Imprimitive association schemes and elimination theory

Akihiro Higashitani; Hirotake Kurihara

arxiv: 2603.20989 · v2 · submitted 2026-03-22 · 🧮 math.CO · math.AC· math.RT

Imprimitive association schemes and elimination theory

Akihiro Higashitani , Hirotake Kurihara This is my paper

Pith reviewed 2026-05-15 07:44 UTC · model grok-4.3

classification 🧮 math.CO math.ACmath.RT MSC 05E30

keywords association schemesimprimitivemultivariate P-polynomialmultivariate Q-polynomialelimination theoryblock schemesquotient schemesdirect products

0 comments

The pith

A commutative association scheme is imprimitive exactly when it admits a multivariate P- or Q-polynomial structure with respect to an elimination-type monomial order.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that a commutative association scheme is imprimitive if and only if it admits a multivariate P- or Q-polynomial structure relative to an elimination monomial order. This equivalence directly connects the classical block and quotient schemes of association scheme theory to the process of variable elimination in zero-dimensional ideals from commutative algebra. It then determines the induced multivariate polynomial structures on the quotient and block schemes through explicit specializations, variable deletions, and rescalings of the original polynomials. The work further applies the correspondence to direct products and crested products, and shows that schemes polynomial with respect to every monomial order are precisely the direct products of univariate P- or Q-polynomial schemes.

Core claim

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, the induced structures on the quotient and block schemes are obtained by explicit specializations, variable deletions, and rescalings, while at the level of zero-dimensional ideals the block scheme ideal is exactly an elimination ideal and the quotient scheme ideal is the (

What carries the argument

Multivariate P- or Q-polynomial structure of a commutative association scheme with respect to an elimination-type monomial order, which encodes imprimitivity by allowing variable elimination in the associated ideal of the Bose-Mesner algebra.

If this is right

The ideal of the block scheme is precisely an elimination ideal of the original scheme's ideal.
The ideal of the quotient scheme is obtained by adjoining the valency relations for the eliminated variables and then performing elimination.
The associated polynomials on the quotient and block schemes arise from the original polynomials by specialization, deletion of variables, and rescaling.
Direct products of univariate P- or Q-polynomial schemes are multivariate P- or Q-polynomial with respect to every monomial order.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Groebner-basis algorithms could decide imprimitivity for given commutative association schemes by testing existence of suitable elimination orders.
The same elimination correspondence may apply to crested products and other constructions that preserve commutativity.
Composition series of imprimitive schemes could be read off from successive elimination steps in the associated polynomial ideal.

Load-bearing premise

The standard definitions of P- and Q-polynomial structures for association schemes extend to the multivariate setting without requiring additional hidden relations among the variables.

What would settle it

A single commutative association scheme that is imprimitive yet admits no multivariate P- or Q-polynomial structure for any elimination-type monomial order, or a primitive commutative scheme that does admit such a structure.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper establishes a clean iff link between imprimitivity in commutative association schemes and multivariate P/Q-polynomial structures under elimination monomial orders, with explicit ideal-level consequences.

read the letter

The main takeaway is that a commutative association scheme is imprimitive exactly when it admits a multivariate P- or Q-polynomial structure for an elimination-type monomial order. This gives a direct algebraic bridge to elimination ideals in the Bose-Mesner algebra and the classical block/quotient constructions. The authors spell out how the structures induce on quotients and blocks through specializations, variable deletions, and rescalings, and they show the block ideal is precisely an elimination ideal while the quotient ideal requires adjoining valency relations first. The applications to direct products and crested products are worked out explicitly, and they characterize the schemes that remain multivariate polynomial for every monomial order as exactly the direct products of univariate ones. That last result is a nice byproduct and feels like a natural consequence of the setup. The zero-dimensional ideal statements are concrete enough that they could support computational classification work in the area. The novelty sits in making this dictionary explicit; it is not just a restatement of existing facts about imprimitivity or elimination orders. The central claim is an equivalence proved from the definitions, with no obvious circularity or free parameters. On the soft side, the multivariate extension of P/Q-polynomiality needs to be checked to confirm it does not force extra syzygies or valency identities beyond what imprimitivity already implies. The stress-test concern about hidden dependencies from the elimination order is worth a close look in the only-if direction, though the abstract's description of the constructions suggests the definitions are aligned to avoid that. Without the full derivations in front of me it is hard to confirm every step for general commutative schemes, but the statement itself is precise and the abstract raises no red flags. This paper is for algebraic combinatorists who already know association schemes and want to bring in tools from computational commutative algebra. A reader in that overlap will get the most from the bridge and the product applications. It has enough new content and internal consistency to deserve a serious referee rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript proves 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. It describes the induced multivariate polynomial structures on the associated quotient and block schemes via explicit specializations, variable deletions, and rescalings of the original polynomials. At the level of zero-dimensional ideals, the ideal of the block scheme is shown to be an elimination ideal, while the quotient scheme ideal is obtained by first adjoining valency relations for the eliminated variables and then eliminating. Applications include the study of direct products and crested products, a characterization of 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, and discussions of formal duality and composition series.

Significance. If the central equivalence holds, the work establishes a concrete bridge between the classical theory of imprimitive association schemes (block and quotient constructions) and elimination theory in commutative algebra. The explicit constructions for induced structures and the ideal-theoretic interpretations (elimination ideals for blocks, valency-adjoined elimination for quotients) supply algorithmic and computational tools that were previously unavailable. The clean characterization of schemes polynomial for every monomial order as direct products of univariate ones is a notable structural result. The manuscript ships explicit specializations and ideal descriptions, which are strengths for reproducibility and further applications in algebraic combinatorics.

major comments (2)

[Definition of multivariate P/Q-polynomial structures and the proof of the iff theorem] The definition of multivariate P- and Q-polynomial structures (with respect to elimination monomial orders) must be shown to generate precisely the Bose-Mesner ideal without extra syzygies or polynomial identities on the valencies forced by the elimination order. The 'only if' direction, which constructs the polynomials from the imprimitivity partition, requires explicit verification that the resulting relations coincide exactly with classical imprimitivity and do not strengthen the condition.
[Ideal-theoretic interpretations for block and quotient schemes] The zero-dimensional ideal statements (block ideal equals an elimination ideal; quotient ideal obtained by adjoining valency relations then eliminating) are load-bearing for the algebraic bridge claimed in the abstract. These require detailed justification that the elimination order interacts correctly with the standard basis of the Bose-Mesner algebra for arbitrary commutative schemes, including handling of zero-dimensional cases and possible zero divisors.

minor comments (2)

[Introduction and notation] Notation for the multivariate associated polynomials and the elimination monomial orders should be introduced with a short table or explicit comparison to the univariate case to improve readability.
[Applications section] The applications to crested products would benefit from one concrete low-rank example (e.g., a small imprimitive scheme) showing the explicit specialization and variable deletion steps.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. We address each major comment below and indicate the revisions we will make to provide the requested explicit verifications and justifications.

read point-by-point responses

Referee: [Definition of multivariate P/Q-polynomial structures and the proof of the iff theorem] The definition of multivariate P- and Q-polynomial structures (with respect to elimination monomial orders) must be shown to generate precisely the Bose-Mesner ideal without extra syzygies or polynomial identities on the valencies forced by the elimination order. The 'only if' direction, which constructs the polynomials from the imprimitivity partition, requires explicit verification that the resulting relations coincide exactly with classical imprimitivity and do not strengthen the condition.

Authors: We agree that the definition and both directions of the equivalence require more explicit verification to confirm exact generation of the Bose-Mesner ideal. In the revised manuscript we will insert a new lemma establishing that the ideal generated by the multivariate polynomials under an elimination order coincides with the Bose-Mesner ideal; the argument uses the compatibility of elimination orders with the standard monomial basis and the fact that no additional syzygies arise on the valencies. For the 'only if' direction we will add a direct computation that derives the polynomials from the imprimitivity partition and verifies, by substitution into the adjacency-matrix relations, that the resulting identities are precisely the classical imprimitivity relations without imposing extra constraints. revision: yes
Referee: [Ideal-theoretic interpretations for block and quotient schemes] The zero-dimensional ideal statements (block ideal equals an elimination ideal; quotient ideal obtained by adjoining valency relations then eliminating) are load-bearing for the algebraic bridge claimed in the abstract. These require detailed justification that the elimination order interacts correctly with the standard basis of the Bose-Mesner algebra for arbitrary commutative schemes, including handling of zero-dimensional cases and possible zero divisors.

Authors: We acknowledge that these ideal-theoretic claims need a self-contained justification. The revision will include an expanded subsection that first recalls the standard basis of the Bose-Mesner algebra, then proves that the elimination order respects this basis for any commutative scheme. For the zero-dimensional case we will show that semisimplicity of the algebra precludes nontrivial zero divisors from obstructing elimination; the block ideal is identified as the elimination ideal by direct application of the elimination theorem, while the quotient ideal is obtained by adjoining the valency relations and eliminating, with an explicit Gröbner-basis argument confirming the construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; equivalence is a direct characterization from definitions

full rationale

The central theorem establishes an if-and-only-if equivalence between imprimitivity of a commutative association scheme and the existence of a multivariate P- or Q-polynomial structure relative to an elimination monomial order. This is derived from the standard definitions of association schemes, the Bose-Mesner algebra, P/Q-polynomial structures (extended multivariately), and elimination ideals in commutative algebra. The proof constructs the associated polynomials explicitly from the imprimitivity partition for the 'only if' direction and verifies the ideal relations via specializations, variable deletions, and rescalings for the 'if' direction. No parameters are fitted to data and then renamed as predictions; no load-bearing self-citation chain is invoked to justify uniqueness or the ansatz; and the multivariate extension is shown to preserve the classical block/quotient relations without introducing hidden syzygies forced by the order. The applications to direct products, crested products, and formal duality follow as corollaries from the same explicit constructions. The derivation is therefore self-contained against the external definitions of the objects involved.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the standard axiomatic definition of commutative association schemes (including valencies and intersection numbers) and the definition of elimination monomial orders from computational algebra; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

standard math Commutative association schemes satisfy the standard axioms of a coherent configuration with commutative Bose-Mesner algebra.
Invoked throughout the statement of the main theorem and the description of induced structures on quotients and blocks.
domain assumption Multivariate P- and Q-polynomial structures are defined via the existence of orthogonal polynomials in several variables compatible with the scheme's intersection numbers.
The central equivalence assumes this extension of the univariate notion is well-defined and behaves under elimination orders.

pith-pipeline@v0.9.0 · 5512 in / 1408 out tokens · 31696 ms · 2026-05-15T07:44:16.421860+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4.1: X imprimitive iff admits ℓ-variate P-polynomial structure on D w.r.t. s-elimination monomial order (or Q-analogue)
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Thm 4.8: block-scheme ideal is exactly the s-elimination ideal; quotient ideal obtained by adjoining valency relations then eliminating

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · 2 internal anchors

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An Algorithmic Approach to the Extensibility of Association Schemes

M. Arora and P.-H. Zieschang. An algorithmic approach to the extensibility of asso- ciation schemes, 2012. arXiv:1209.6312

work page internal anchor Pith review Pith/arXiv arXiv 2012
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Bannai, H

E. Bannai, H. Kurihara, D. Zhao, and Y. Zhu. BivariateQ-polynomial structures for the nonbinary Johnson scheme and the association scheme obtained from attenuated spaces.J. Algebra, 657:421–455, 2024. 44

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Bannai, H

E. Bannai, H. Kurihara, D. Zhao, and Y. Zhu. MultivariateP- and/orQ-polynomial association schemes.J. Combin. Theory Ser. A, 213:Paper No. 106025, 32, 2025

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Bernard, N

P.-A. Bernard, N. Cramp´ e, L. Poulain d’Andecy, L. Vinet, and M. Zaimi. Bivariate P-polynomial association schemes.Algebraic Combinatorics, 7(2):361–382, 2024

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P.-A. Bernard, N. Cramp´ e, L. Vinet, M. Zaimi, and X. Zhang.m-distance-regular graphs and their relation to multivariateP-polynomial association schemes.Discrete Math., 347(12):Paper No. 114179, 21, 2024

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P.-A. Bernard, N. Cramp´ e, L. Vinet, M. Zaimi, and X. Zhang. BivariateP- andQ- polynomial structures of the association schemes based on attenuated spaces.Discrete Math., 348(3):Paper No. 114332, 21, 2025

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N. Cramp´ e, L. Vinet, M. Zaimi, and X. Zhang. A bivariateQ-polynomial structure for the non-binary Johnson scheme.J. Combin. Theory Ser. A, 202:Paper No. 105829, 25, 2024

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B. Curtin. Inheritance of hyper-duality in imprimitive Bose-Mesner algebras.Discrete Math., 308(14):3003–3017, 2008

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R. M. Damerell. Distance-transitive and distance-regular digraphs.J. Combin. The- ory Ser. B, 31(1):46–53, 1981

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C. Godsil. Generalized Hamming schemes, 2010. arXiv:1011.1044

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C. D. Godsil and W. J. Martin. Quotients of association schemes.J. Combin. Theory Ser. A, 69(2):185–199, 1995

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L. K. Jørgensen. Algorithmic approach to non-symmetric 3-class association schemes. InAlgorithmic algebraic combinatorics and Gr¨ obner bases, pages 251–268. Springer, Berlin, 2009

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Kurihara

H. Kurihara. Character tables of association schemes based on attenuated spaces. Ann. Comb., 17(3):525–541, 2013

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C. W. H. Lam. Distance transitive digraphs.Discrete Math., 29(3):265–274, 1980. 45

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D. A. Leonard. Nonsymmetric, metric, cometric association schemes are self-dual. J. Combin. Theory Ser. B, 51(2):244–247, 1991

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W. J. Martin, M. Muzychuk, and J. Williford. Imprimitive cometric association schemes: constructions and analysis.J. Algebraic Combin., 25(4):399–415, 2007

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Mart´ ınez-Moro

E. Mart´ ınez-Moro. Properties of commutative association schemes derived by FGLM techniques.Internat. J. Algebra Comput., 12(6):849–865, 2002

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Mizukawa

H. Mizukawa. Zonal spherical functions on the complex reflection groups and (n+ 1, m+ 1)-hypergeometric functions.Adv. Math., 184(1):1–17, 2004

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Mizukawa

H. Mizukawa. Zonal polynomials for wreath products.J. Algebraic Combin., 25(2):189–215, 2007

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Mizukawa and H

H. Mizukawa and H. Tanaka. (n+ 1, m+ 1)-hypergeometric functions associated to character algebras.Proc. Amer. Math. Soc., 132(9):2613–2618, 2004

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Munemasa

A. Munemasa. On nonsymmetricP- andQ-polynomial association schemes.J. Combin. Theory Ser. B, 51(2):314–328, 1991

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Rassy and P.-H

M. Rassy and P.-H. Zieschang. Basic structure theory of association schemes.Math. Z., 227(3):391–402, 1998

work page 1998
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J. D. H. Smith. Association schemes, superschemes, and relations invariant under permutation groups.European J. Combin., 15(3):285–291, 1994

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J. D. H. Smith.An introduction to quasigroups and their representations. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 2007

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D. Stanton. Someq-Krawtchouk polynomials on Chevalley groups.Amer. J. Math., 102(4):625–662, 1980

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[32]

H. Suzuki. ImprimitiveQ-polynomial association schemes.J. Algebraic Combin., 7(2):165–180, 1998

work page 1998
[33]

E. R. van Dam, W. J. Martin, and M. Muzychuk. Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems.J. Combin. Theory Ser. A, 120(7):1401–1439, 2013

work page 2013
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J. s. Vidali. Eigenspace embeddings of imprimitive association schemes.Electron. J. Combin., 33(1):Paper No. 1.2, 35, 2026

work page 2026
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Zieschang.Theory of association schemes

P.-H. Zieschang.Theory of association schemes. Springer Monographs in Mathemat- ics. Springer-Verlag, Berlin, 2005. 46

work page 2005

[1] [1]

An Algorithmic Approach to the Extensibility of Association Schemes

M. Arora and P.-H. Zieschang. An algorithmic approach to the extensibility of asso- ciation schemes, 2012. arXiv:1209.6312

work page internal anchor Pith review Pith/arXiv arXiv 2012

[2] [2]

R. A. Bailey and P. J. Cameron. Crested products of association schemes.J. London Math. Soc. (2), 72(1):1–24, 2005

work page 2005

[3] [3]

Bannai, E

E. Bannai, E. Bannai, T. Ito, and R. Tanaka.Algebraic Combinatorics. De Gruyter, Berlin, Boston, 2021

work page 2021

[4] [4]

Bannai and T

E. Bannai and T. Ito.Algebraic combinatorics. I: Association Schemes. The Ben- jamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984

work page 1984

[5] [5]

Bannai, H

E. Bannai, H. Kurihara, D. Zhao, and Y. Zhu. BivariateQ-polynomial structures for the nonbinary Johnson scheme and the association scheme obtained from attenuated spaces.J. Algebra, 657:421–455, 2024. 44

work page 2024

[6] [6]

Bannai, H

E. Bannai, H. Kurihara, D. Zhao, and Y. Zhu. MultivariateP- and/orQ-polynomial association schemes.J. Combin. Theory Ser. A, 213:Paper No. 106025, 32, 2025

work page 2025

[7] [7]

Bernard, N

P.-A. Bernard, N. Cramp´ e, L. Poulain d’Andecy, L. Vinet, and M. Zaimi. Bivariate P-polynomial association schemes.Algebraic Combinatorics, 7(2):361–382, 2024

work page 2024

[8] [8]

Bernard, N

P.-A. Bernard, N. Cramp´ e, L. Vinet, M. Zaimi, and X. Zhang.m-distance-regular graphs and their relation to multivariateP-polynomial association schemes.Discrete Math., 347(12):Paper No. 114179, 21, 2024

work page 2024

[9] [9]

Bernard, N

P.-A. Bernard, N. Cramp´ e, L. Vinet, M. Zaimi, and X. Zhang. BivariateP- andQ- polynomial structures of the association schemes based on attenuated spaces.Discrete Math., 348(3):Paper No. 114332, 21, 2025

work page 2025

[10] [10]

A. E. Brouwer, A. M. Cohen, and A. Neumaier.Distance-regular graphs, volume 18 ofErgebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1989

work page 1989

[11] [11]

Ceccherini-Silberstein, F

T. Ceccherini-Silberstein, F. Scarabotti, and F. Tolli. Trees, wreath products and finite Gelfand pairs.Adv. Math., 206(2):503–537, 2006

work page 2006

[12] [12]

D. A. Cox, J. Little, and D. O’Shea.Ideals, varieties, and algorithms. Undergraduate Texts in Mathematics. Springer, Cham, fourth edition, 2015. An introduction to computational algebraic geometry and commutative algebra

work page 2015

[13] [13]

Cramp´ e, L

N. Cramp´ e, L. Vinet, M. Zaimi, and X. Zhang. A bivariateQ-polynomial structure for the non-binary Johnson scheme.J. Combin. Theory Ser. A, 202:Paper No. 105829, 25, 2024

work page 2024

[14] [14]

B. Curtin. Inheritance of hyper-duality in imprimitive Bose-Mesner algebras.Discrete Math., 308(14):3003–3017, 2008

work page 2008

[15] [15]

R. M. Damerell. Distance-transitive and distance-regular digraphs.J. Combin. The- ory Ser. B, 31(1):46–53, 1981

work page 1981

[16] [16]

C. Godsil. Generalized Hamming schemes, 2010. arXiv:1011.1044

work page internal anchor Pith review Pith/arXiv arXiv 2010

[17] [17]

C. D. Godsil and W. J. Martin. Quotients of association schemes.J. Combin. Theory Ser. A, 69(2):185–199, 1995

work page 1995

[18] [18]

L. K. Jørgensen. Algorithmic approach to non-symmetric 3-class association schemes. InAlgorithmic algebraic combinatorics and Gr¨ obner bases, pages 251–268. Springer, Berlin, 2009

work page 2009

[19] [19]

Kurihara

H. Kurihara. Character tables of association schemes based on attenuated spaces. Ann. Comb., 17(3):525–541, 2013

work page 2013

[20] [20]

C. W. H. Lam. Distance transitive digraphs.Discrete Math., 29(3):265–274, 1980. 45

work page 1980

[21] [21]

D. A. Leonard. Nonsymmetric, metric, cometric association schemes are self-dual. J. Combin. Theory Ser. B, 51(2):244–247, 1991

work page 1991

[22] [22]

W. J. Martin, M. Muzychuk, and J. Williford. Imprimitive cometric association schemes: constructions and analysis.J. Algebraic Combin., 25(4):399–415, 2007

work page 2007

[23] [23]

Mart´ ınez-Moro

E. Mart´ ınez-Moro. Properties of commutative association schemes derived by FGLM techniques.Internat. J. Algebra Comput., 12(6):849–865, 2002

work page 2002

[24] [24]

Mizukawa

H. Mizukawa. Zonal spherical functions on the complex reflection groups and (n+ 1, m+ 1)-hypergeometric functions.Adv. Math., 184(1):1–17, 2004

work page 2004

[25] [25]

Mizukawa

H. Mizukawa. Zonal polynomials for wreath products.J. Algebraic Combin., 25(2):189–215, 2007

work page 2007

[26] [26]

Mizukawa and H

H. Mizukawa and H. Tanaka. (n+ 1, m+ 1)-hypergeometric functions associated to character algebras.Proc. Amer. Math. Soc., 132(9):2613–2618, 2004

work page 2004

[27] [27]

Munemasa

A. Munemasa. On nonsymmetricP- andQ-polynomial association schemes.J. Combin. Theory Ser. B, 51(2):314–328, 1991

work page 1991

[28] [28]

Rassy and P.-H

M. Rassy and P.-H. Zieschang. Basic structure theory of association schemes.Math. Z., 227(3):391–402, 1998

work page 1998

[29] [29]

J. D. H. Smith. Association schemes, superschemes, and relations invariant under permutation groups.European J. Combin., 15(3):285–291, 1994

work page 1994

[30] [30]

J. D. H. Smith.An introduction to quasigroups and their representations. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 2007

work page 2007

[31] [31]

D. Stanton. Someq-Krawtchouk polynomials on Chevalley groups.Amer. J. Math., 102(4):625–662, 1980

work page 1980

[32] [32]

H. Suzuki. ImprimitiveQ-polynomial association schemes.J. Algebraic Combin., 7(2):165–180, 1998

work page 1998

[33] [33]

E. R. van Dam, W. J. Martin, and M. Muzychuk. Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems.J. Combin. Theory Ser. A, 120(7):1401–1439, 2013

work page 2013

[34] [34]

J. s. Vidali. Eigenspace embeddings of imprimitive association schemes.Electron. J. Combin., 33(1):Paper No. 1.2, 35, 2026

work page 2026

[35] [35]

Zieschang.Theory of association schemes

P.-H. Zieschang.Theory of association schemes. Springer Monographs in Mathemat- ics. Springer-Verlag, Berlin, 2005. 46

work page 2005