| Title:
|
The upper triangular algebra loop of degree $4$ (English) |
| Author:
|
Johnson, K. W. |
| Author:
|
Munywoki, M. |
| Author:
|
Smith, Jonathan D. H. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
55 |
| Issue:
|
4 |
| Year:
|
2014 |
| Pages:
|
457-470 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A natural loop structure is defined on the set $U_4$ of unimodular upper-triangular matrices over a given field. Inner mappings of the loop are computed. It is shown that the loop is non-associative and nilpotent, of class 3. A detailed listing of the loop conjugacy classes is presented. In particular, one of the loop conjugacy classes is shown to be properly contained in a superclass of the corresponding algebra group. (English) |
| Keyword:
|
algebra group |
| Keyword:
|
quasigroup |
| Keyword:
|
loop |
| Keyword:
|
supercharacter |
| Keyword:
|
fusion scheme |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 06391554 |
| idMR:
|
MR3269008 |
| . |
| Date available:
|
2014-10-09T09:48:21Z |
| Last updated:
|
2017-01-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143970 |
| . |
| Reference:
|
[1] Aguiar M. et al.: Supercharacters, symmetric functions in non-commuting variables, and related Hopf algebras.Adv. Math. 229 (2012), 2310–2337. MR 2880223, 10.1016/j.aim.2011.12.024 |
| Reference:
|
[2] André C.A.M.: Basic characters of the unitriangular group.Algebra 175 (1995), 287–319. Zbl 1064.20005, MR 1338979, 10.1006/jabr.1995.1187 |
| Reference:
|
[3] Bruck R.H.: Contributions to the theory of loops.Trans. Amer. Math. Soc. 60 (1946), 245–354. Zbl 0061.02201, MR 0017288, 10.1090/S0002-9947-1946-0017288-3 |
| Reference:
|
[4] Bruck R.H.: A Survey of Binary Systems.Springer, Berlin, 1958. Zbl 0141.01401, MR 0093552 |
| Reference:
|
[5] Diaconis P., Isaacs I.M.: Supercharacters and superclasses for algebra groups.Trans. Amer. Math. Soc. 360 (2008), 2359–2392. Zbl 1137.20008, MR 2373317, 10.1090/S0002-9947-07-04365-6 |
| Reference:
|
[6] 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, 10.1016/S0195-6698(89)80032-0 |
| Reference:
|
[7] Smith J.D.H.: An Introduction to Quasigroups and Their Representations.Chapman and Hall/CRC, Boca Raton, FL, 2007. Zbl 1122.20035, MR 2268350 |
| Reference:
|
[8] Smith J.D.H., Romanowska A.B.: Post-Modern Algebra.Wiley, New York, NY, 1999. Zbl 0946.00001, MR 1673047 |
| . |
