Difference between revisions of "Schur ring"
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For each $ D \subset G $, | For each $ D \subset G $, | ||
let $ \overline{D}\; = \sum _ {g \in D } g $ | let $ \overline{D}\; = \sum _ {g \in D } g $ | ||
| − | and $ D ^ {-} | + | and $ D ^ {-1} = \{ {g ^ {-1} } : {g \in D } \} $. |
Suppose that for each $ D \in \pi $, | Suppose that for each $ D \in \pi $, | ||
| − | $ D ^ {-} | + | $ D ^ {-1} \in \pi $, |
and for all $ D _ {i} , D _ {j} \in \pi $, | and for all $ D _ {i} , D _ {j} \in \pi $, | ||
| − | $ \overline{D}\; _ {i} \overline{D}\; _ {j} = \sum _ {k=} | + | $ \overline{D}\; _ {i} \overline{D}\; _ {j} = \sum _ {k=1} ^ {n} c _ {ij} ^ {k} \overline{D}\; _ {k} $ |
for certain $ c _ {ij} ^ {k} \in \mathbf Z $. | for certain $ c _ {ij} ^ {k} \in \mathbf Z $. | ||
Then the $ \overline{D}\; _ {1} \dots \overline{D}\; _ {n} $ | Then the $ \overline{D}\; _ {1} \dots \overline{D}\; _ {n} $ | ||
| Line 33: | Line 33: | ||
of $ \mathbf Z [ G] $ | of $ \mathbf Z [ G] $ | ||
is a Schur ring over $ G $ | is a Schur ring over $ G $ | ||
| − | if and only if $ x ^ {(- | + | if and only if $ x ^ {(-1 )} \in S $ |
for all $ x \in S $( | for all $ x \in S $( | ||
| − | where $ x ^ {(- | + | where $ x ^ {(- 1)} = \sum a _ {g} g ^ {-1} $ |
if $ x = \sum a _ {g} g $) | 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 $. | 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 $ | A symmetric Schur ring $ S $ | ||
| − | is a Schur ring for which $ x ^ {(- | + | is a Schur ring for which $ x ^ {(-1)} = x $ |
for all $ x \in S $. | for all $ x \in S $. | ||
Latest revision as of 18:57, 11 January 2024
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 |
Schur ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schur_ring&oldid=48626