Difference between revisions of "Coherent algebra"
m (link) |
m (link) |
||
Line 1: | Line 1: | ||
Algebras introduced by D.G. Higman, first in relational language under the name coherent configuration [[#References|[a4]]] and later in terms of matrices [[#References|[a6]]]. The slightly different axiomatics of cellular algebras were independently suggested by B.Yu. Weisfeiler and A.A. Leman (cf. also [[Cellular algebra|Cellular algebra]]). | Algebras introduced by D.G. Higman, first in relational language under the name coherent configuration [[#References|[a4]]] and later in terms of matrices [[#References|[a6]]]. The slightly different axiomatics of cellular algebras were independently suggested by B.Yu. Weisfeiler and A.A. Leman (cf. also [[Cellular algebra|Cellular algebra]]). | ||
− | Like | + | Like [[association scheme]]s and Bose–Mesner algebras, coherent algebras provide a wide and solid foundation for investigations in various areas of algebraic combinatorics. |
A coherent algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c1301401.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c1301402.png" /> and rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c1301403.png" /> is a matrix subalgebra of the full matrix algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c1301404.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c1301405.png" />-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c1301406.png" /> such that: | A coherent algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c1301401.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c1301402.png" /> and rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c1301403.png" /> is a matrix subalgebra of the full matrix algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c1301404.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c1301405.png" />-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c1301406.png" /> such that: | ||
Line 11: | Line 11: | ||
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014013.png" /> is the all-one matrix; | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014013.png" /> is the all-one matrix; | ||
− | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014014.png" /> is closed with respect to Schur–Hadamard multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014016.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014019.png" />. Each coherent algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014020.png" /> has a unique basis of zero-one matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014021.png" /> such that: | + | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014014.png" /> is closed with respect to [[Schur–Hadamard multiplication]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014016.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014019.png" />. Each coherent algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014020.png" /> has a unique basis of zero-one matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014021.png" /> such that: |
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014022.png" />; | 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130140/c13014022.png" />; |
Revision as of 12:29, 30 December 2014
Algebras introduced by D.G. Higman, first in relational language under the name coherent configuration [a4] and later in terms of matrices [a6]. The slightly different axiomatics of cellular algebras were independently suggested by B.Yu. Weisfeiler and A.A. Leman (cf. also Cellular algebra).
Like association schemes and Bose–Mesner algebras, coherent algebras provide a wide and solid foundation for investigations in various areas of algebraic combinatorics.
A coherent algebra of order
and rank
is a matrix subalgebra of the full matrix algebra
of
-matrices over
such that:
is closed with respect to the Hermitian adjoint, which is defined by
for
;
, where
is the unit matrix;
, where
is the all-one matrix;
is closed with respect to Schur–Hadamard multiplication
, where
for
,
,
. Each coherent algebra
has a unique basis of zero-one matrices
such that:
1) ;
2) , where
is the matrix transposed to
;
3) . Property 1) implies that the basis
consists of mutually orthogonal idempotents with respect to the Schur–Hadamard product. This basis is called the standard basis of
. The non-negative integer structure constants
are important numerical invariants of
. The notation
indicates that
is a coherent algebra with standard basis
.
Let and denote by
a binary relation over
.
is called the support of the zero-one matrix
(or, in other words,
is the adjacency matrix of the graph
with vertex set
and set
of directed edges). The system of relations
obtained in this way from a coherent algebra
is called a coherent configuration.
The structure constants of
are sometimes called the intersection numbers of
. They have the following combinatorial interpretation:
![]() |
![]() |
A coherent configuration is called homogeneous if one of its basic relations, say
, coincides with the diagonal relation
. In terms of matrices, a coherent algebra
is called a Bose–Mesner algebra (briefly BM-algebra) if
. Note that according to E. Bannai and T. Ito [a1], a homogeneous coherent configuration is also called an association scheme (not necessarily commutative; cf. also Association scheme).
Let be a coherent configuration. A subset
is called a fibre of
if
![]() | (a1) |
and is a minimal (with respect to inclusion) subset satisfying condition (a1). The coherent algebras with one fibre are exactly the BM-algebras. Coherent algebras with few fibres may be used for a unified presentation and investigation of various combinatorial objects, see, for example, [a3], [a7], [a9].
An important class of coherent algebras consists of the centralizer algebras of permutation groups (not necessarily transitive) [a2], [a10] (cf. also Permutation group; Centralizer). This leads to many important applications of coherent algebras.
It was Higman [a5], [a8] who developed the foundations of the representation theory of coherent algebras as a generalization of the representation theory of finite permutation groups (cf. also Finite group, representation of a).
References
[a1] | E. Bannai, T. Ito, "Algebraic combinatorics" , I , Benjamin/Cummings (1984) |
[a2] | I.A. Faradžev, M.H. Klin, M.E. Muzichuk, "Cellular rings and groups of automorphisms of graphs" I.A. Faradžev (ed.) et al. (ed.) , Investigations in Algebraic Theory of Combinatorial Objects , Kluwer Acad. Publ. (1994) pp. 1–152 |
[a3] | W.H. Haemers, D.G. Higman, "Strongly regular graphs with strongly regular decomposition" Linear Alg. & Its Appl. , 114/115 (1989) pp. 379–398 |
[a4] | D.G. Higman, "Coherent configurations I" Rend. Sem. Mat. Univ. Padova , 44 (1970) pp. 1–25 |
[a5] | D.G. Higman, "Coherent configurations, Part I: Ordinary representation theory" Geom. Dedicata , 4 (1975) pp. 1–32 |
[a6] | D.G. Higman, "Coherent algebras" Linear Alg. & Its Appl. , 93 (1987) pp. 209–239 |
[a7] | D.G. Higman, "Strongly regular designs and coherent configurations of type ![]() |
[a8] | D.G. Higman, "Computations related to coherent configurations" Congr. Numer. , 75 (1990) pp. 9–20 |
[a9] | M.E. Muzychuk, M. Klin, "On graphs with three eigenvalues" Discr. Math. , 189 (1998) pp. 191–207 |
[a10] | H. Wielandt, "Finite permutation groups" , Acad. Press (1964) |
Coherent algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Coherent_algebra&oldid=35934