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A certain kind of subring of the [[Group algebra|group algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s0834701.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s0834702.png" />.
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$#C+1 = 28 : ~/encyclopedia/old_files/data/S083/S.0803470 Schur ring
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s0834703.png" /> be a [[Finite group|finite group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s0834704.png" /> a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s0834705.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s0834706.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s0834707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s0834708.png" />. Suppose that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s0834709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347010.png" />, and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347012.png" /> for certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347013.png" />. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347014.png" /> form the basis (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347015.png" />) of a subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347016.png" />. These subrings are called Schur rings. A unitary Schur ring is one which contains the unit element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347017.png" />.
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A subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347019.png" /> is a Schur ring over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347020.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347021.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347022.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347023.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347024.png" />) and it is closed under the Hadamard product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347025.png" />.
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A certain kind of subring of the [[Group algebra|group algebra]]  $  \mathbf Z [ G] $
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of a group  $  G $.
  
A symmetric Schur ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347026.png" /> is a Schur ring for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083470/s08347028.png" />.
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Let  $  G $
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be a [[Finite group|finite group]] and  $  \pi = ( D _ {1} \dots D _ {n} ) $
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a partition of  $  G $.
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For each  $  D \subset  G $,
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let  $  \overline{D}\; = \sum _ {g \in D }  g $
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and  $  D  ^ {-1} = \{ {g  ^ {-1} } : {g \in D } \} $.
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Suppose that for each  $  D \in \pi $,
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$  D  ^ {-1} \in \pi $,
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and for all  $  D _ {i} , D _ {j} \in \pi $,
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$  \overline{D}\; _ {i} \overline{D}\; _ {j} = \sum _ {k=1} ^ {n} c _ {ij}  ^ {k} \overline{D}\; _ {k} $
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for certain  $  c _ {ij}  ^ {k} \in \mathbf Z $.  
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Then the  $  \overline{D}\; _ {1} \dots \overline{D}\; _ {n} $
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form the basis (over  $  \mathbf Z $)
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of a subring of  $  \mathbf Z [ G] $.  
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These subrings are called Schur rings. A unitary Schur ring is one which contains the unit element of  $  \mathbf Z [ G] $.
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A subring  $  S $
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of  $  \mathbf Z [ G] $
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is a Schur ring over  $  G $
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if and only if  $  x  ^ {(-1 )} \in S $
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for all  $  x \in S $(
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where  $  x  ^ {(- 1)} = \sum a _ {g} g  ^ {-1} $
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if  $  x = \sum a _ {g} g $)
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and it is closed under the Hadamard product  $  ( \sum a _ {g} g ) ( \sum b _ {g} g ) = \sum a _ {g} b _ {g} g $.
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A symmetric Schur ring  $  S $
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is a Schur ring for which  $  x  ^ {(-1)} = x $
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for all $  x \in S $.
  
 
Historically, Schur rings were first studied by I. Schur [[#References|[a1]]] and H. Wielandt [[#References|[a2]]], who coined the name, in connection with the study of permutation groups; cf. [[#References|[a3]]]–[[#References|[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, [[#References|[a6]]], [[#References|[a7]]].
 
Historically, Schur rings were first studied by I. Schur [[#References|[a1]]] and H. Wielandt [[#References|[a2]]], who coined the name, in connection with the study of permutation groups; cf. [[#References|[a3]]]–[[#References|[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, [[#References|[a6]]], [[#References|[a7]]].

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
How to Cite This Entry:
Schur ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schur_ring&oldid=15134