No Access

Classification of Table Algebras with Two Basis Elements Whose Product Is a Linear Combination of Their Conjugates

Zvi Arad

Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

Netanya Academic College, Kiryat Rabin, University St. Netanya, Israel

Search for more papers by this author

and

Efi Cohen

Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

Search for more papers by this author

https://doi.org/10.1142/S1005386706000526Cited by:1 (Source: Crossref)

Abstract

Table Algebras (A,B) with distinct non-real basis elements a, b∈ B satisfying , (m, n∈ ℝ⁺) are studied in this paper. The two cases where either a is real and b is non-real or a, b are both real were completely classified by Arad and Blau. Here, we study the remaining case where a and b are non-real.

This research was carried out as master's thesis of the second author under the supervision of Prof. Zvi Arad at Bar-Ilan University, Israel.

Keywords:

AMSC: 20C05, 20C15, 05E30

References

Z. Arad, On products of conjugacy classes and irreducible characters in finite groups, A.M.S. Lecture Notes prepared with the Summer Institute on Representations of Finite Groups, S6-ZA-1-12 (1986) . Google Scholar
Z. Arad, Proc. Simpos. Pure Math. 47, 3 (1987). Google Scholar
Z. Arad, Survey on table algebras and applications to finite group theory, Proceeding Israel Mathematics Conference (Weizmann Science Press, Jerusalem, 1989) pp. 96–110. Google Scholar
Z. Arad, Groups St. Andrews 1997 in Bath, London Math. Soc. Lecture Note Ser. 260 (Cambridge Univ. Press, Cambridge, 1999) pp. 20–29. Crossref, Google Scholar
Z. Aradet al., J. Algebra 231(2), 473 (2000). Crossref, Web of Science, Google Scholar
Z. Arad and H. Blau, J. Algebra 138, 137 (1991). Crossref, Web of Science, Google Scholar
Z. Aradet al., J. Algebra 168, 615 (1994). Crossref, Web of Science, Google Scholar
Z. Arad, H. Blau and E. Fisman, J. Algebra 138, 186 (1991). Crossref, Web of Science, Google Scholar
Z. Arad et al. , Homogenous Integral Table Algebras of Degree Three: A Trilogy , Memoirs of Amer. Math. Soc. 144 ( Providence, RI. , 2000 ) . Google Scholar
Z. Arad, D. Chillag and M. Herzog, J. Algebra 103, 241 (1986). Crossref, Web of Science, Google Scholar
Z. Arad, E. Fisman, On products of irreducible characters and conjugacy classes of finite groups in complementary sets, A.M.S. Lecture Notes prepared with the Summer Institute on Representations of Finite Groups, S5-AZ/EF-1-26 (1986) . Google Scholar
Z. Arad and E. Fisman, Proc. Edinburgh Math. Soc. 30(1), 7 (1987). Crossref, Web of Science, Google Scholar
Z. Arad and E. Fisman, Proc. Sympos. Pure Math. 47, 11 (1987). Web of Science, Google Scholar
Z. Arad and E. Fisman, J. Algebra 114, 466 (1988). Crossref, Web of Science, Google Scholar
Z. Arad, E. Fisman and M. Muzychuck, Israel J. Math. 114, 29 (1999). Crossref, Web of Science, Google Scholar
Z. Arad and M. Herzog, Survey on products of conjugacy classes in groups, Proceedings of the Fourth International Conference on Representations of Algebras, Mathematical Lecture Notes Series 2 (1984) pp. 2.01–2.04. Google Scholar
Z. Arad and M. Herzog (eds.) , Products of Conjugacy Classes in Groups , Lecture Notes in Mathematics 1112 ( Springer-Verlag , Berlin , 1985 ) . Crossref, Google Scholar
Z. Arad and H. Lipman, Houston J. Math. 15, 305 (1989). Web of Science, Google Scholar
Z. Arad and M. Muzychuck , Standard Integral Table Algebras Generated by a Nonreal Element of Small Degree , Lecture Notes in Mathematics 1173 ( Springer-Verlag , Berlin , 2002 ) . Crossref, Google Scholar
H. I. Blau, J. Algebra 177, 297 (1995). Crossref, Web of Science, Google Scholar
H. I. Blau and D. Chillag, Proc. Amer. Math. Soc. 298, 7 (1986). Google Scholar
H. I. Blau and B. Xu, J. Algebra 199, 393 (1998). Google Scholar
G. Hoheisel and Uber Charaktere, Monastsch. Math. Phys. 48, 448 (1939). Web of Science, Google Scholar
Y. Kawadw, Proc. Phys. Math. Soc. Japan 24(3), 97 (1942). Web of Science, Google Scholar