Norton algebras of the Hamming Graphs via linear characters

The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful in group theory due to its interesting automorphism group. We provide a formula for the Norton product on each eigenspace of a Hamming graph using linear characters. We construct a large subgroup of automorphisms of the Norton algebra of a Hamming graph and completely describe the automorphism group in some cases. We also show that the Norton product on each eigenspace of a Hamming graph is as nonassociative as possible, except for some special cases in which it is either associative or equally as nonassociative as the so-called double minus operation previously studied by the author, Mickey, and Xu. Our results restrict to the hypercubes and extend to the halved and/or folded cubes, the bilinear forms graphs, and more generally, all Cayley graphs of finite abelian groups.


Introduction
Distance regular graphs have many nice algebraic and combinatorial properties and have been extensively studied. For instance, (the adjacency matrix A of ) a distance regular graph Γ = (X, E) with vertex set X and edge set E has d + 1 distinct eigenvalues θ 0 > θ 1 > · · · > θ d and the corresponding eigenspaces V 0 , V 1 , . . . , V d form a direct sum decomposition of the vector space R X := { f : X → R} ∼ = R |X| , where d is the diameter of Γ. Furthermore, there is a general method to obtain the eigenvalues and eigenspaces of a distance regular graph; see, for example, Brouwer, Cohen and Neumaier [3, §4.1].
One can define an interesting product on each eigenspace V i of a distance regular graph Γ by doing the entry-wise product of two eigenvectors in V i and projecting the resulting vector back to V i . The gives an algebra, known as the Norton algebra, which is commutative but not necessarily associative. It was studied in group theory due to its interesting automorphism group [6,25].
When Γ belongs to certain important families of distance regular graphs (i.e., the Johnson graphs, Grassmann graphs, dual polar graphs, and hypercube graphs), Levstein, Maldonado and Penazzi [19,21] constructed the eigenspaces from a filtration of vector spaces corresponding to a graded lattice associated with Γ, and derived an explicit formula for the Norton product on the eigenspace of V 1 . Recently Terwilliger [27] obtained a more general formula for Q-polynomial distance-regular graphs. But for i ≥ 2 the Norton algebra structure on V i has not been determined.
In this paper we focus on the Hamming graph H(n, e), whose vertex set X consists of all words of length n on the alphabet {0, 1, . . . , e − 1} and whose edge set E consists of all unordered pairs of vertices differing in exactly one position. As an important family of distance regular graphs, the Hamming graphs are useful in multiple branches of mathematics and computer science. Their eigenvalues are well known [3, §9.2] and their eigenspaces have been investigated from various perspectives. For example, Valyuzhenich and Vorob'ev [29] studied the minimum cardinality of the support of an eigenvector of a Hamming graph, and for certain Hamming graphs with special parameters, Sander [23] constructed bases for their eigenspaces using vectors over {0, 1, −1}.
Key words and phrases. Hamming graph, halved and folded cube, bilinear forms graphs, Norton algebra, wreath product, nonassociativity. 1 If we allow extension of scalars to the complex field C, there is a nice complex eigenbasis of the Hamming graph H(n, e) consisting of all linear characters of its vertex set X = Z n e viewed as a group, and a real eigenbasis can be obtained by taking the real and imaginary parts of these characters. In fact, this is valid for any Cayley graph of a finite abelian group; see, for example, Lovász [13,Exercise 11.8] and DeVos-Goddyn-Mohar-Šámal [11]. It will be an interesting problem to determine whether an eigenbasis over Z (or even over {0, 1, −1}) can be obtained.
As an application of the linear character approach, we provide a formula for the Norton product on each eigenspace V i of the Hamming graph H(n, e), and use this formula to study the automorphism group Aut(V i ) of the Norton algebra V i . It is known that the automorphisms of the Hamming graph H(n, e) form a group isomorphic to the wreath product S e ≀ S n [3, Theorem 9.2.1]. We show that Aut(V 1 ) ∼ = S e ≀ S n by constructing all idempotents in V 1 , but Aut(V i ) could be much smaller or bigger than this group is i = 1. In general, we construct a large subgroup of Aut(V i ), which is related to the wreath product of the semidirect product Z e ⋊ Z × e of the group Z e and its multiplicative group Z × e with the symmetric group S n . A complete description of Aut(V i ) for i ≥ 2 will be a problem for future study.
We also determine the extent to which the Norton product on V i is nonassociative. For a given binary operation * on a set Z, let C * ,m be the number of distinct results obtained by inserting parentheses into the expression z 0 * z 1 * · · · * z m , where z 0 , z 1 , . . . , z m are Z-valued indeterminates. It is well known that C * ,m is bounded above by the ubiquitous Catalan number C m := 1 m+1 ( 2m m ). We have C * ,m = 1 for all m ≥ 0 if and only if * is associative, and we say * is totally nonassociative if C * ,m = C m for all m ≥ 0. In general, C * ,m measures how far the operation * is from being associative. The sequence (C * ,m ) m≥0 was called the associative spectrum of the binary operation * by Csákány and Waldhauser [7]. Braitt and Silberger [2] studied this sequence for a groupoid (G, * ) and called it the subassociativity type of (G, * ). Independently, Hein and the author [15] also proposed the study of C * ,m for a binary operation * . For further investigations of this nonassociativity measurement, see, e.g., Hein and the author [16] and Liebscher and Waldhauser [20].
We show that the Norton product ⋆ on each eigenspace V i is totally nonassociative except for some special cases, in which it is either associative for some trivial reasons or equally as nonassociative as the double minus operation ⊖ defined by a ⊖ b := −a − b for all a, b ∈ C, in the sense that any two ways to parenthesize z 0 ⊖ z 1 ⊖ · · · ⊖ z m produce distinct results if and only if so do the same two ways to parenthesize of z 0 ⋆ z 1 ⋆ · · · ⋆ z m . Therefore in the last case we have C ⋆,m = C ⊖,m given by the sequence A000975 [26] in OEIS [24], according to Csákány and Waldhauser [7] and work of the author, Mickey, and Xu [18].
Below is a summary of our results on the Norton algebra of the Hamming graph H(n, e). Theorem 1.1. For i = 0, 1, . . . , n, the (complex) Norton algebra V i of H(n, e) satisfies the following.
• It has a basis {χ u : u ∈ X i }, where X i is the set of elements in X = Z n e with exactly i nonzero entries, such that if u, v ∈ X i then • For e ≥ 3, its automorphism group is trivial if i = 0, is isomorphic to S e ≀ S n if i = 1 or S 3 ≀ S 2 n−1 if i = n and e = 3, and admits a subgroup isomorphic to (Z e ⋊ Z × e ) ≀ S n if i ≥ 1. • Its product ⋆ is associative if i = 0, equally as nonassociative as the double minus operation ⊖ if e = 3 and i ∈ {1, n}, or totally nonassociative if e = 3 and 1 < i < n or if e ≥ 4 and 1 ≤ i ≤ n. The nonassociativity measurement in Theorem 1.1 is very similar to what we obtained in previous work [17] on the Norton product on the eigenspace V 1 of the Johnson graphs, Grassmann graphs and dual polar graphs based on the formulas by Levstein, Maldonado and Penazzi [19,21]. The results on the automorphism group and nonassociativity in Theorem 1.1 do not include the case e = 2 as it is somewhat different from the case e ≥ 3. 2 In fact, the Hamming graph H(n, 2) is the well-known hypercube Q n , and the linear characters of its vertex set X = Z n 2 are all real (actually over {0, 1, −1}). Furthermore, as the hypercube Q n is both bipartite and antipodal, it can be halved and folded [3, §9.2.D]. The halved cube or half-cube 1 2 Q n can be obtained from the hypercube Q n by selecting vertices with an even number of ones and drawing edges between pairs of vertices differing in exactly two positions. The folded cube n can be obtained from the hypercube graph Q n by identifying each pair of vertices at distance n from each other. Applying both constructions above to the hypercube Q n gives the folded half-cube Since these graph are also Cayley graphs of finite abelian groups, we can study their Norton algebras using the same method as for the Hamming graphs, with the linear characters naturally indexed by certain sets and the Norton product of two linear characters χ S and χ T determined by the symmetric difference S△T := (S − T) ∪ (T − S) of the indexing sets S and T. The automorphism groups of their Norton algebras are related to the hyperoctahedral group S B n ∼ = Z 2 ≀ S n (the Coxeter group of type B n ) consisting of all bijections f on the set {±1, . . . , ±n} satisfying f (−j) = − f (j) for all j ∈ [n], its center {±1} consisting of the constant functions f = ±1, and its subgroup S D n (the Coxeter group of type D n ) consisting of all f ∈ S B n with |{j ∈ [n] : f (j) < 0}| even. Our results are summarized below, where V i (Γ) denotes the ith eigenspace and the corresponding Norton algebra of a distance regular graph Γ. Theorem 1.2. For i = 0, 1, . . . , n, the (real) Norton algebra V i (Q n ) satisfies the following.
• It has a basis {χ S : S ⊆ [n], |S| = i} such that for all S, T ⊆ [n] with |S| = |T| = i, • Its automorphism group is trivial if i = 0, equals the general linear group of the underlying vector space if i > ⌊2n/3⌋ or i is odd, and admits S B n /{±1} as a subgroup if 1 ≤ i < n is even. • Its product is associative if i = 0, i > ⌊2n/3⌋ or i is odd, but totally nonassociative otherwise. For i = 0, 1, . . . , ⌊n/2⌋, there is an algebra isomorphism V i ( n ) ∼ = V 2i (Q n ).
. • Its automorphism group admits a subgroup isomorphic to S D n if i and n are not both even and (i, n) = (1, 2), or S D n /{±1} if i and n are both even and (i, n) = (2, 4). • Its product ⋆ is associative if i = 0, i < ⌈n/3⌉ is odd, i and n − i are both odd, or n − i is odd and i > ⌊2n/3⌋, but totally nonassociative otherwise. For i = 0, 1, . . . , d = ⌊n/4⌋, there is an algebra isomorphism V i ( 1 2 n ) ∼ = V 2i ( 1 2 Q n ). Our method is also valid for the bilinear forms graph H q (d, e), whose vertex set X = Mat d,e (F q ) consists of all d × e matrices over a finite field F q and whose edge set E consists of unordered pairs xy of vertices x, y ∈ X with rank(x − y) = 1. This is a distance regular graph of diameter d (assuming d ≤ e) and can be viewed as a q-analogue of the Hamming graph H(d, e) [3, §9.5.A]. If d = 1 then H q (d, e) is a complete graph isomorphic to the Hamming graph H(1, q e ). In general, H q (d, e) is a Cayley graph of the finite abelian group X, so the linear character approach applies.
• It has a basis {χ u : u ∈ X, rank(u) = i} such that • Its automorphism group admits subgroups isomorphic to This paper is structured as follows. In Section 2 we review the linear character eigenbasis for any Cayley graph of a finite abelian group and use this basis to establish a formula for the Norton product on each eigenspace. Then we carefully examine the Norton algebras of the Hamming graphs in Section 3, investigate their automorphisms groups in Section 4, and measure their nonassociativity in Section 5. We extend our results to the halved and/or folded cubes in Section 6 and to the bilinear forms graphs in Section 7. We conclude this paper with some remarks and questions in Section 8.

Cayley graphs of finite abelian groups
In this section we study the Norton algebras of a Cayley graph of a finite abelian group using the linear characters of the group. First recall that a distance regular graph is a graph with distance d(−, −) such that the number of vertices z with d(x, z) = i and d(y, z) = j depends only on i, j, and k = d(x, y), but not on the choices of the vertices x and y. Such a graph satisfies many nice properties and has been an important topic in algebraic combinatorics; see, for example, Brouwer-Cohen-A. Neumaier [3] and van Dam-Koolen-Tanaka [8].
For some important families of distance regular graphs (the Johnson graphs, Grassmann graphs, dual polar graphs, and hypercube graphs), Levstein, Maldonado and Penazzi [19,21] used a graded lattice to construct a filtration of vector spaces and obtained a formula for the Norton product on the eigenspace of the second largest eigenvalue. Based on this formula we [17] studied the nonassociativity of the Norton product, with the result on the hypercube graphs extended to the Hamming graphs. For the other eigenspaces of these graphs and for the eigenspaces of other distance regular graphs, the Norton algebra structure has not been determined yet. Now we use linear characters to give a complete description of all Norton algebras of a distance regular graph if it is also a Cayley graph of a finite abelian group, with the ground field R extended to the complex field C. To this end, we review some basic properties of linear characters.
Let G be a group and let R × denote the multiplicative group of all units in a ring R. A linear character of G is a group homomorphism χ : G → C × . The linear characters of G form an abelian group G * under the entry-wise product defined by (χ · χ ′ )(g) := χ(g)χ ′ (g) for all χ, χ ′ ∈ G * and g ∈ G.
The following result is well known and we include a short proof here for completeness. Theorem 2.1. The group G * of all linear characters of a finite abelian group G is isomorphic to G and is an orthonormal basis for the space C G := {φ : G → C} ∼ = C |G| endowed with the inner product Proof. For any χ ∈ G * and any g ∈ G with order n, we have χ(g) n = 1 which implies |χ(g)| = 1 and χ(g) = χ(g) −1 . Thus a finite cyclic group Z e has e distinct linear characters given by χ a (b) := ω ab for all a, b ∈ Z e , where ω := exp(2πi/e) is a primitive eth root of unity. Moreover, if G 1 and G 2 are finite abelian groups then χ is a linear character of G 1 × G 2 if and only if χ = χ 1 · χ 2 for some linear characters χ 1 ∈ G * 1 and χ 2 ∈ G * 2 . Therefore G * ∼ = G for any finite abelian group G; in particular, |G * | = |G| = dim(C G ). If χ ∈ G * then χ, χ = 1 since χ(g)χ(g) = |χ(g)| = 1 for all g ∈ G. For distinct χ, χ ′ ∈ G * , we have χ(h) = χ ′ (h) for some h ∈ G and thus which implies that χ, χ ′ = 0. Therefore G * is an orthonormal basis for C G .
Let G be a finite abelian group expressed additively, and let S be a subset of G − {0} such that s ∈ S ⇒ −s ∈ S. The Cayley graph Γ = Γ(G, S) of G with respect to S has vertex set X = G and edge set E = {xy : y − x ∈ S}. The eigenvalues and eigenvectors of Γ are those of its adjacency matrix A = [a xy ] x,y∈X , where a xy is one if xy ∈ E or zero otherwise. An eigenbasis of Γ is a basis for the vector space C X consisting of eigenvectors of Γ. An explicit construction of such a basis is well known in the abelian case [13,Exercise 11.8]; for the nonabelian case, see, for example, Babai [1], Brouwer-Haemers [4, Proposition 6.3.1], and Lovász [14]. Proof. By Theorem 2.1, the linear characters of X form a basis for the space C X . Each linear character χ of X is an eigenvector of Γ corresponding to the eigenvalue χ(S) since Since any Cayley graph Γ(X, S) of a finite abelian group X has an eigenbasis X * consisting of all linear characters of X, we can define the Norton product χ ⋆ χ ′ of two linear characters χ and χ ′ in the same eigenspace of Γ by projecting the entry-wise product χ · χ ′ back to this eigenspace. We provide a formula for this product below. Theorem 2.3. For any Cayley graph Γ(X, S) of a finite abelian group X, if two linear characters χ and χ ′ of X correspond to the same eigenvalue χ(S) = χ ′ (S) then Proof. Given linear characters χ and χ ′ of X with χ(S) = χ ′ (S), the entry-wise product χ · χ ′ is still a linear character of X with the corresponding eigenvalue (χ · χ ′ )(S). The projection onto the eigenspace containing χ and χ ′ fixes χ · χ ′ if (χ · χ ′ )(S) = χ(S) or annihilates it otherwise.
In the remainder of this paper we elaborate the aforementioned linear character approach to the Norton algebras in the context of the Hamming graphs, halved/folded cubes, and bilinear forms graphs, as these graphs are simultaneously distance regular graphs and Cayley graphs of finite abelian groups. Even though their eigenspaces can be realized over R, we allow extension of scalars to the complex field C so that we can apply Theorem 2.2 and Theorem 2.3 to obtain linear character bases and explicit formulas of the Norton product for all eigenspaces. When there are multiple graphs involved, we let V i (Γ) denote the ith eigenspace of Γ and the corresponding Norton algebra.

Hamming graphs
In this section we study the Norton algebras of the Hamming graphs. Given integers n ≥ 1 and e ≥ 2, the finite abelian group X = Z n e consists of all words of length n on the alphabet Z e = {0, 1, . . . , e − 1} with addition performed entry-wise modulo e. We can write an element x ∈ X as either a word x = x 1 · · · x n or a function x : The Hamming graph H(n, e) is the Cayley graph Γ(Z n e , X 1 ), which is a distance regular graph of diameter d = n. For i = 0, 1, . . . , n, the ith eigenvalue of H(n, e) and its multiplicity are [3, §9.2] Maldonado and Penazzi [21] showed that the Norton product on the eigenspace V 1 of the Hamming graph H(n, 2) is constantly zero. For e ≥ 3, we showed in previous work [17] that the Norton algebra V 1 (H(1, e)) has a spanning set {v 1 , . . . ,v e } such that and the direct product of n copies this algebra is isomorphic to the Norton algebra V 1 (H(n, e)). Now using linear characters we can determine all Norton algebras of H(n, e).

Basis and Norton product.
For every u ∈ X we define a linear character χ u of X by Here ω := exp(2πi/e) ∈ C is a primitive eth root of unity, which satisfies the following: The proof of the above identity is an easy exercise. We are ready to provide our first main result on the Hamming graphs.
Proof. By Theorem 2.1 and its proof, {χ u : u ∈ X} is a complete set of distinct linear characters of the abelian group X = Z n e and thus a basis for C X . For each u ∈ X, the linear character χ u is an eigenvector of H(n, e) corresponding to the eigenvalue χ u (X 1 ) by Theorem 2.2.
To compute the eigenvalue χ u (X 1 ), suppose u ∈ X i and let x ∈ X 1 . Then we have x(j) = 0 for a unique j ∈ [n] and thus χ u (x) = ω u(j)x(j) . If j / ∈ supp(u) then u(j) = 0 and χ u (x) = 1. If j ∈ supp(u) then ω u(j) is a nontrivial eth root of unity and the sum of χ u (x) = ω u(j)x(j) as x(j) runs through {1, 2, . . . , e − 1} with j fixed is −1, thanks to Equation (3). Thus Example 3.2. The complex eigenbasis for the Hamming graph H(2, 3) given in Theorem 3.1 consists of the rows of the matrix below, which are indexed by the vertices (written as words).
X 00 10 01 20 11 02 21 12 22 1 ω 2 The first row spans V 0 with χ 00 ⋆ χ 00 = χ 00 , the next four rows span V 1 , and the last four rows span V 2 . The Norton algebras V 1 and V 2 are isomorphic to each other by the following charts (a coincidence between Corollary 3.5 and Proposition 3.7).
⋆ χ 01 χ 02 χ 10 χ 20 χ 01 χ 02 0 0 0 χ 02 0 χ 01 0 0 Although the basis given in Theorem 3.1 consists of complex vectors, we can obtain a real basis by taking real and imaginary parts of the vectors. Let X 0 i := {u ∈ X i : 2u = 0} and let X + i and X − i be the sets of all u ∈ X i − X 0 i with u(j) > e/2 or u(j) < e/2, respectively, where j is the smallest integer such that u(j) = e/2. For each u ∈ X, define ξ u : X → R by for all x ∈ X. One sees that for i = 0, 1, . . . , d = n, the eigenspace V i of H(n, e) has a real basis {ξ u : u ∈ X i }. Below is an example for the Hamming graph H(2, 3), which can be further normalized to an eigenbasis over Z.
X 00 10 01 20 11 02 21 12 22 Recently, there has been work on the minimum cardinality of the support of eigenfunctions of a Hamming graph; see for example, Valyuzhenich and Vorob'ev [29]. The linear characters may provide another possible approach to such problems.
3.2. Algebra structure. Theorem 3.1 leads to the following result on the structure of the Norton algebras of H(n, e).

Corollary 3.4.
For i = 0, 1, . . . , n, we have a direct sum decomposition of vector spaces ) as an algebra. This direct sum becomes a direct product of algebras if i = 0, 1, n or if e = 2 and either i > ⌊2n/3⌋ or i is odd.
Proof. Using the basis provided in Theorem 3.1, one sees that the eigenspace V i (H(n, e)) is the direct sum of subspaces V S for all i-sets S ⊆ [n]. By the formula for the Norton product on V i (H(n, e)) in Theorem 3.1, each direct summand V S is a subalgebra isomorphic to V i (H(i, e)) by sending χ u to χū for all u ∈ X i with supp(u) = S, whereū is obtained from u by deleting all zero entries. Finally, the above direct sum becomes a direct product of algebras in the following cases.
• If i = 0 or i = n then this direct sum has only one summand.
13, which will be provided later.
The above corollary includes a previous result as a special case.

Corollary 3.5 ([17]
). The Norton algebra V 1 (H(n, e)) is isomorphic to the direct product of n copies of the Norton algebra V 1 (H(1, e)).
We have another similar result on the Norton algebra V n (H(n, 3)).
Lemma 3.6. For any u ∈ X i with supp(u) = S and u(S) ⊆ Z × e , the span V u of {χ ku : k ∈ [e − 1]} is a subalgebra of the subalgebra V S of V i (H(n, e)) satisfying V u ∼ = V 1 (H(1, e)).
Proof. We have supp(ku) = S for all k ∈ [e − 1] since u(S) ⊆ Z × e . Thus V u is a subalgebra of V S and is isomorphic to the Norton algebra V 1 (H(1, e)) via χ ku → χ k . Proposition 3.7. The Norton algebra V n (H(n, 3)) is isomorphic to the direct product of 2 n−1 copies of the Norton algebra V 1 (H(1, 3)).
Proof. By Lemma 3.6, there is a subalgebra V u ∼ = V 1 (H(1, 3)) spanned by χ u and χ 2u for all u ∈ X n . Let u and v be distinct elements in X n . We have {u(j), v(j)} = {1, 2} for some j ∈ [n] and thus 3)) is isomorphic to the direct product of 2 n−1 copies of V 1 (H(1, 3)).
We also observe that there is no identity for the Norton product ⋆ on V i (H(n, e)) unless i = 0. Proof. Any element of V i (H(n, e)) can be written as ∑ u∈X i c u χ u . For any v ∈ X i we have if and only if 0 ∈ X i and c 0 = 1. The result follows immediately. 8 3.3. Hypercube. Finally, we focus on the case e = 2. In this case the Hamming graph H(n, 2) is known as the hypercube Q n . The basis given in Theorem 3.1 for each eigenspace of Q n is real and actually over {0, 1, −1} since ω = −1 when e = 2. Moreover, each element u ∈ X i is uniquely determined by its support S = supp(u), which is an i-subset of [n], and the linear character χ u = χ S is given by Using the symmetric difference S△T := (S − T) ∪ (T − S) of two sets S and T we can rephrase Theorem 3.1 for the hypercube Q n below.
Proof. This follows immediately from Theorem 3.1 with e = 2.
This agrees with the cross product on the three-dimensional space, except for the anticommutativity.
Proof. By Lemma 3.12, there exist i-sets S, T ⊆ [n] such that |S△T| = i if and only if i ≤ ⌊2n/3⌋ and i is even. Thus if i > ⌊2n/3⌋ or i is odd then χ S ⋆ χ T = 0 for all i-sets S, T ⊆ [n].

Automorphisms
Recall that an automorphism of an algebra (V, ⋆) is an automorphism φ of the underlying vector Note that we do not assume an algebra is associative or unital. In this section we study the automorphisms of the Norton algebras of the Hamming graph H(n, e). We begin with some special cases. Proof. The Norton algebra V i (H(n, e)) is spanned by an idempotent χ 0 if i = 0, or has a zero product if e = 2 and either i > ⌊2n/3⌋ or i is odd by Proposition 3.13. The result follows.
To deal with the remaining cases, we review some basic concepts from group theory. If there is a homomorphism φ : H → Aut(N) from a group H to the automorphism group of a group N, then the semidirect product N ⋊ H is the Cartesian product N × H endowed with the operation (n 1 , h 1 )(n 2 , h 2 ) = (n 1 φ(h 1 )(n 2 ), h 1 h 2 ) for all n 1 , n 2 ∈ N and h 1 , h 2 ∈ H. In particular, the 9 semidirect product Z e ⋊ Z × e of the group Z e and its multiplicative group Z × e is the Cartesian product Z e × Z × e equipped with an operation defined by (a, b)(a ′ , b ′ ) := (a + b −1 a ′ , bb ′ ) for all a, a ′ ∈ Z e and b, b ′ ∈ Z × e . The identity element of this group is (0, 1), where 0 and 1 are the identity elements of Z e and Z × e , respectively. Next, the symmetric group S n consists of all permutations on the set [n], and we let id denote its identity. The wreath product G ≀ S n of a group G with S n is the semidirect product G n ⋊ S n , where G n is the direct product of n copies of G and S n permutes these n copies. More precisely, the elements of G ≀ S n are of the form (g, σ) with g ∈ G n and σ ∈ S n , and with h j denoting the jth component of an n-tuple h ∈ G n , the operation of G ≀ S n is defined by (g, σ) ). The automorphism group of the Hamming graph H(n, e) is isomorphic to the wreath product S e ≀ S n [3, Theorem 9.2.1], which acts on X = Z n e in a natural way. However, not all automorphisms of H(n, e) induce automorphisms of the Norton algebra V i (H(n, e)) nor can they give all automorphisms of V i (H(n, e)). This can be seen in our next result, which involves the group (Z e × Z × e ) ⋊ S n . The elements in this group are of the form (a, b, σ) with a ∈ Z n e , b ∈ (Z × e ) n , and σ ∈ S n . If (a, b, σ) and (a ′ , b ′ , σ ′ ) are in this group then Here the dot "·" denotes the entry-wise product on Z n e , which makes Z n e become a ring with (Z × e ) n as its group of units, and σσ ′ is the composition of permutations. Theorem 4.2. Each element φ = (a, b, σ) ∈ (Z e ⋊ Z × e ) ≀ S n induces an automorphism of the Norton algebra V i (H(n, e)) by sending χ u to χ a (b · σ(u))χ b·σ(u) for all u ∈ X i . Such automorphisms form a group isomorphic to (Z e ⋊ Z × e ) ≀ S n if e ≥ 3 and i ≥ 1 or if e = 2 and 1 ≤ i < n is odd, but isomorphic to (Z 2 ≀ S n )/{(0, 1, id), (1, 1, id)} if e = 2 and 1 ≤ i < n is even.
Proof. Let φ = (a, b, σ) ∈ (Z e ⋊ Z × e ) ≀ S n . Then u and b · σ(u) have the same support for any Therefore φ induces an automorphism of the Norton algebra V i (H(n, e)).
On the other hand, we have Then we obtain φ(φ ′ (χ u )) = (φφ ′ )(u) as Equation (2) implies It follows that we have a homomorphism from (Z e ⋊ Z × e ) ≀ S n to the automorphism group of V i (H(n, e)), and we need to show that its kernel is trivial if e ≥ 3 and i ≥ 1 or if e = 2 and 1 ≤ i < n is odd, or equals {(0, 1, id), (1, 1, id)} if e = 2 and 1 ≤ i < n is even. To this end, suppose that φ(χ u ) = χ u , i.e., b · σ(u) = u and χ a (b · σ(u)) = χ a (u) = 1 for all u ∈ X i . If 1 ≤ i < n then S n acts faithfully on i-subsets of [n] and thus taking the support of both sides of the equality b · σ(u) = u gives σ = id. If i = n and e ≥ 3 then we also have σ = id since if σ(j) = k = j for some j ∈ [n] then b · σ(u) = u for some u ∈ X i with u(j) = 1 and u(k) = b k .
Next, if σ = id then b · σ(u) = b · u = u for all u ∈ X i and thus b = 1. Finally, we consider the condition that χ a (u) = 1 for all u ∈ X i .
• If e ≥ 3 then a = 0 as we can obtain v ∈ X i from u by changing the jth entry to a different nonzero number and then χ a (u) = χ a (v) implies a(j) = 0 for any j ∈ [n]. • Assume e = 2 and 1 ≤ i < n. For any distinct j, k ∈ [n], there exists u ∈ X i such that u(j) = 1 and u(k) = 0. We obtain v ∈ X i from u by switching the jth and kth entries. Then χ a (u) = χ a (v) implies a(j) = a(k). Thus a is either 0 or 1, and the latter is possible if and only if i is even in order to have χ 1 (u) = 1 for all u ∈ X i .
When e = 2 the group in Theorem 4.2 becomes Z 2 ≀ S n . This group is often interpreted as the hyperoctahedral group S B n , which consists of bijections f on {±1, . . . , ±n} with f (−j) = − f (j) for all j ∈ [n]. The group S B n is the Coxeter group of type B n and its elements are called signed permutations of [n] as they are determined by their values on [n]. We can write an element in S B n as f = (σ, ǫ) for unique σ ∈ S n and ǫ : [n] → {±1} such that f (j) = ǫ(σ(j))σ(j) = (ǫσ)(j)σ(j) for all j ∈ [n], where ǫσ is the composition of ǫ and σ.
, where the dot "·" is the entry-wise product, since Comparing this with the operation in the group Z 2 ≀ S n as a special case (b = b ′ = 1) of Equation (4), we have a group isomorphism Z 2 ≀ S n ∼ = S B n by sending a ∈ Z n 2 to ǫ :  . Such automorphisms form a group isomorphic to S B n if 1 ≤ i < n is odd or S B n /{±1} if 1 ≤ i < n is even. In general, the automorphism group of the Norton algebra V i (H(n, e)) does not equal the subgroup given in Theorem 4.2, as shown in Example 4.4 below. To better understand it, we try to construct all of the idempotents in V i (H(n, e)), as any algebra automorphism must permute the idempotents. Here an idempotent is an element η satisfying η 2 = η ⋆ η = η. We provide some small examples below.
Remark 4.6. In our previous work [17] we found a real basis {v 1 , . . . ,v e } for the Norton algebra V 1 (H(1, e)); see also Equation (1). By Equation (3) and Equation (6), for all j, k ∈ Z e we have Comparing this with our previous work [17] we have η j =v e−j ∈ R X for all j ∈ Z e . Furthermore, a similar argument as in the proof of Proposition 4.5 shows that η ∈ V 1 (H(1, e)) is a nilpotent element of order 2, i.e., η = 0 and η 2 = 0, if and only if e = c(η j 1 + · · · + η j ℓ ) with ℓ = e/2 for some nonzero scalar c ∈ C.
The first case occurs by our hypothesis on η and η ′ . Assume that the second case also occurs. Since the coefficients in these two cases are nonzero, they must coincide, i.e., ℓ = ℓ ′ by Proposition 4.5. Then supp(η) ∩ supp(η ′ ) = ∅ since the coefficient in the third case is nonzero but distinct from the coefficient in the first two cases. It follows that the support of the idempotent c(η + η ′ ) has cardinality 4ℓ and Proposition 4.5 implies c(e−2) e−2ℓ = e−2 e−4ℓ , i.e., c = e−2ℓ e−4ℓ . Now assume that the second case does not occur. Then the third case must occur since supp(η ′ ) = ∅. As the coefficient in the third case differs from the first, we have e−2 e−2ℓ + e−2 e−2ℓ ′ = 0, i.e., ℓ ′ = e − ℓ by Proposition 4.5. Then the support of c(η + η ′ ) has ℓ − ℓ ′ = 2ℓ − e elements and thus c(e−2) e−2ℓ = e−2 e−2(2ℓ−e) , i.e., c = e−2ℓ 3e−4ℓ .
Proposition 4.8. For e ≥ 3, the automorphism group of the Norton algebra V 1 (H(1, e)) is isomorphic to the symmetric group S e .
Proof. By Corollary 3.4, the Norton algebra V 1 (H(n, e)) is isomorphic to the direct product of its subalgebras V S for all 1-sets S = {k} ⊆ [n], i.e., V 1 (H(n, e)) ∼ = V {1} × · · · × V {n} . Thus any idempotent in V 1 (H(n, e)) can be expressed as η = η (1) + · · · + η (n) with η (j) being an idempotent in V {j} for all j ∈ [n]. By Proposition 4.7, the idempotent η is primitive if and only if all but one of η (1) , . . . , η (n) are zero. Moreover, two primitive idempotents in V 1 (H(n, e)) are orthogonal if and only if they are from V {j} and V {k} for some distinct j, k ∈ [n] by Proposition 4.7. Thus any automorphism φ of the Norton algebra V 1 (H(n, e)) must permute the subalgebras V {1} , . . . , V {n} . It follows from this and Proposition 4.8 that the automorphism group of V 1 (H(n, e)) is the wreath product S e ≀ S n . The result on V n (H(n, 3)) is similar, thanks to Proposition 3.7.
Remark 4.11. The above theorem shows that the automorphism group of a Norton algebra may agree with the automorphism group of the underlying graph in some cases but could be much larger in some other cases. It is also possible to have the former smaller than the latter. In fact, one can check that the nonzero idempotents in the Norton algebra V 2 (Q 3 ) (see Example 3.11) are χ 12 + χ 13 + χ 23 , χ 12 − χ 13 − χ 23 , −χ 12 + χ 13 − χ 23 , and −χ 12 − χ 13 + χ 23 . Thus its automorphism group is a subgroup of S 4 , and it actually equals S 4 since it has a subgroup isomorphic to S B 3 /{±1} by Corollary 4.3, whose order is 2 3 · 3!/2 = 24 = |S 4 |. This is smaller than the automorphism group S 2 ≀ S 3 of the graph Q 3 , whose order is 2 3 · 3! = 48. It would be nice to generalize this example to the Norton algebra V i (H(n, e)) with i ≥ 2.

Nonassociativity
In this section we study the nonassociativity of the Norton product ⋆ on each eigenspace V i the Hamming graph H(n, e) based on the results in Section 3.
Given a binary operation * defined on a set Z, define C * ,m to be the number of distinct results that one can obtain from the expression z 0 * z 1 * · · · * z m by inserting parentheses in all possible ways, where z 0 , z 1 , . . . , z m are indeterminates taking values from Z. We have C * ,m ≥ 1 and the equality holds for all m ≥ 0 if and only if * is associative. On the other hand, it is well known that the number of ways to insert parentheses into the expression z 0 * z 1 * · · · * z m is the ubiquitous Catalan number C m := 1 m+1 ( 2m m ), giving an upper bound for C * ,m . If C * ,m = C m for all m ≥ 0 then * is said to be totally nonassociative. In general, the number C * ,m is between 1 and C m and can be viewed as a quantitative measure for how far the operation * is from being associative or totally nonassociative [7,15] There is a natural bijection between the ways to insert parentheses into z 0 * z 1 * · · · * z m and binary trees with m + 1 leaves. Here a binary tree is a rooted plane tree in which every node 14 has exactly two children except the leaves. Let T m be the set of all binary trees with m + 1 leaves. Any t ∈ T m naturally corresponds to a parenthesization of z 0 * z 1 * · · · * z m , and we let (z 0 * z 1 * · · · * z m ) t to denote the result. With the leaves of t labeled 0, 1, . . . , m from left to right, the depth sequence of t is d(t) := (d 0 (t), d 1 (t), . . . , d m (t)), where the depth d j (t) of a leaf j in T is the number of steps in the unique path from the root of t to the leaf j. See Figure 1 for some examples.

Figure 1. Parenthesizations, binary trees, and leaf depths
Two binary trees s, t ∈ T m are * -equvalent if (z 0 * z 1 * · · · * z m ) s = (z 0 * z 1 * · · · * z m ) t . The number of equivalence classes in T m is exactly the nonassociativity measure C * ,m . Two binary operations are said to be equally nonassociative if their corresponding equivalence relations agree on T m for all m ≥ 0.
Previously, the author, Mickey, and Xu [18] defined the double minus operation a ⊖ b := −a − b for all a, b ∈ C and discovered that the nonassociativity measure C ⊖,m agrees with an interesting sequence A000975 [26] in OEIS [24]; see also Csákány and Waldhauser [7].
Theorem 5.1 ([18]). For any s, t ∈ T m , we have (z 0 ⊖ z 1 ⊖ · · · ⊖ z m ) s = (z 0 ⊖ z 1 ⊖ · · · ⊖ z m ) t if and only if d(s) ≡ d(t) (mod 2), i.e., d j (s) ≡ d j (t) (mod 2) for all j = 0, 1, . . . , m. Moreover, the nonassociativity measure C ⊖,m is given by the the sequence A000975 in OEIS except for m = 0. Now we turn to the Norton algebras of the Hamming graph H(n, e), which are related to the double minus operation in certain cases. The Norton algebra V 0 (H(n, e)) is one dimensional and must be associative. In recent work [17] we obtained the following result on the nonassociativity of the Norton algebra V 1 (H(n, e)).
We extend this result to the Norton algebras V i (H(n, e)) for i ≥ 2 in the next few subsections. 5.1. The case e = 3. Assume e = 3 in this subsection. We begin with the subcase i = n. Proposition 5.3. The Norton product ⋆ on V n (H(n, 3)) is equally as nonassociative as the double minus operation ⊖.
Proof. By Proposition 3.7, the Norton algebra V n (H(n, 3)) is isomorphic to the direct product of 2 n−1 copies of V 1 (H(1, 3)). Thus the result holds by Proposition 5.2. Now assume 1 < i < n. We need to recall some terminology for binary trees. Given a vertex x in a binary tree t, the (maximal) subtree of t rooted at x consists of all vertices and edges weakly below x. In particular, the left/right subtree of t is the subtree rooted at the left/right child of the root of t.
Proof. We induct on m. For m = 0, the result is trivial. Assume m ≥ 1 below.
Proof. Let s and t be any two distinct binary trees in T m . We need to show that (H(n, 3)).
We may assume that d s (j) ≡ d j (t) (mod 2) for all j = 0, 1, . . . , m; if not, then there exist z 0 , z 1 , . . . , z m in the subalgebra V u ∼ = V 1 (H(1, 3)) of V i (H(n, 3)) (see Lemma 3.6) for any u ∈ X i such that (z 0 ⋆ · · · ⋆ z m ) s = (z 0 ⋆ · · · ⋆ z m ) t by Theorem 5.1 and Proposition 5.2. We proceed by induction on m. For m = 2, Equation (8) holds since any u ∈ X i satisfies ( Now assume m ≥ 3. Let j be the leftmost leaf with the largest depth among all leaves in s. Then j is a left leaf, j + 1 is a right leaf, and they share a common parent in s. We distinguish some cases below for the positions of j and j + 1 in t.

Case 1.
Suppose that j is a left leaf and j + 1 is a right leaf, so they share a common parent in t. Then deleting j and j + 1 from s and t gives two distinct trees s ′ and t ′ in T m−1 . Applying the inductive hypothesis to s ′ and t ′ gives We have z ′ j = χ v for some v ∈ X i and we can define z j = z j+1 := χ 2v . Also let z k := z ′ k for k = 0, . . . , j − 1 and z ℓ := z ′ ℓ−1 for ℓ = j + 2, . . . , m. Since Case 2. Suppose that j and j + 1 are both left leaves in t. Then j + 1 is contained in the subtree r of t rooted at the right sibling of j. Since d j (t) ≡ d j (s) = d j+1 (s) ≡ d j+1 (t) (mod 2), the depth of j + 1 in r must be even. Thus the left subtree of r has two left and right subtrees r 1 and r 2 and j + 1 is in r 1 with an even depth. Define u, v, w ∈ X i below such that u + v / ∈ X i , u + w ∈ X i , and v + w ∈ X i .
By Lemma 5.4, the subtree r 1 can produce χ v with z j+1 = χ v , the subtree r 2 can produce χ w , and the right subtree of r can product χ 2v . Combining these with z j = χ u gives See the left picture in Figure 2, where j and j + 1 are in red. Then applying Lemma 5.4 to the tree obtained from t by contracting j and r to their parent gives (z 0 ⋆ · · · ⋆ z m ) t = χ c(u+w) = 0, where c ∈ {1, 2}. On the other hand, we have (z 0 ⋆ · · · ⋆ z m ) s = 0 since z j ⋆ z j+1 = χ u ⋆ χ v = 0. Thus we are done with this case.

Case 3.
Suppose that j is a right leaf in t. Then j is contained in the subtree r 1 of t rooted at the parent of j, and j + 1 is contained in the subtree r 2 of t rooted at the right sibling of the parent , the depth of j + 1 in r 2 must be odd. Thus j + 1 must be a left leaf in the left subtree of r 2 with an even depth. By Lemma 5.4, we can obtain χ u from r 1 with z j = χ 2u , obtain χ 2v from the left subtree of r 2 with z j+1 = χ 2v , and obtain χ v+w from the right subtree of r 2 . Combining r 1 and r 2 gives See the right picture in Figure 2, where j and j + 1 are in red. Contracting r 1 and r 2 and applying Lemma 5.4 again gives The case e ≥ 4. We study case e ≥ 4 similarly as the case e = 3. We may assume i ≥ 1.
Remark 5.10. The above proof replies on the fact that χ A ⋆ χ B = χ C for any permutation χ A , χ B , χ C of the triple χ S , χ T , χ S∆T . Thus one can use the same proof to show that the cross product on the n-dimensional space is totally nonassociative for all n ≥ 3, and that the octonions (hence all higher dimensional Cayley-Dickson constructions) have a totally nonassociative multiplication as well, thanks to the existence of a triple that behaves in the same way as above, except for the anticommutativity which does not affect the proof.

Halved and/or folded cubes
In this section we study the Norton algebras of the halved and/or folded cubes via the same linear character approach used for the hypercube.
6.1. Halved cube. Let Γ be a distance regular graph of diameter d. For i = 0, 1, . . . , d, let Γ i be the graph with the same vertex set as Γ but with edge set consisting of all unordered pairs of vertices at distance i from each other in Γ. If the graph Γ is bipartite then Γ 2 has two connected components, giving two distance regular graphs known as the halved graphs of Γ [8, Proposition 2.13].
In particular, the halved cube or half-cube 1 2 Q n has vertex set X consisting of all binary strings of length n with even weight and edge set E consisting of all unordered pairs of vertices differing in exactly two positions. This is a distance regular graph of diameter d = ⌊n/2⌋. For i = 0, 1, . . . , d, the ith eigenvalue and its multiplicity are [3, §9.2D] With the vertex set X viewed as a subgroup of Z n 2 , the halved cube 1 2 Q n becomes the Cayley graph Γ(X, X 2 ) of X with respect to X 2 , where X i := {x ∈ X : |supp(x)| = i}. For every S ⊆ [n] we define a linear character χ S of X by Proof. For each x ∈ X, since |supp(x)| is even, we have Thus χ S = χ S c . Conversely, suppose that χ S = χ T for two distinct sets S, T ⊆ [n]. Then there exists j ∈ S − T. If there exists k ∈ [n] − S△T then we have a contradiction that χ S (x) = χ T (x) for x ∈ X with x(j) = x(k) = 1 and all other entries zero. Thus S△T = [n], which implies S c = T.
any u ∈ X i we have where the second last equality holds by Equation (3).
, and the projection onto V i either fixes χ u+v if i = j or annihilates it otherwise.
Remark 7.2. The eigenvalue χ u (X 1 ) is a special case of the computation by Delsarte [9, Theorem A.2] using recursion, but our calculation in the above proof is more direct.
Next, we study the automorphisms of the Norton algebra V i (H q (d, e)). If i = 0 then this algebra is spanned by an idempotent χ 0 and thus has a trivial automorphism group. For i ≥ 1 we have the following result, where I n denotes the n × n identity matrix.
Proof. First, every x ∈ X = Mat d,e (F q ) induces an automorphism φ x of V i (H q (d, e)) by sending χ u to χ x (u)χ u for all u ∈ X i , as one can check the following for all u, v ∈ X i .
, χ x (u) = 1 for all u ∈ X i . For any j ∈ [d] and k ∈ [e], we can construct u ∈ X i in such a way that changing its (j, k)-entry gives another matrix u ′ with the same rank, and χ x (u) = χ x (u ′ ) implies that the (j, k)-entry of w is zero. Therefore x → φ x gives a monomorphism from X to the automorphism group of V i (H q (d, e)).
Next, every a ∈ GL d (F q ) induces an automorphism λ a of V i (H q (d, e)) by sending χ u to χ au for all u ∈ X i as the following holds for all u, v ∈ X i .
Suppose that λ a (χ u ) = χ au = χ u , i.e., au = u for all u ∈ X i . For any j ∈ [d], there exists u ∈ X i such that its (1, k)-entry is one if k = j or zero if k = j, and au = u implies that the jth column of a coincides with the first column of u. This shows that a = I d . Hence a → λ a gives a monomorphism from GL e (F q ) to the automorphism group of V i (H q (d, e)).
Similarly, every b ∈ GL d (F q ) induces an automorphism ρ b of V i (H q (d, e)) by sending χ u to χ ub −1 for all u ∈ X i , and b → ρ b gives a monomorphism from GL e (F q ) to the automorphism group of V i (H q (d, e)). It is clear that λ a and ρ b commute. Suppose that λ a = ρ b , i.e., au = ub −1 for all u ∈ X i . Below we distinguish two cases to show that a = cI d and b = cI e for some c ∈ F × q . • Assume i = d = e. Taking u = I d gives a = b −1 . Then au = ub −1 = ua for all u ∈ X i means that a is in the center of GL d (F q ), that is, a = cI d for some c ∈ F × q . • Assume i < d or d < e. For any j, j ′ ∈ [d] and any k, k ′ ∈ [e], let a jj ′ and b −1 k,k ′ denote the (j, j ′ )-entry of a and (k, k ′ )-entry of b −1 . If j = j ′ and k = k ′ then there exists u ∈ X i such that its (j, k)-entry is 1 and all other entries on the jth and j ′ th rows and on the kth and k ′ th columns are zero, and taking the (j, k)-entry, (j, k ′ )-entry, and (j ′ , k)-entry of au = ub −1 gives a jj = b −1 kk and a j ′ j = b −1 kk ′ = 0. Thus a = cI d and b −1 = cI e for some c ∈ F × q . It follows that the group G generated by {λ a : a ∈ GL d (F q )} and {ρ b : b ∈ GL e (F q )} is isomorphic to (GL d (F q ) × GL e (F q ))/{(cI d , cI e ) : c ∈ F × q }. This group has a trivial intersection with the group H := {φ x : x ∈ X}, since if φ x = λ a ρ b then χ x (u)χ u = χ aub −1 for all u ∈ X i and φ x must be the identity mapping. Moreover, we have ρ where the last equality holds by Equation (9) and the fact that tr(x t aub −1 ) = tr(b −1 x t au). Hence HG contains H as a normal subgroup and must be isomorphic to H ⋊ G.
Now we measure the nonassociativity of the Norton product on V i (H q (d, e)). The case d = 1 is already done as H q (1, e) is a complete graph isomorphic to the Hamming graph H(1, q e ). Proof. The Norton product ⋆ on V 0 is associative as V 0 is one-dimensional. Assume i ≥ 1 below.
For q ≥ 4, the proof of Theorem 5.7 is still valid since any u ∈ X i gives cu ∈ X i if c ∈ [q − 1]. For q = 3, the proof of Theorem 5.5 is still valid since one can find matrices u, v, w ∈ X i such that u + v / ∈ X i , u + w ∈ X i , and v + w ∈ X i . For q = 2, the proof of Theorem 5.9 is still valid since there exist matrices u, v, w ∈ X i such that u + v = w, which implies u + w = v and v + w = u.
We leave the existence of u, v, w in the above cases as an exercise to the reader.

Remarks and questions
In this paper we construct a basis for each eigenspace V i of the Hamming graph H(n, e) using the linear characters of the vertex set X = Z n e treated as a finite abelian group. Our basis is complex and can be converted to a real basis, but the existence of a basis over Z or even {0, 1, −1} needs further investigation.
We use our basis to provide a formula for the Norton product ⋆ on V i and obtain the following result on the automorphism group of the Norton algebra (V i , ⋆).
• It is the trivial group if i = 0.
• It is isomorphic to S e ≀ S n if i = 1.
• It is isomorphic to S 3 ≀ S 2 n−1 if e = 3 and i = n.
• It is the general linear group of V i if e = 2 and either i > ⌊2n/3⌋ or i is odd.
• It admits a subgroup is isomorphic to (Z e ⋊ Z × e ) ≀ S n if e ≥ 3 and i ≥ 1. • It admits a subgroup is isomorphic to S B n /{±1} if e = 2 and 1 ≤ i < n is even. 25 The groups mentioned above are all different, although most of them are related to the wreath product with symmetric groups. It will be interesting to see a complete description of the automorphism groups of all Norton algebras of the Hamming graph H(n, e). We also measure the nonassociativity of the Norton product on V i (H(n, e)) and show that this commutative product as nonassociative as possible, except for some special cases in which it is either associative for trivial reasons (being zero or defined on a one-dimensional space) or equally as nonassociative as the double minus operation a ⊖ b := −a − b. We are curious about whether the Norton algebras of other distance regular graphs are totally nonassociative in most cases and whether they could be related to the double minus operation or other elementary operations in some special cases.
Our results restrict to the hypercube Q n = H(n, 2) and extend to the halved and/or folded cubes. We have algebra isomorphisms V i ( n ) ∼ = V 2i (Q n ) and V i ( 1 2 n ) ∼ = V 2i ( 1 2 Q n ). More generally, the folded graph Γ can be defined for any antipodal distance regular graph Γ and is still distance regular [8, Proposition 2.14]. We would like to know if the Norton algebras of Γ and Γ are related in the same way as above.
For other distance regular graphs, the linear character approach should be valid as long as they are also Cayley graphs of finite abelian groups, such as the square graph of the hypercube [22] and the alternating forms graphs [3, §9.5B], but a different method may be necessary otherwise. For instance, the Johnson graphs are not Cayley graphs in general, but they have been heavily studied and their spectra can be obtained by linear algebra (see Burcroff [5]) or representation theory (see Krebs and Shaheen [12]). It would be interesting to see whether the existing constructions of the eigenspaces of the Johnson graphs could be used to determine their Norton algebras.