aa r X i v : . [ m a t h . C O ] J a n A NOTE ON p ′ -VALENCED ASSOCIATION SCHEMES YU JIANG
Abstract.
Let p be a prime. In this note, we establish an optimal lower bound forthe dimensions of Jacobson radicals of modular Terwilliger algebras of associationschemes. As a byproduct, we characterize the p ′ -valenced association schemes bya property of their modular Terwilliger algebras. Keywords.
Association scheme; Modular Terwilliger algebra; p ′ -valenced scheme Mathematics Subject Classification 2020.
Introduction
Let X be a nonempty finite set and call an association scheme on X a scheme.Fix a field F of positive characteristic p . In [2], Hanaki introduced the Terwilligeralgebras of schemes over F and called these algebras the modular Terwilliger algebrasof schemes. By the definition of a modular Terwilliger algebra of a scheme, noticethat the modular Terwilliger algebras of schemes contain the adjacency algebras ofschemes over F . Therefore the modular Terwilliger algebras of schemes may containmore combinatorial information than the adjacency algebras of schemes over F . Tounderstand the combinatorial information in the modular Terwilliger algebras ofschemes, it is very necessary to study the modular Terwilliger algebras of schemes.For a finite-dimensional F -algebra A , A is semisimple if and only if its Jacobsonradical contains exactly the zero element of A . The modular Terwilliger algebras ofschemes are finite-dimensional F -algebras. By Theorem 2.3, the modular Terwilligeralgebras of schemes may not be semisimple. To study the nonsemisimple modularTerwilliger algebras of schemes, it is very necessary to study the Jacobson radicals ofmodular Terwilliger algebras of schemes. In [3], the author obtained the Jacobsonradicals of modular Terwilliger algebras of quasi-thin schemes (see Theorem 3.8).In general, it is difficult to determine the Jacobson radical of a modular Terwilligeralgebra of a scheme. Fix x ∈ X and a scheme S . let T be the modular Terwilligeralgebra of S with respect to x . Let J be the Jacobson radical of T . In this note,we get an optimal lower bound for the F -dimension of J . Our result is as follows. Theorem A.
Let S p ⊆ S contain exactly all relations whose valencies are divisibleby p . Then | S p | + 2 | S p || S \ S p | is an optimal lower bound for the F -dimension of J . Theorem 2.3 is a property of the p ′ -valenced schemes. This property motivates usto characterize the p ′ -valenced schemes by the properties of their modular Terwilligeralgebras. As a byproduct of Theorem A, in this note, we characterize the p ′ -valencedschemes by a property of their modular Terwilliger algebras (see Corollary 3.12).The outline of this note is as follows. In Section 2, we set up the notation andgive some preliminary results. In Section 3, we prove the main results of this note. Notation and preliminaries
In this section, we gather the notation and present some preliminary results. Weassume that the reader is familiar with the theory of association schemes. For ageneral background on association schemes, the reader may refer to [1], [5], or [6].2.1.
General conventions.
Throughout the whole note, fix a field F of positivecharacteristic p and a nonempty finite set X . Let N be the set of all natural numbersand N = N ∪ { } . For a nonempty set Y of an F -linear space, write h Y i F for the F -linear space generated by Y . By convention, h ∅ i F is the zero space. The addition,the multiplication, and the scalar multiplication of matrices displayed in this noteare the usual matrix operations. A scheme means an association scheme on X .2.2. Schemes.
Let S = { R , R , . . . , R d } denote a partition of the cartesian product X × X , where R a = ∅ for every R a ∈ S . Then S is called a scheme of class d if thefollowing conditions hold:(i) R = { ( b, b ) : b ∈ X } ;(ii) For every 0 ≤ c ≤ d , we have 0 ≤ c ′ ≤ d , where { ( e, f ) : ( f, e ) ∈ R c } = R c ′ ∈ S ;(iii) For any 0 ≤ i, j, k ≤ d and ( m, n ) , ( ˜ m, ˜ n ) ∈ R k , the following equality holds: |{ ℓ ∈ X : ( m, ℓ ) ∈ R i , ( ℓ, n ) ∈ R j }| = |{ ˜ ℓ ∈ X : ( ˜ m, ˜ ℓ ) ∈ R i , (˜ ℓ, ˜ n ) ∈ R j }| . In this note, S = { R , R , . . . , R d } is a fixed scheme of class d . Every member of S is called a relation of S . Let g ∈ X and R h ∈ S . We set gR h = { q ∈ X : ( g, q ) ∈ R h , ( q, g ) ∈ R h ′ } = { q ∈ X : ( g, q ) ∈ R h } . By the definition of S , | gR h | = | rR h | for every r ∈ X . So there exists k h ∈ N suchthat k h = | rR h | for every r ∈ X . The number k h is called the valency of R h .We are interested in some special schemes. Let us state their definitions as follows.Let S p = { R s ∈ S : p | k s } . The scheme S is called a p ′ -valenced scheme if S p = ∅ .Therefore we have S is not a p ′ -valenced scheme if and only if S p = ∅ .Set O ϑ ( S ) = { R u ∈ S : k u = 1 } and observe that R ∈ O ϑ ( S ). The scheme S iscalled a quasi-thin scheme if k v = 2 for every R v ∈ S \ O ϑ ( S ).2.3. Algebras.
Let A denote a finite-dimensional F -algebra with the identity 1 and b ∈ A . Then we call b a left-invertible element of A if there exists c ∈ A such that cb = 1. Let J be the Jacobson radical of A . Recall that J is the sum of all two-sidednilpotent ideals of A . In particular, J is a two-sided ideal of A . The relationshipbetween J and the left-invertible elements of A are described as follows. Lemma 2.1. [4, Lemma 4.1]
Let e ∈ A . Then we have e ∈ J if and only if − f e is a left-invertible element of A for every f ∈ A . Let M X ( F ) denote the full matrix algebra of F -square matrices whose rows andcolumns are labeled by the members of X . For every M ∈ M X ( F ) and N ⊆ M X ( F ),let M t denote the transpose of M and N t = { L t : L ∈ N } . We close this subsectionby the following lemma. Lemma 2.2. [3, Lemma 2.3]
Assume that A is a unital F -subalgebra of M X ( F ) . If A t = A , then J t = J . NOTE ON p ′ -VALENCED ASSOCIATION SCHEMES 3 Modular Terwilliger algebras of schemes.
Let I , J , O denote the identitymatrix, the all-one matrix, the all-zero matrix in M X ( F ) respectively. Let a, b ∈ X .The matrix E ab is defined to be the (0 , M X ( F ) whose unique nonzeroentry is the ( a, b )-entry. Note that { E rs : r, s ∈ X } is an F -basis of M X ( F ).Let R c , R e ∈ S . The adjacency F -matrix with respect to R c , denoted by A c , isdefined to be the (0 , a fg ) of M X ( F ), where we have a fg = 1 if and only if( f, g ) ∈ R c . Notice that A = I and A tc = A c ′ .The dual F -idempotent with respect to a and R c , denoted by E ∗ c ( a ), is defined tobe the matrix X h ∈ aR c E hh . (2.1)Notice that E ∗ c ( a ) is a diagonal matrix. In particular, E ∗ c ( a ) is a symmetric matrix.Let δ αβ denote the Kronecker delta of integers α, β whose values are in F . Note that E ∗ c ( a ) E ∗ e ( a ) = δ ce E ∗ c ( a ). Since we have | aR c | = k c and J E ab J = J , by (2.1), noticethat J E ∗ ( a ) J = J and J E ∗ c ( a ) J = O if R c ∈ S p .Let T ( a ) be the modular Terwilliger algebra of S with respect to a . It is the unital F -subalgebra of M X ( F ) generated by A , A , . . . , A d and E ∗ ( a ) , E ∗ ( a ) , . . . , E ∗ d ( a ).Note that T ( a ) t = T ( a ) by the definition of T ( a ). Throughout the remaining partof this note, we fix x ∈ X and let T = T ( x ). Write E ∗ u = E ∗ u ( x ) for every R u ∈ S .Let J denote the Jacobson radical of T . Let m ∈ N and R i n , R j n , R ℓ n ∈ S for every0 ≤ n ≤ m . We set m Y v =0 ( E ∗ i v A j v E ∗ ℓ v ) = E ∗ i A j E ∗ ℓ E ∗ i A j E ∗ ℓ · · · E ∗ i m A j m E ∗ ℓ m ∈ T . For example, if m = 2, then Q v =0 ( E ∗ i v A j v E ∗ ℓ v ) = E ∗ i A j E ∗ ℓ E ∗ i A j E ∗ ℓ E ∗ i A j E ∗ ℓ . Let S m = { m Y v =0 ( E ∗ i v A j v E ∗ ℓ v ) : R i v , R j v , R ℓ v ∈ S for every 0 ≤ v ≤ m } \ { O } . and set S − = ∅ . Let w ∈ N ∪ {− } and put T w = hS w i F ⊆ T . We thus can set P wy = − T y = h S wy = − S w i F . We conclude the whole section by listing some results.These results shall be used in the next section. Theorem 2.3. [2, Theorem 3.4 and Page 6]
We have T is semisimple only if S is a p ′ -valenced scheme. Moreover, there exists an example such that S is a p ′ -valencedscheme and T is not semisimple. Lemma 2.4.
The following assertions hold. (i) [3, Lemma 3.2]
Assume that O = E ∗ a A b E ∗ c ∈ T . If k a = 1 or k c = 1 , then E ∗ a A b E ∗ c = E ∗ a J E ∗ b . (ii) [3, Lemma 2.4 (iii)] The set { E ∗ a J E ∗ b : R a , R b ∈ S } is an F -linearly independentsubset of T . In particular, O / ∈ { E ∗ a J E ∗ b : R a , R b ∈ S } . (iii) We have T t = T and J t = J .Proof. (iii) follows by the definition of T and Lemma 2.2. (cid:3) YU JIANG Main results
In this section, we prove our main results. We first present some needed results.
Lemma 3.1. [3, Lemma 3.11 (iii)]
Let M ∈ T . Then there exist c, a , a , . . . , a c ∈ N such that M = P cb =0 e b M b , where we have = e b ∈ F , M b ∈ S a b , and M b / ∈ P a b − h = − T h for every ≤ b ≤ c . Moreover, let ≤ u ≤ c . If a u > and M u = Q a u v =0 ( E ∗ i v A j v E ∗ ℓ v ) ,then min { k i v , k j v , k ℓ v } > for every ≤ v ≤ a u . Lemma 3.2.
Let M ∈ T and N ∈ M X ( F ) . If E ∗ a N = N for some R a ∈ O ϑ ( S ) ,then there exist c ∈ N , R i , R i , . . . , R i c ∈ S , and e , e , . . . , e c ∈ F such that M N = c X b =0 e b E ∗ i b J N.
Proof. If M N = O , we set c = i = 0 and e = 0 ∈ F . Notice that the desiredequality holds. We thus assume further that M N = O . As E ∗ a N = N , observe that O = M N = M ( E ∗ a N ) = ( M E ∗ a ) N . So we have M E ∗ a = O . By Lemma 3.1, thereexist g, a , a , . . . , a g ∈ N such that M = g X f =0 m f M f , (3.1)where we have 0 = m f ∈ F , M f ∈ S a f , and M f / ∈ P a f − h = − T h for every 0 ≤ f ≤ g .Let 0 ≤ k ≤ g . If a k > M k = Q a k m =0 ( E ∗ r m A s m E ∗ t m ), by Lemma 3.1, notice that M k E ∗ a = O as R a ∈ O ϑ ( S ) and min { k r m , k s m , k t m } > ≤ m ≤ a k . So wededuce that O = M E ∗ a = g X f =0 m f M f E ∗ a = g X f =0 a f =0 m f M f E ∗ a ∈ T (3.2)by the inequality M E ∗ a = O , (3.1), and the definition of T . According to (3.2) andthe definition of T , there exist ˜ c ∈ N and R j , R j , . . . , R j ˜ c , R ℓ , R ℓ , . . . , R ℓ ˜ c ∈ S such that O = M E ∗ a = ˜ c X b =0 ˜ e b E ∗ j b A ℓ b E ∗ a , (3.3)where we have 0 = ˜ e b ∈ F and E ∗ j b A ℓ b E ∗ a = O for every 0 ≤ b ≤ ˜ c . Since R a ∈ O ϑ ( S )and E ∗ j b A ℓ b E ∗ a = O for every 0 ≤ b ≤ ˜ c , by (3.3) and Lemma 2.4 (i), we have O = M E ∗ a = ˜ c X b =0 ˜ e b E ∗ j b A ℓ b E ∗ a = ˜ c X b =0 ˜ e b E ∗ j b J E ∗ a . (3.4)We now set c = ˜ c , i b = j b , and e b = ˜ e b for every 0 ≤ b ≤ c . As E ∗ a N = N , by (3.4),we have M N = M ( E ∗ a N ) = ( M E ∗ a ) N = P cb =0 e b E ∗ i b J ( E ∗ a N ) = P cb =0 e b E ∗ i b J N . Sothe desired equality also holds. The proof is now complete. (cid:3)
The following corollary is a special case of Lemma 3.2.
NOTE ON p ′ -VALENCED ASSOCIATION SCHEMES 5 Corollary 3.3.
Let M ∈ T and R a , R b ∈ S . If R a ∈ O ϑ ( S ) , then there exist c ∈ N , R i , R i , . . . , R i c ∈ S , and e , e , . . . , e c ∈ F such that M E ∗ a J E ∗ b = c X f =0 e f E ∗ i f J E ∗ b . Proof.
Observe that E ∗ a E ∗ a J E ∗ b = E ∗ a J E ∗ b ∈ T . As R a ∈ O ϑ ( S ), according to Lemma3.2, there exist c ∈ N , R i , R i , . . . , R i c ∈ S , and e , e , . . . , e c ∈ F such that M E ∗ a J E ∗ b = c X f =0 e f E ∗ i f J E ∗ a J E ∗ b . (3.5)As R a ∈ O ϑ ( S ), by (3.5), we have J E ∗ a J = J and M E ∗ a J E ∗ b = c X f =0 e f E ∗ i f J E ∗ a J E ∗ b = c X f =0 e f E ∗ i f ( J E ∗ a J ) E ∗ b = c X f =0 e f E ∗ i f J E ∗ b . The desired corollary thus follows. (cid:3)
For further discussion, recall that S p = { R a ∈ S : p | k a } . Lemma 3.4.
Let
M, N ∈ T and R a , R b , R g , R h ∈ S . Assume that R a , R g ∈ O ϑ ( S ) and R b ∈ S p , then M E ∗ a J E ∗ b N E ∗ g J E ∗ h = O .Proof. According to the hypotheses and Corollary 3.3, note that there are c , c ∈ N , R i , R i , . . . , R i c , R j , R j , . . . , R j c ∈ S , and u , u , . . . , u c , v , v , . . . , v c ∈ F suchthat M E ∗ a J E ∗ b = c X r =0 u r E ∗ i r J E ∗ b and N E ∗ g J E ∗ h = c X s =0 v s E ∗ j s J E ∗ h . (3.6)To get the desired equality, by (3.6), it is enough to prove that E ∗ i r J E ∗ b E ∗ j s J E ∗ h = O for every 0 ≤ r ≤ c and 0 ≤ s ≤ c . Let 0 ≤ y ≤ c and 0 ≤ z ≤ c . If b = j z , it isclear that E ∗ i y J E ∗ b E ∗ j z J E ∗ h = O . If b = j z , as R b ∈ S p , we have p | k b , which impliesthat J E ∗ b J = O . So we have E ∗ i y J E ∗ b E ∗ j z J E ∗ h = E ∗ i y ( J E ∗ b J ) E ∗ h = O . Since y, z arechosen arbitrarily, we obtain that E ∗ i r J E ∗ b E ∗ j s J E ∗ h = O for every 0 ≤ r ≤ c and0 ≤ s ≤ c . Therefore we get the desired equality. The proof is now complete. (cid:3) Corollary 3.5.
Let M ∈ T and R a , R b ∈ S . If R a ∈ O ϑ ( S ) and R b ∈ S p , then wehave ( I + M E ∗ a J E ∗ b )( I − M E ∗ a J E ∗ b ) = I .Proof. According to the hypotheses and Lemma 3.4, the desired equality follows bydirect computation. (cid:3)
For further discussion, recall that J is the Jacobson radical of T . Corollary 3.6.
Assume that R a ∈ O ϑ ( S ) and R b ∈ S p . Then E ∗ a J E ∗ b , E ∗ b J E ∗ a ∈ J .Proof. Since R a ∈ O ϑ ( S ) and R b ∈ S p , by Corollary 3.5 and Lemma 2.1, we have E ∗ a J E ∗ b ∈ J . Moreover, we also have E ∗ b J E ∗ a = ( E ∗ a J E ∗ b ) t ∈ J t = J by Lemma 2.4(iii). The desired corollary thus follows. (cid:3) YU JIANG
The following corollary gives us a lower bound of the F -dimension of J . Corollary 3.7.
The following assertions hold. (i)
We have h{ E ∗ a J E ∗ b : R a , R b ∈ S, p | k a k b }i F ⊆ J . (ii) The F -dimension of h{ E ∗ a J E ∗ b : R a , R b ∈ S, p | k a k b }i F is | S p | + 2 | S p || S \ S p | . (iii) The F -dimension of J is no less than | S p | + 2 | S p || S \ S p | .Proof. If S is a p ′ -valenced scheme, it is obvious that (i), (ii), (iii) hold trivially. Wethus assume further that S is not a p ′ -valenced scheme.For (i), let R c , R e ∈ S and p | k c k e . To get the desired containment, it is enoughto check that E ∗ c J E ∗ e ∈ J . As p | k c k e , notice that p | k c or p | k e . If p | k e , byCorollary 3.6, note that E ∗ J E ∗ e ∈ J . As E ∗ c J E ∗ ∈ T and J is a two-sided ideal,we have J E ∗ J = J and E ∗ c J E ∗ e = E ∗ c ( J E ∗ J ) E ∗ e = E ∗ c J E ∗ E ∗ J E ∗ e ∈ J . If p | k c , by the analysis of the case p | k e , we have E ∗ e J E ∗ c ∈ J and E ∗ c J E ∗ e = ( E ∗ e J E ∗ c ) t ∈ J t = J by Lemma 2.4 (iii). So we always have E ∗ c J E ∗ e ∈ J . (i) thus follows.For (ii), as { E ∗ a J E ∗ b : R a , R b ∈ S, p | k a k b } ⊆ { E ∗ g J E ∗ h : R g , R h ∈ S } , we have { E ∗ a J E ∗ b : R a , R b ∈ S, p | k a k b } is an F -linearly independent subset of T by Lemma2.4 (ii). Let S = { ( R i , R j ) : R i , R j ∈ S, p | k i k j } . Note that |S| = | S p | +2 | S p || S \ S p | .Let φ be the map from S to { E ∗ a J E ∗ b : R a , R b ∈ S, p | k a k b } that sends every ( R i , R j )to E ∗ i J E ∗ j . It is clear that φ is a surjective map. Let R k , R ℓ , R m , R n ∈ S . It is obviousto see that E ∗ k J E ∗ ℓ = E ∗ m J E ∗ n if and only if k = m and ℓ = n . So φ is a bijection,which implies that |{ E ∗ a J E ∗ b : R a , R b ∈ S, p | k a k b }| = | S p | + 2 | S p || S \ S p | . So the F -dimension of h{ E ∗ a J E ∗ b : R a , R b ∈ S, p | k a k b }i F is | S p | + 2 | S p || S \ S p | . (ii) is nowproved. (iii) follows by (i) and (ii). The proof is now complete. (cid:3) The following theorem describes the semisimplicity and the Jacobson radical of amodular Terwilliger algebra of a quasi-thin scheme.
Theorem 3.8. [3, Theorems B, C, Corollary 6.7 (iii)]
Assume that S is a quasi-thinscheme. Then T is semisimple if and only if S is a p ′ -valenced scheme. Moreover,if T is not semisimple, then we have p = 2 , J = h{ E ∗ a J E ∗ b : R a , R b ∈ S, | k a k b }i F ,and M = O for every M ∈ J . Remark 3.9. If p = 2 and S is a quasi-thin scheme, by Theorem 3.8 and Corollary3.7 (ii), the lower bound given in Corollary 3.7 (iii) is optimal.Theorem A is now proved by Corollary 3.7 (iii) and Remark 3.9. Remark 3.10. As T is semisimple if and only if J = { O } , by Theorem 2.3, thereexists an example such that S is a p ′ -valenced scheme and J 6 = { O } . In this example,the lower bound established in Theorem A is 0. So there is an example such thatthe F -dimension of J is strictly larger than the lower bound established in TheoremA.We close this note by two corollaries of Corollary 3.6. NOTE ON p ′ -VALENCED ASSOCIATION SCHEMES 7 Corollary 3.11.
Let a ∈ N . Assume that S is not a p ′ -valenced scheme and M a = O for every M ∈ J , then a ≥ .Proof. It suffices to check that a = 2. As S is not a p ′ -valenced scheme, we have S p = ∅ . Choose R b ∈ S p and note that E ∗ b J E ∗ , E ∗ J E ∗ b ∈ J by Corollary 3.6. So E ∗ b J E ∗ + E ∗ J E ∗ b ∈ J . Observe that J E ∗ J = J and J E ∗ b J = O as p | k b . So we have( E ∗ b J E ∗ + E ∗ J E ∗ b ) = E ∗ b J E ∗ b = O by Lemma 2.4 (ii). Therefore a = 2 as M a = O for every M ∈ J . The desired corollary follows. (cid:3) Our final corollary gives a characterization of the p ′ -valenced schemes. Corollary 3.12.
The scheme S is a p ′ -valenced scheme if and only if we have E ∗ a M = M E ∗ a = O for every R a ∈ O ϑ ( S ) and M ∈ J .Proof. If S is a p ′ -valenced scheme, by [3, Lemma 5.1], we have E ∗ a M = M E ∗ a = O for every R a ∈ O ϑ ( S ) and M ∈ J . Conversely, assume that E ∗ a M = M E ∗ a = O for every R a ∈ O ϑ ( S ) and M ∈ J . If S is not a p ′ -valenced scheme, notice that S p = ∅ . Choose R b ∈ S p and notice that E ∗ J E ∗ b ∈ J by Corollary 3.6. So we have E ∗ E ∗ J E ∗ b = E ∗ J E ∗ b = O by Lemma 2.4 (ii). The inequality E ∗ E ∗ J E ∗ b = O gives usa contradiction as we have assumed that E ∗ a M = M E ∗ a = O for every R a ∈ O ϑ ( S )and M ∈ J . So S is a p ′ -valenced scheme. The proof is now complete. (cid:3) References [1] E. Bannai, T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings,Menlo Park, CA, 1984.[2] A. Hanaki, Modular Terwilliger algebras of association schemes, arxiv: 2004.09692, 2020.[3] Y. Jiang, On Terwilliger F -algebras of quasi-thin association schemes, arxiv: 2012.14811, 2020.[4] T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, ,Springer-Verlag, New York, 1991.[5] P.-H. Zieschang, An Algebraic Approach to Association Schemes, Lecture Notes in Math., vol. , Springer-Verlag, Berlin, 1996.[6] P.-H. Zieschang, Theory of Associaiton Schemes, Spring Monogr. Math., Springer-Verlag,Berlin, 2005.(Y. Jiang) Division of Mathematical Sciences, Nanyang Technological University,SPMS-MAS-05-34, 21 Nanyang Link, Singapore 637371.
Email address , Y. Jiang:, Y. Jiang: