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Imprimitive association schemes and elimination theory

Akihiro Higashitani, Hirotake Kurihara

Abstract

We prove that a commutative association scheme is imprimitive if and only if it admits a multivariate $P$- or $Q$-polynomial structure with respect to an elimination-type monomial order. This provides a direct bridge between the classical theory of block and quotient schemes for imprimitive association schemes and elimination theory in computational commutative algebra. For an imprimitive multivariate $P$- or $Q$-polynomial association scheme, we determine the induced multivariate polynomial structures on the quotient and block schemes and describe their associated polynomials via explicit specializations, variable deletions, and rescalings of the original associated polynomials. At the level of zero-dimensional ideals, we show that the ideal of the block scheme is exactly an elimination ideal, whereas the ideal of the quotient scheme is obtained by adjoining the valency relations for the eliminated variables and then eliminating. As applications, we study direct products and crested products from the viewpoint of multivariate polynomiality, and we characterize the schemes that are multivariate $P$- or $Q$-polynomial with respect to every monomial order as precisely the direct products of univariate $P$- or $Q$-polynomial schemes. We also discuss formal duality, composition series, and several related open problems.

Imprimitive association schemes and elimination theory

Abstract

We prove that a commutative association scheme is imprimitive if and only if it admits a multivariate - or -polynomial structure with respect to an elimination-type monomial order. This provides a direct bridge between the classical theory of block and quotient schemes for imprimitive association schemes and elimination theory in computational commutative algebra. For an imprimitive multivariate - or -polynomial association scheme, we determine the induced multivariate polynomial structures on the quotient and block schemes and describe their associated polynomials via explicit specializations, variable deletions, and rescalings of the original associated polynomials. At the level of zero-dimensional ideals, we show that the ideal of the block scheme is exactly an elimination ideal, whereas the ideal of the quotient scheme is obtained by adjoining the valency relations for the eliminated variables and then eliminating. As applications, we study direct products and crested products from the viewpoint of multivariate polynomiality, and we characterize the schemes that are multivariate - or -polynomial with respect to every monomial order as precisely the direct products of univariate - or -polynomial schemes. We also discuss formal duality, composition series, and several related open problems.
Paper Structure (24 sections, 33 theorems, 145 equations)

This paper contains 24 sections, 33 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Association schemes
  4. Imprimitive association schemes
  5. Monomial orders, Gröbner bases and elimination orders
  6. Multivariate $P$- and $Q$-polynomial association schemes
  7. Characterization via the first eigenmatrix of multivariate $P$-polynomial association schemes
  8. Main theorems
  9. Imprimitive association schemes and elimination orders
  10. Quotient and block schemes and elimination orders
  11. Examples
  12. Commutative association schemes of class $2$
  13. Nonbinary Johnson schemes
  14. Imprimitive distance-regular graphs
  15. Multivariate polynomial association schemes, direct products, and crested products
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $\mathfrak{X}=(X,\mathcal{R},\mathcal{I})$ be an association scheme. Then the following equalities hold for the intersection numbers and Krein numbers of $\mathfrak{X}$:

Theorems & Definitions (63)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3: cf. Zieschang
  • Definition 2.4
  • Lemma 2.5: cf. curtin2008
  • Lemma 2.6: cf. BI1984curtin2008Zieschang
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 53 more