Bose–Mesner algebra
E886600
The Bose–Mesner algebra is a commutative matrix algebra arising from association schemes in algebraic combinatorics, fundamental for studying symmetric relations and distance-regular graphs.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bose–Mesner algebra canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10803780 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Bose–Mesner algebra Context triple: [Raj Chandra Bose, notableWork, Bose–Mesner algebra]
-
A.
Racah algebra
Racah algebra is a mathematical structure in representation theory and quantum mechanics that encodes the symmetries and coupling properties of angular momenta, particularly through Racah coefficients and related special functions.
-
B.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
-
C.
Hadamard matrices
Hadamard matrices are square matrices with entries ±1 whose rows are mutually orthogonal, playing a key role in combinatorics, coding theory, and signal processing.
-
D.
Pólya enumeration theorem
The Pólya enumeration theorem is a fundamental result in combinatorics that counts distinct configurations of objects under group actions by using cycle index polynomials and generating functions.
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E.
Graham–Pollak theorem
The Graham–Pollak theorem is a result in graph theory that states the edges of a complete graph on n vertices cannot be partitioned into fewer than n−1 complete bipartite subgraphs.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Bose–Mesner algebra Target entity description: The Bose–Mesner algebra is a commutative matrix algebra arising from association schemes in algebraic combinatorics, fundamental for studying symmetric relations and distance-regular graphs.
-
A.
Racah algebra
Racah algebra is a mathematical structure in representation theory and quantum mechanics that encodes the symmetries and coupling properties of angular momenta, particularly through Racah coefficients and related special functions.
-
B.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
-
C.
Hadamard matrices
Hadamard matrices are square matrices with entries ±1 whose rows are mutually orthogonal, playing a key role in combinatorics, coding theory, and signal processing.
-
D.
Pólya enumeration theorem
The Pólya enumeration theorem is a fundamental result in combinatorics that counts distinct configurations of objects under group actions by using cycle index polynomials and generating functions.
-
E.
Graham–Pollak theorem
The Graham–Pollak theorem is a result in graph theory that states the edges of a complete graph on n vertices cannot be partitioned into fewer than n−1 complete bipartite subgraphs.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure
ⓘ
commutative algebra ⓘ finite-dimensional algebra ⓘ matrix algebra ⓘ |
| appearsIn | theory of symmetric relations on a finite set ⓘ |
| arisesFrom | association scheme ⓘ |
| closedUnder |
matrix addition
ⓘ
matrix multiplication ⓘ scalar multiplication ⓘ |
| contains |
all adjacency matrices of the association scheme
ⓘ
identity matrix ⓘ |
| context | finite association scheme ⓘ |
| dimensionEquals | number of associate classes in the association scheme ⓘ |
| encodes |
Krein parameters of an association scheme
ⓘ
eigenvalues of adjacency matrices ⓘ intersection numbers of an association scheme ⓘ |
| field |
complex numbers
ⓘ
real numbers ⓘ |
| generalizes | adjacency algebra of a regular graph ⓘ |
| hasApplication |
analysis of linear codes
ⓘ
classification of distance-regular graphs ⓘ construction of combinatorial designs ⓘ eigenvalue bounds for graphs ⓘ |
| hasBasis |
adjacency matrices of an association scheme
ⓘ
primitive idempotents of an association scheme ⓘ |
| hasDecomposition | simultaneous eigenspace decomposition ⓘ |
| hasDualBasis | primitive idempotent basis ⓘ |
| hasDualStructureConstants | Krein parameters ⓘ |
| hasProperty |
all basis adjacency matrices are 0–1 matrices
ⓘ
basis adjacency matrices are pairwise disjoint in support ⓘ basis adjacency matrices sum to the all-ones matrix ⓘ |
| hasStructureConstants | intersection numbers ⓘ |
| isCommutative | true ⓘ |
| isSemisimple | true ⓘ |
| isSimultaneouslyDiagonalizable | true ⓘ |
| namedAfter |
D. M. Mesner
NERFINISHED
ⓘ
R. C. Bose NERFINISHED ⓘ |
| relatedTo |
Terwilliger algebra
NERFINISHED
ⓘ
distance-regular graph ⓘ strongly regular graph ⓘ symmetric association scheme ⓘ |
| typicalReference |
Bannai–Ito theory of association schemes
NERFINISHED
ⓘ
Brouwer–Cohen–Neumaier distance-regular graphs NERFINISHED ⓘ |
| usedIn |
algebraic combinatorics
ⓘ
coding theory ⓘ design theory ⓘ spectral graph theory ⓘ study of distance-regular graphs ⓘ theory of association schemes ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Bose–Mesner algebra Description of subject: The Bose–Mesner algebra is a commutative matrix algebra arising from association schemes in algebraic combinatorics, fundamental for studying symmetric relations and distance-regular graphs.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.