Eigenspace embeddings of imprimitive association schemes

Vidali, Janoš (author)

For a given symmetric association scheme $\mathcal{A}$ and its eigenspace $S_j$ there exists a mapping of vertices of $\mathcal{A}$ to unit vectors of $S_j$, known as the spherical representation of $\mathcal{A}$ in $S_j$, such that the inner products of these vectors only depend on the relation between the corresponding vertices; furthermore, these inner products only depend on the parameters of $\mathcal{A}$. We consider parameters of imprimitive association schemes listed as open cases in the list of parameters for quotient-polynomial graphs recently published by Herman and Maleki, and study embeddings of their substructures into some eigenspaces consistent with spherical representations of the putative association schemes. Using this, we obtain nonexistence for two parameter sets for $4$-class association schemes and one parameter sets for a $5$-class association scheme passing all previously known feasibility conditions, as well as uniqueness for two parameter sets for $5$-class association schemes.

• Language: English
• Keywords: association scheme, imprimitivity, spherical representation, nonexistence, uniqueness
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2026
• Year: 2026
• Number of pages: 35 str.
• Numbering: Vol. 33, iss. 1, article no. P1.2
• PID: 20.500.12556/RUL-177899
• UDC: 519.17
• ISSN on article: 1077-8926
• DOI: 10.37236/14071
• COBISS.SI-ID: 264381187
• Publication date in RUL: 12.01.2026

Record is a part of a journal

• Title: The Electronic journal of combinatorics
• Shortened title: Electron. j. comb.
• Publisher: N.J. Calkin and H.S. Wilf
• ISSN: 1077-8926
• COBISS.SI-ID: 6973785

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: asociativne sheme, imprimitivnost, sferična reprezentacija, neobstoj, enoličnost