Universalis 48 (2002), 427-438. Zbl 1068.20070, MR 1967091 |
| . |
