Algebraic Combinatorics (ALCO)

Eberhard, Sean

Muzychuk, Mikhail

ALGEBRAIC COMBINATORICSSean Eberhard & Mikhail MuzychukFusions of tensor powers of Johnson schemesVolume 6, issue 2 (2023), p. 533-538.https://doi.org/10.5802/alco.271© The author(s), 2023.This article is licensed under theCREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.http://creativecommons.org/licenses/by/4.0/Algebraic Combinatorics is published by The Combinatorics Consortiumand is a member of the Centre Mersenne for Open Scientific Publishingwww.tccpublishing.org www.centre-mersenne.orge-ISSN: 2589-5486Algebraic CombinatoricsVolume 6, issue 2 (2023), p. 533–538https://doi.org/10.5802/alco.271Fusions of tensor powers of Johnson schemesSean Eberhard & Mikhail MuzychukAbstract This paper is a follow-up to [5], in which the first author studied primitive associationschemes lying between a tensor power T dm of the trivial association scheme and the Hammingscheme H(d, m). A question which arose naturally in that study was whether all primitivefusions of T dm lie between Td/eme and H(d/e, me) for some e | d. This note answers this questionpositively provided that m is large enough. We similarly classify primitive fusions of the dthtensor power of a Johnson scheme on(mk)points when m is large enough in terms of k and d.1. IntroductionAssociation schemes are objects of central importance in algebraic combinatorics. Foran introduction to association schemes, the reader could refer to any of [4, 2, 13, 1, 5].All our association schemes are symmetric. If X,Y are association schemes on acommon vertex set then we write X ⩽ Y if X refines Y as a partition. A fusion of anassociation scheme is a coarsening which is again an association scheme. Notice thatsome authors, for example [6], use an opposite order on the set of association schemes.Our choice was motivated by keeping notation consistent with [1, 5]. Notice that withthis choice of ordering the inclusion X ⩽ Y implies a similar inclusion between theautomorphism groups Aut(X) ⩽ Aut(Y).We denote the Hamming scheme of order md and rank d + 1 by H(d, m) andthe Johnson scheme of order(mk)and rank k + 1 by J (m, k). The special caseH(1, m) ∼= J (m, 1) is the trivial scheme, denoted Tm. The dth tensor power of anassociation scheme X is denoted Xd. The symmetrized dth tensor power of the John-son scheme J (m, k) is called the Cameron scheme and denoted C(m, k, d).In this paper we classify primitive fusions of J (m, k)d assuming m is sufficientlylarge in terms of k and d.Theorem 1.1. For any positive integers k, d there exists a constant m0(k, d) suchthat any primitive fusion X of J (m, k)d with m ⩾ m0(k, d) belongs, up to permutingcoordinates, to one of the following intervals:(1) J (m, k)d ⩽ X ⩽ C(m, k, d),(2) T d/eMe ⩽ X ⩽ H(d/e, Me) for some integer e | d where M =(mk).Manuscript received 30th May 2022, revised 21st September 2022 and 27th September 2022, accepted27th September 2022.Keywords. association schemes, permutation groups.Acknowledgements. SE has received funding from the European Research Council (ERC), via theEU’s Horizon 2020 grant No. 803711. MM was supported by the Israeli Ministry of Absorption.ISSN: 2589-5486 http://algebraic-combinatorics.org/S. Eberhard & M. MuzychukThe special cases k = 1 and d = 1 are worth highlighting individually. In [9]it was shown that H(d, m) has no nontrivial fusions for m > 4. The case k = 1 ofTheorem 1.1 more generally classifies primitive fusions of T dm (for m sufficiently large).Corollary 1.2. Let X be a primitive fusion of T dm, where m ⩾ m0(1, d). Then, up topermuting coordinates, T d/eme ⩽ X ⩽ H(d/e, me) for some integer e | d.In [10] it was shown that J (m, k) has no nontrivial fusions for m ⩾ 3k + 4. Thisresult was improved to m ⩾ 3k − 1 in [11]. The case k = 1 of Theorem 1.1 recoversthis result, except for the precise lower bound.Corollary 1.3. Let X be a fusion of J (m, k), where m ⩾ m0(k, 1). Then eitherX = J (m, k) or X is trivial.Association schemes of the type appearing in the conclusion of Theorem 1.1 arestudied in [5], where they are called “Cameron sandwiches” and “Hamming sand-wiches,” respectively. The main result of [5] is that there are infinite families of non-schurian Hamming sandwiches.Remark 1.4. Imprimitive fusions of J (m, k)d are not so easily classified. Certainlyone must allow arbitrary tensor products of the cases appearing in Theorem 1.1, butthere are still many others. For example, the imprimitive wreath product Tm ≀ Tm(see [4, Section 3.4.1]) is an imprimitive fusion of T 2m not fitting this description.As an application we give an elementary classification of primitive groups contain-ing (A(k)m )d. Here A(k)m denotes the image of the alternating group Am in its permu-tation action on k-sets, and below S(k)m is defined similarly.The statement below is a special case of Cameron’s theorem [3, 7, 8], which moregenerally classifies all large primitive permutation groups. However, while the proofof Cameron’s theorem depends on the classification of finite simple groups, our proofdoes not.Corollary 1.5. Let G ⩽ Sn be a primitive permutation group containing (A(k)m )d,where n =(mk)d and m ⩾ m0(k, d). Then either(1) (A(k)m )d ⩽ G ⩽ S(k)m ≀ Sd or(2) (AMe)d/e ⩽ G ⩽ SMe ≀ Sd/e for some integer e | d where M =(mk).Proof. Let X be the orbital scheme of G. It follows from (A(k)m )d ⩽ G that X is a fusionof the orbital scheme of (A(k)m )d, which coincides with J (m, k)d. Thus J (m, k)d ⩽ Xand, by Theorem 1.1, either J (m, k)d ⩽ X ⩽ C(m, k, d) or T d/eMe ⩽ X ⩽ H(d/e, Me).In the first case, X has a constituent graph equal to the Cameron graph C(m, k, d). Inthe second case, X has a constituent graph equal to the Hamming graph H(d/e, Me)(which is a special case of a Cameron graph). Applying [12, Theorem 8.2.1], either theclaimed conclusion holds or G is small: |G| ⩽ exp(c(log n)3). Since |G| ⩾ |(A(k)m )d| =m!d and n =(mk)d⩽ mkd, we get d log m! ⩽ c(log n)3 ⩽ c(kd log m)3, in contradictionto the hypothesis m ⩾ m0(k, d). □Algebraic Combinatorics, Vol. 6 #2 (2023) 534Fusions of tensor powers of Johnson schemes2. NotationLet [0, k]d = {0, . . . , k}d. We use the following notation for vectors a, b, c ∈ [0, k]d:a ⩽ b ⇐⇒ ai ⩽ bi for all i (in this case we say that b dominates a),|a − b| = (|a1 − b1|, . . . , |ad − bd|),min(a, b) = (min(a1, b1), . . . , min(ad, bd)),max(a, b) = (max(a1, b1), . . . , max(ad, bd)),a! = a1! · · · ad!,(ab)=(a1b1)· · ·(adbd),wt(a) = a1 + · · · + ad,supp(a) = {i : ai > 0},[a] = {b ∈ [0, k]d : b ⩽ a},(x)d = (x, . . . , x),ei = (0, . . . , 0, 1, 0, . . . , 0).We understand(ab)to be zero unless (0)d ⩽ b ⩽ a. We call wt(a) the weight of a andsupp(a) the support of a.If p is a polynomial in one variable we write deg(p) for its degree and λ(p) for itsleading term.3. ProofThe structure constants of J (m, k) are given bypab,c(m) =∑i(k − ai)(ak − b − i)(ak − c − i)(m − k − ab + c + i − k)(as in [5, 6]). Here 0 ⩽ a, b, c ⩽ k, and i can be restricted to the rangemax(0, k − a − b, k − b − c, k − a − c) ⩽ i ⩽ min(k − a, k − b, k − c, m − a − b − c).In the following we always assume m ⩾ 3k. Under this assumption we have pab,c(m) > 0if and only if a, b, c satisfy the triangle inequalities [5, Lemma 4.1]. More precisely wehave the following.Lemma 3.1. We have pab,c(m) > 0 if and only if |b − c| ⩽ a ⩽ b + c. Assumingthat 0 ⩽ b − c ⩽ a ⩽ b + c, the leading term of pab,c(m) isλ(pab,c(m)) ={(ab)(ac) 1(b+c−a)! mb+c−a : a ⩾ b,(k−ak−b)(ab−c) 1c! mc : a ⩽ b.In particular, deg(pab,c(m)) ⩽ min(b, c), and equality holds if and only if a ⩽ max(b, c).Now consider J (m, k)d. For a, b, c ∈ [0, k]d, letpab,c(m) =d∏i=1paibi,ci(m).These are the structure constants of J (m, k)d. The following lemma generalizes theprevious one.Algebraic Combinatorics, Vol. 6 #2 (2023) 535S. Eberhard & M. MuzychukLemma 3.2. Let a, b, c ∈ [0, k]d. Then pab,c(m) > 0 if and only if |b − c| ⩽ a ⩽ b + cMoreover deg(pab,c(m)) ⩽ wt(min(b, c)), with equality if and only if a ⩽ max(b, c).If deg(pab,c(m)) = wt(b) = wt(c) then a ⩽ b = c and the leading term of pab,c(m) isλ(pab,c(m)) =((k)d − a(k)d − b)1b!mwt(b).Proof. It follows from pab,c(m) =∏di=1 paibici(m) that pab,c(m) > 0 iff each triple ai, bi, cisatisfies the triangle condition |bi − ci| ⩽ ai ⩽ bi + ci. In this casedeg(pab,c(m)) =d∑i=1deg(paibi,ci(m)) ⩽d∑i=1min(bi, ci) = wt(min(b, c)).Equality holds if and only if ai ⩽ max(bi, ci) for all i.If deg(pab,c(m)) = wt(b) = wt(c) then wt(min(b, c)) = wt(b) = wt(c), which impliesb = c, and moreover we have seen that we must have a ⩽ b. Multiplying the leadingterms of paibi,ci(m) given by the previous lemma, we get the claimed formula. □Let X be a fusion of J (m, k)d. Since X is a coarsening of J (m, k)d there is apartition S of [0, k]d such that X = {Rα : α ∈ S}, where(u, v) ∈ Rα ⇐⇒ (|u1 ∖ v1|, . . . , |ud ∖ vd|) ∈ α.In this situation we write X = J (m, k)S [5, Section 4]. For β, γ ∈ S and a ∈ [0, k]ddefinepaβ,γ(m) =∑b∈β,c∈γpab,c(m).For X = J (m, k)S to be an association scheme, S must satisfy two conditions:(1) {(0)d} ∈ S,(2) paβ,γ(m) = pa′β,γ(m) for all α, β, γ ∈ S and a, a′ ∈ α.We may denote the common value of paβ,γ(m) (a ∈ α) by pαβ,γ(m); these are thestructure constants of X.We call the sets α ∈ S the basic S-sets; their unions are called S-sets. Fornonempty S ⊆ [0, k]d let wt(S) = max{wt(b) | b ∈ S}. For α ∈ S let α∗ be theset of a ∈ α of maximal weight. Let Dα =⋃a∈α∗ [a]. Note that α∗ ⊆ Dα.For any α, β ⊆ [0, k]d and a ∈ [0, k]d the structure constant paα,β(m) is a real poly-nomial in m the coefficients of which depend on α, β and a. For every pair of distinctreal polynomials f, g ∈ R[x] there exists a real number cf,g ∈ R such that f(x) ̸= g(x)holds for all x > cf,g. Therefore, there exists a constant m0(k, d) such that, providedm ⩾ m0(k, d), the following condition holds for all α, β ⊆ [0, k]d and a, b ∈ [0, k]d:paα,β(m) = pbα,β(m) =⇒ paα,β(m) = pbα,β(m) as polynomials in m(1)=⇒ λ(paα,β(m)) = λ(pbα,β(m)).In what follows we assume that m ⩾ m0(k, d) and hence paβ,γ(m) = pa′β,γ(m) onlyif paβ,γ(m) and pa′β,γ(m) are equal as polynomials in m.Proposition 3.3. Let β ∈ S be a basic S-set.(1) Dβ is an S-set, and wt(Dβ ∖ β) < wt(β).(2) For every basic set α ∈ S there is a constant Nβα such that∑b:a⩽b∈β∗((k)d − a(k)d − b)= Nβα (a ∈ α).(3) Every element of β is dominated by a unique element of β∗.Algebraic Combinatorics, Vol. 6 #2 (2023) 536Fusions of tensor powers of Johnson schemes(4) Either wt(β) = wt(Dβ ∖ β) + 1 or β∗ ⊆ {0, k}d.Proof. Let w = wt(β) and D = Dβ . Then, since the polynomials pab,c(m) have positiveleading coefficient whenever they are nonzero, the degree of paβ,β(m) is, by the previouslemma,deg(paβ,β(m)) = maxb,c∈βdeg(pab,c(m)) ⩽ maxb,c∈βwt(min(b, c)) ⩽ w,with equality holding if and only if a ⩽ b = c ∈ β∗, i.e., if and only if a ∈ D.Since paβ,β(m) should depend only on the cell of S containing a, it follows that D isan S-set. Since β is basic, β ⊆ D, and since β ⊇ β∗ we have wt(D ∖ β) < wt(β).Moreover if a ∈ D then the leading term of paβ,β isλ(paβ,β(m)) =∑b:a⩽b∈β∗((k)d − a(k)d − b)1b!mw,and again this should depend only on the cell α containing a. Taking α = β anda ∈ β∗, it follows that a! is a constant for a ∈ β∗. Hence (2) holds (if α is notcontained in D then Nβα = 0).Next apply (2) with α = β. Taking a ∈ β∗ shows Nββ = 1, so (3) holds.Finally let b ∈ β∗ and suppose a = b − ei ⩾ 0. If a ∈ β then we get(k − bi + 1k − bi)= Nββ = 1,so bi = k. Hence either wt(b) = wt(β)+1 or bi ∈ {0, k} for all i, which implies (4). □We can define a partial ordering on S by saying α ⪯ β if every a ∈ α is dominatedby some b ∈ β. By part (3) of the proposition, this is equivalent to Nβα > 0. It isobvious that ⪯ is reflexive and transitive on S. To verify antisymmetry, note that ifα ⪯ β ⪯ α and a ∈ α∗ then there are b ∈ β and a′ ∈ α such that a ⩽ b ⩽ a′, whichby maximality of a implies a = b = a′ and hence α = β since S is a partition. Wesay α ∈ S is minimal if it is minimal in (S∖ {(0)d}, ⪯).Corollary 3.4. Let α ∈ S be a minimal basic set. Then Dα = α∪{(0)d}. Moreover,the elements of α∗ have disjoint equal-sized supports, and we either have wt(α) = 1or α∗ ⊆ {0, k}d.Proof. Apply the proposition. Note that if β is a basic subset of Dα∖α then β ≺ α. Byminimality of α this implies β = {(0)d}. Hence Dα = α ∪{(0)d}. Next, for any a ∈ α∗and i ∈ supp(a) we have ei ∈ Dα. By part (3) of Proposition 3.3, a is the uniqueelement of α∗ dominating ei. Therefore the elements of α∗ have disjoint supports. Bypart (4), either wt(α) = 1 or α∗ ⊆ {0, k}d. If wt(α) = 1 then all a ∈ α∗ have singletonsupport, and otherwise |supp(a)| = wt(a)/k = wt(α)/k for all a ∈ α∗. □Until now X could be imprimitive. Now we specialize to the primitive case to com-plete the proof of Theorem 1.1. Let α ∈ S be minimal. If X is primitive then Rαmust be connected, which implies that α∗ covers {1, . . . , d}. Hence α∗ is an equipar-tition of {1, . . . , d}. Let w = wt(α). If w = 1 then α∗ must be the set of elementsof weight 1, so Rα is the Cameron graph. Since the Weisfeiler–Leman stabilizationof the Cameron graph is the Cameron scheme, we find X ⩽ C(m, k, d). If w > 1then α∗ ⊆ {0, k}d. If the elements of α∗ have support size e then Rα is the Hamminggraph H(Me, d/e), so X ⩽ H(d/e, Me). To finish we must show T d/eMe ⩽ X. For this itsuffices to prove that for every β ∈ S, the elements of β∗ are sums of elements of α∗(and in particular β∗ ⊆ {0, k}d).We apply Proposition 3.3(2) to α and β. Let i ∈ {1, . . . , d}. Taking a = ei, we findthat Nβα is at least the number of b ∈ β∗ such that bi > 0. On the other hand, taking aAlgebraic Combinatorics, Vol. 6 #2 (2023) 537S. Eberhard & M. Muzychukto be the unique element of α∗ dominating ei, since a ∈ {0, k}d we find that Nβα isequal to the number of b ∈ β∗ such that a ⩽ b. Hence bi > 0 implies a ⩽ b. Thisimplies that b is the sum of those a ∈ α∗ such that a ⩽ b, as required.References[1] R. A. Bailey, Association schemes, Cambridge Studies in Advanced Mathematics, vol. 84, Cam-bridge University Press, Cambridge, 2004, Designed experiments, algebra and combinatorics.[2] Eiichi Bannai and Tatsuro Ito, Algebraic combinatorics. I: Association schemes, The Ben-jamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984.[3] Peter J. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc.13 (1981), no. 1, 1–22.[4] Gang Chen and Ilia Ponomarenko, Lectures on coherent configurations, online lecture notes,2019, http://www.pdmi.ras.ru/~inp/ccNOTES.pdf.[5] Sean Eberhard, Hamming sandwiches, preprint, 2022, https://arxiv.org/abs/2203.03687, toappear in Combinatorica.[6] L. A. Kalužnin and M. H. Klin, Certain maximal subgroups of symmetric and alternating groups,Mat. Sb. (N.S.) 87(129) (1972), 91–121.[7] Martin W. Liebeck, On minimal degrees and base sizes of primitive permutation groups, Arch.Math. (Basel) 43 (1984), no. 1, 11–15.[8] Attila Maróti, On the orders of primitive groups, J. Algebra 258 (2002), no. 2, 631–640.[9] M. E. Muzychuk, Subschemes of Hamming association schemes H(n, q), q ⩾ 4, in Interactionsbetween algebra and combinatorics (A. A. Ivanov and M. H. Klin, eds.), Springer, Dordrecht,1992, Acta Appl. Math. 29 (1992), no. 1-2, pp. 119–128.[10] , Subschemes of the Johnson scheme, European J. Combin. 13 (1992), no. 3, 187–193.[11] Daiyu Uchida, On the subschemes of the Johnson scheme, Mem. Fac. Sci. Kyushu Univ. Ser. A46 (1992), no. 1, 85–92.[12] John Wilmes, Structure, automorphisms, and isomorphisms of regular combinatorial objects,ProQuest LLC, Ann Arbor, MI, 2016, Ph.D. thesis – The University of Chicago.[13] Paul-Hermann Zieschang, Theory of association schemes, Springer Monographs in Mathematics,Springer-Verlag, Berlin, 2005.Sean Eberhard, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB(U.K.)E-mail : eberhard@maths.cam.ac.ukMikhail Muzychuk, Department of Mathematics, Ben Gurion University, P.O.B. 653, Beer Sheva8410501 (Israel)E-mail : muzychuk@bgu.ac.ilAlgebraic Combinatorics, Vol. 6 #2 (2023) 538

Fusions of tensor powers of Johnson schemes

Nakamura, Hiroaki

Demi-shuffle duals of Magnus polynomials in a free associative algebra

Abram, Antoine

Reutenauer, Christophe

Saliola, Franco V.

Quivers of stylic algebras

Ayyer, Arvind

Mandelshtam, Olya

Martin, James B

Modified Macdonald polynomials and the multispecies zero-range process: I

Fayers, Matthew

Comparing Fock spaces in types $A^{(1)}$ and $A^{(2)}$

Donten-Bury, Maria

Escobar, Laura

Portakal, Irem

Complexity of the usual torus action on Kazhdan–Lusztig varieties

DuBose, Sage

Vallières, Daniel

On $\mathbb{Z}_{\ell }^{d}$-towers of graphs

Green, R. M.

Xu, Tianyuan

Kazhdan–Lusztig cells of $\protect \mathbf{a}$-value 2 in $\protect \mathbf{a}(2)$-finite Coxeter systems

Tran, Tan N.

Tsujie, Shuhei

MAT-free graphic arrangements and a characterization of strongly chordal graphs by edge-labeling

Renteln, Paul

A natural idempotent in the descent algebra of a finite Coxeter group

Not a member yet