Skip to main content
SpringerLink
Log in

The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4

  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

In a cyclotomic scheme over a finite field, there are some relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the field. These relations were investigated in [3]. In this paper, we replace the finite field by a commutative local ring which is called a Galois ring of characteristic 4. Hence we want to find similar relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the local ring. Specifically, if we let ℛ be a Galois ring of characteristic 4,X a cyclotomic scheme over ℛ with classD and ℐ the Terwilliger algebra ofX, then we show that most of the irreducible ℐ-modules have standard forms; otherwise, certain relations of the Jacobi sums hold. When the classD is three, we can completely determine the irreducible ℐ-modules using Jacobi sums.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bannai, E., Ito, T.: Algebraic Combinatorics I: Association Schemes. Benjamin/Cumming, New York, 1984

    MATH  Google Scholar 

  2. Claasen, H.L., Goldbach, R.W.: Cyclotomy and association schemes. Report of the Faculty of Technical Mathematics and Informatics 89-73, Delft 1989

  3. Ishibashi, H., Ito, T., Munemasa, A.: The Terwilliger algebras of cyclotomic schemes and the rationality of Jacobi sums. Algebraic Combinatorics (Fukuoka, 1993) 43–44

    Google Scholar 

  4. Ito, T., Munemasa, A., Yamada, M.: Amorphous association schemes over the Galois Rings of characteristic 4. Europ. J. Comb.12, 513–526 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  5. McDonald, B.R.: Finite Rings with Identity, Marcel Dekker, INC, New York, Pure and Applied Mathematics, A Series of Monographs and Textbooks

  6. Terwilliger, P.: The subconstituent algebra of an association scheme (Part I). J. Alge. Comb.1, 363–388 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. Terwilliger, P.: The subconstituent algebra of an association scheme (Part II). J. Alge. Comb.2, 73–103 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Terwilliger, P.: The subconstituent algebra of an association scheme (Part III). J. Alge. Comb.2, 177–210 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. Yamada, M.: Distance-regular digraphs of girth 4 over an extension ring ofZ/4Z. Graphs and Combinatorics6, 381–394 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Mathematics, Kyushu University, 812, Hakozaki, Fukuoka, Japan

    Haruo Ishibashi

Authors
  1. Haruo Ishibashi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ishibashi, H. The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4. Graphs and Combinatorics 12, 39–54 (1996). https://doi.org/10.1007/BF01858443

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01858443

Keywords

Search

Navigation