Schur ring

A certain kind of subring of the group algebra of a group .

Let be a finite group and a partition of . For each , let and . Suppose that for each , , and for all , for certain . Then the form the basis (over ) of a subring of . These subrings are called Schur rings. A unitary Schur ring is one which contains the unit element of .

A subring of is a Schur ring over if and only if for all (where if ) and it is closed under the Hadamard product .

A symmetric Schur ring is a Schur ring for which for all .

Historically, Schur rings were first studied by I. Schur [a1] and H. Wielandt [a2], who coined the name, in connection with the study of permutation groups; cf. [a3]–[a5] for applications of Schur rings to group theory. More recently it was discovered that they are also related to certain combinatorial structures, such as association schemes and strongly regular graphs, [a6], [a7].

References

[a1]	I. Schur, "Zur Theorie der einfach transitiven Permutationsgruppen" Sitzungsber. Preuss. Akad. Wissenschaft. Berlin. Phys.-Math. Kl. (1933) pp. 598–623
[a2]	H. Wielandt, "Zur Theorie der einfach transitiven Permutationsgruppen II" Math. Z. , 52 (1949) pp. 384–393
[a3]	O. Tamaschke, "Schur-Ringe" , B.I. Wissenschaftsverlag Mannheim (1970)
[a4]	W.R. Scott, "Group theory" , Prentice-Hall (1964)
[a5]	H. Wielandt, "Finite permutation groups" , Acad. Press (1964) (Translated from German)
[a6]	E. Bannai, T. Ito, "Algebraic combinatorics I: Association schemes" , Benjamin/Cummings (1984)
[a7]	S.L. Ma, "On association schemes, Schur rings, strongly regular graphs and partial difference sets" Ars Comb. , 27 (1989) pp. 211–220