Schur ring

A certain kind of subring of the group algebra $ \mathbf Z [ G] $ of a group $ G $.

Let $ G $ be a finite group and $ \pi = ( D _ {1} \dots D _ {n} ) $ a partition of $ G $. For each $ D \subset G $, let $ \overline{D}\; = \sum _ {g \in D } g $ and $ D ^ {-} 1 = \{ {g ^ {-} 1 } : {g \in D } \} $. Suppose that for each $ D \in \pi $, $ D ^ {-} 1 \in \pi $, and for all $ D _ {i} , D _ {j} \in \pi $, $ \overline{D}\; _ {i} \overline{D}\; _ {j} = \sum _ {k=} 1 ^ {n} c _ {ij} ^ {k} \overline{D}\; _ {k} $ for certain $ c _ {ij} ^ {k} \in \mathbf Z $. Then the $ \overline{D}\; _ {1} \dots \overline{D}\; _ {n} $ form the basis (over $ \mathbf Z $) of a subring of $ \mathbf Z [ G] $. These subrings are called Schur rings. A unitary Schur ring is one which contains the unit element of $ \mathbf Z [ G] $.

A subring $ S $ of $ \mathbf Z [ G] $ is a Schur ring over $ G $ if and only if $ x ^ {(-} 1) \in S $ for all $ x \in S $( where $ x ^ {(-} 1) = \sum a _ {g} g ^ {-} 1 $ if $ x = \sum a _ {g} g $) and it is closed under the Hadamard product $ ( \sum a _ {g} g ) ( \sum b _ {g} g ) = \sum a _ {g} b _ {g} g $.

A symmetric Schur ring $ S $ is a Schur ring for which $ x ^ {(-} 1) = x $ for all $ x \in S $.

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