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 |
Schur ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schur_ring&oldid=15134