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Modular Terwilliger algebras of association schemes

We define modular Terwilliger algebras of association schemes, Terwilliger algebras over a positive characteristic field, and consider basic properties. We give a condition for the modular Terwilliger algebra to be non-semisimple. We show that the dimension of a Terwilliger algebra of a Johnson scheme depends on the characteristic of the coefficient field. We also give some other examples.


Introduction
In a series of papers [6,7,8], P. Terwilliger defined and studied subconstituent algebras of commutative association schemes for the theory of distance regular graphs. Now the algebras are called Terwilliger algebras. Terwilliger algebras are finite-dimensional semisimple algebras over the complex number field. Since the Terwilliger algebra is defined for an association scheme and a fixed point, we can expect that it has more combinatorial information than the adjacency algebra. The Terwilliger algebra is defined as a matrix algebra generated by some matrices all whose entries are in {0, 1}. Thus, for a commutative ring R with the identity 1, we can define an R-algebra generated by these matrices. We call it the Terwilliger algebra over R. Especially, we call it the modular Terwilliger algebra when R is a field of positive characteristic.
Let (X, S) be an association scheme, x ∈ X, and K be a field. We denote by KT (x) the Terwilliger algebra of (X, S) at the point x over K. If K is of positive characteristic, KT (x) is not necessary semisimple. A natural question is when it is semisimple. It is known that the dimension of CT (x) is depending on the choice of the point x [9]. By example, we can see that the semisimplicity of KT (x) also depends on the choice of the point x. Thus the question seems to be difficult, in general. We will give a sufficient condition for KT (x) to be non-semisimple in Theorem 4.1, KT (x) is not semisimple if the characteristic of K divides a valency of some element of S.
In Section 4, we will consider Johnson schemes J(n, 2). Let K be a field of characteristic p. We will see that the dimension of the Terwilliger algebra is 16 if p = 0 or p ∤ n − 4 and 15 if p | n − 4. These examples give a negative answer to Terwilliger's small question in [8, Conjecture 10]. In Section 5, we will give some other examples.

Preliminaries and definitions
Let X be a finite set, and let R be a commutative ring with the identity 1. We denote by M X (R) the full matrix ring over R, rows and columns of whose matrices are indexed by the set X. For sin M X (R), we denote the transposed matrix of σ by σ T .
For s ⊂ X × X, the adjacency matrix σ s ∈ M X (Z) is defined by (σ s ) xy = 1 if (x, y) ∈ s and 0 otherwise. We often regard σ s is in M X (R) for a suitable R.
2.1. Association schemes. Let X be a finite set, and let X × X = s∈S s be a partition of X × X. We call the pair (X, S) an association scheme if (1) 1 : for s, t, u ∈ S, there is a non-negative integer p u st such that p u st = |xs ∩ ty| when (x, y) ∈ u. The condition (3) means that σ s σ t = u∈S p u st σ u by the usual matrix multiplication. For s ∈ S, n s := p 1 ss * = |xs| is independent of the choice of x ∈ X, and we call this number the valency of s.
We say an association scheme (X, S) is commutative if p u st = p u ts for all s, t, u ∈ S, symmetric if s * = s for all s ∈ S. Symmetric association schemes are commutative.
By the condition (3), RS := s∈S Rσ s is an R-algebra. We call RS the adjacency algebra of (X, S) over R.
It is well known that strongly regular graphs correspond to symmetric association schemes with |S| = 3. We often identify them.
2.2. Terwilliger algebras. Let (X, S) be an association scheme. Fix x ∈ X. We have a partition X = s∈S xs. We set the diagonal matrix E * s ∈ M X (Z) whose (y, y)-entry is 1 if y ∈ xs and 0 otherwise.
Let R be a commutative ring with 1. We regard σ s and E * s are elements in M X (R) and set RT (x) the R-algebra generated by {E * s σ t E * u | s, t, u ∈ S}. We call RT (x) the Terwilliger algebra of (X, S) over R at x. The original definition of a Terwilliger algebra in [6] is CT (x). When R is a field of positive characteristic we call RT (x) a modular Terwilliger algebra. Remark that E * s (RT (x))E * s is a subalgebra of RT (x) with the identity element E * s . If K and K ′ have the same characteristic, then the dimensions of the Terwilliger algebras over them are equal. Semisimplicity is also depending only on the characteristic of the coefficient field, because the algebra is defined over the prime field and the prime field is perfect (see [5, Chap. II, Sect. 5], for example).
Let p be a prime number, and let F p be a field of order p. By our definition, Now, the following proposition holds.

Semisimplicity
The Terwilliger algebra over the complex number field is semisimple since it is closed by transposition and complex conjugate. The Terwilliger algebra over a positive characteristic field is not necessary semisimple. Moreover it depends on the choice of the point x ∈ X. For example, (26, 10, 3, 4)-strongly regular graphs (see Subsection 5.4) give such examples in characteristic 7 and 11. Proposition 3.1. Let (X, S) be an association scheme and fix x ∈ X. Let K be a field. If the Terwilliger algebra KT (x) of (X, S) over K is semisimple, then E * s (KT (x))E * s is also semisimple for every s ∈ S.
Proof. Suppose E * s (KT (x))E * s is not semisimple for some s ∈ S. The Jacobson radical J of E * s (KT (x))E * s is non-zero. Since J is nilpotent, assume J ℓ = 0. It is enough to show that ((KT (x))J (KT (x))) ℓ = 0. For a i , b i ∈ KT (x) and j i ∈ J , we have Lemma 3.2. For s, t, u ∈ S, σ u st is an incidence matrix of a tactical configuration. Every row of σ u st contains p s tu * ones and every column of σ u st contains p t su ones. Proof. For y ∈ xs, equivalent to (x, y) ∈ s, the y-th row of σ u st contains ♯{z ∈ xt | (y, z) ∈ u} = ♯{z ∈ X | (x, z) ∈ t, (z, y) ∈ u * } = p s tu * ones. Similarly, for z ∈ xt, the z-th column of σ u st contains ♯{y ∈ xs | (y, z) ∈ u} = ♯{y ∈ X | (x, y) ∈ s, (y, z) ∈ u} = p t su ones.
where J is the square matrix all whose entries are one. Proof. The statement holds by Lemma 3.2.
Theorem 3.4. Let K be a field of positive characteristic p. Suppose that p divides the valency n s for some s ∈ S. Then KT (x) is not semisimple.
Proof. Suppose p divides |xs| for s ∈ S. Then the ideal KE * s JE * s of E * s KT (x)E * s is nilpotent. Thus E * s KT (x)E * s is not semisimple, and so is KT (x) by Proposition 3.1.

Johnson graphs J(n, 2)
The structures of Terwilliger algebras of Johnson schemes J(n, k) were determined in [3,4]. We focus only on the case k = 2 and consider their modular Terwilliger algebras. The structure is independent of the choice of a point x, because the automorphism group acts on points transitively. We will write T instead of T (x) in this section.
Let (X, S) be the Johnson scheme J(n, 2) (n ≥ 5). This gives an (n(n − 1)/2, 2(n − 2), n − 2, 4)-strongly regular graph. By definition, we set X := {{i, j} | 1 ≤ i < j ≤ n}, S := {s 0 , s 1 , s 2 }, s 0 := {(y, y) | y ∈ X}, s 1 := {(y, z) | |y ∩ z| = 1}, and s 2 := {(y, z) | y ∩ z = ∅}. We often write 1 instead of s 1 , and so on. For example, σ 1 is σ s 1 . Fix x := {1, 2}. We have where I n−2 is the identity matrix of degree n − 2, J n−2 is the square matrix of degree (n − 2) all whose entries are one, j 2(n−2) is the row vector of degree 2(n − 2) all whose entries are one, 0 n 2 is the zero row vector of degree n 2 , C is the adjacency matrix of J(n − 2, 2), and D is a incidence matrix of a tactical configuration, every row of D contains n − 3 ones and every column of D contains 4 ones. We set ZT ′ := i,j,k ZE * i σ j E * k . This is not closed by multiplication. It is not so hard to check that all products ( O are zero matrices. Now we can see that On the other hand, This gives a negative answer to Terwilliger's small question "(is generated by?)" in [8, Conjecture 10]. We can get infinitely many such examples in this way.
where K is the field of characteristic 2.