Terwilliger Algebras of Wreath Powers of One-Class Association Schemes

T erwilliger Algebras of W reath P o w ers of One-Class Asso ciation Sc hemes Gargi Bhattac haryya and Sung Y. Song Departmen t of Mathematics, Iow a State Univ ersity , Ames, Io w a, 5 0011, USA Decem b er 11, 2021 Abstract In this pap er, we study the sub constituent algebras, also called a s T erwilliger algebr as, of asso ciatio n schemes that ar e obtained as the wr eath pro duct o f one-cla ss asso ciation schemes K n = H (1 , n ) for n ≥ 2. W e find that the d -cla ss as so ciation scheme K n 1 ≀ K n 2 ≀ · · · ≀ K n d formed by taking the w r eath pro duct of K n i has the triple-r e gularity prop erty . W e determine the dimension of the T erwilliger algebr a for the a sso ciation scheme K n 1 ≀ K n 2 ≀ · · · ≀ K n d . W e give a description of the structur e o f the T erwillige r alg ebra for the wreath p ow er ( K n ) ≀ d for n ≥ 2 by studying its irreducible mo dules. In par ticular, we show that the T erwillig er a lgebra of ( K n ) ≀ d is isomor phic to M d +1 ( C ) ⊕ M 1 ( C ) ⊕ 1 2 d ( d +1) for n ≥ 3, and M d +1 ( C ) ⊕ M 1 ( C ) ⊕ 1 2 d ( d − 1) for n = 2. 1 In tro d uction The sub constituen t a lgebra, whic h is also kno wn as the T erwilliger algebra of an asso ciation sc heme w as in tro duced b y T erwilliger in 1992 as a n ew algebraic to ol for the study of asso ciation sc hemes [19]. The T erwilliger algebra of a commutati v e asso ciation scheme is a fi nite d imensional, semi-simple C -algebra, and is n oncomm utativ e in general. Th is algebra helps und erstanding the structure of the asso ciation sc h emes. It has b een studied extensiv ely for m an y classes of asso cia- tion sc hemes. F or example, the T erwilliger algebra for P - and Q -p olynomial asso ciation schemes has b een studied in [20, 22, 21, 7]. The structure of T erwilliger algebra of group association sc hemes has b een studied in [1] and [2]. In [15] the structure of the T erwilliger algebra of a Hamming scheme H ( d, n ) is giv en as s y m metric d -tensors of the T erwilliger algebra of H (1 , n ) whic h a r e all isomorphic for n > 2. It is also shown that the T erw illiger algebra of H ( d, n ) is decomp osed as a direct sum of T erwilliger algebra of hyp er cu b es H ( d, 2) in [15]. They d ed uce the decomp osition in to simple bilateral id eals usin g the representa tion of classical groups. T here is a detailed study of the irreducible mo du les of the algebra for H ( d, 2) in [11], and for Do ob sc hemes (the sc h emes coming as direct p r o ducts of copies of H (2 , 4) and/or Shrikhan d e graph s ,) in [18]. Both of these studies used elemen tary linear algebra and mo du le theory . In this pap er, we study the T erwilliger algebras of asso ciation schemes w h ic h are obtained as wreath p r o ducts of H (1 , n ), also denoted K n , for n ≥ 2. W e fin d that the d -class asso ciation sc heme K n 1 ≀ K n 2 ≀ · · · ≀ K n d formed by taking the w reath p ro du ct of one-class asso ciation s c hemes K n i has the triple-reg u larit y prop erty in the sense of [16] and [14]. Based on this fact, we 1 determine the dimen sion of the T erwilliger algebra for the asso ciation sc h eme K n 1 ≀ K n 2 ≀ · · · ≀ K n d . W e then find th at the wreath p o wer ( K n ) ≀ d = K n ≀ K n ≀ · · · ≀ K n , d copies of K n , is formally self-dual and the T erwilliger algebra is isomorph ic to M d +1 ( C ) ⊕ M 1 ( C ) ⊕ 1 2 d ( d +1) for n ≥ 3, w hile M d +1 ( C ) ⊕ M 1 ( C ) ⊕ 1 2 d ( d − 1) for n = 2 in the notion of W edderbu rn-Artin’s decomp osition theorem of semisimp le algebra. The case ( K 2 ) ≀ d b ehav es a little differen tly from the general case ( K n ) ≀ d for n ≥ 3. W e also stud y the corresp onding irreducible mo dules for n = 2 in the cour se. W e giv e the decomp osition by follo w ing the elemen tary approac h employ ed in [22 , 18, 11]. The remainder of the pap er is organized as follo ws. In Section 2, w e pr o vide the notation and terminology as w ell as a few b asic facts on the T erwilliger algebra and wr eath p ro duct of asso ciation sc hemes that will b e used thr oughout. In Section 3, w e discu s s the structure of the wreath pro d u ct of one-class asso ciation schemes and compute the dimension of its T erw illiger algebra. I n S ection 4, we study T erw illiger algebras of wr eath p o wers of one-class asso ciation sc hemes and their irreducible mo du les. In S ection 5, we mak e a concluding remark and men tion a few r elated op en problems. 2 Preliminaries In this section we fir st briefly recall the notation and some b asic facts ab out asso ciation s c hemes and the T erw illiger algebra of a sc heme that are needed to dedu ce our results. Th en we recall the definition of wreath pro d uct of asso ciation sc hemes. F or more information on the topics co v ered in this section, w e refer the reader to [3, 5, 19, 20, 17]. 2.1 Asso ciation schemes and their T erwilliger algebras Let X denote an n -elemen t set, and let M X ( C ) denote the C -algebra of matrices w hose rows and columns are ind exed by X . Let X = ( X, { R i } 0 ≤ i ≤ d ) b e a d -class comm utativ e asso ciation sc heme of order n . T he d + 1 relations R i ⊆ X × X := { ( x, y ) : x, y ∈ X } are conv enien tly describ ed b y their { 0 , 1 } -adjacency matrices A 0 , A 1 , . . . , A d defined by ( A i ) xy = 1 if ( x, y ) ∈ R i ; 0 otherwise. The interse ction numb ers p h ij are defined in terms of the relations for the sc h eme by p h ij = |{ z ∈ X : ( x, z ) ∈ R i , ( z , y ) ∈ R j }| where ( x, y ) is a member of the relation R h . The definition of an asso ciation sc heme is equiv alen t to the follo wing four axioms: (1) A 0 = I , (2) A 0 + A 1 + · · · + A d = J , (3) A t i = A i ′ for some i ′ ∈ { 0 , 1 , . . . , d } , (4) A i A j = d P h =0 p h ij A h , for all i, j ∈ { 0 , 1 , . . . , d } where I = I n and J = J n are the n × n identit y matrix and all-ones matrix, resp ectiv ely , and A t denotes the transp ose of the matrix A . The sc heme is symmetric if A i ′ = A i for all i , and is comm utativ e if p h ij = p h j i for all h, i, j ; and thus A i A j = A j A i . 2 If the s c heme is commuta tiv e, the adjacency matrices generate a ( d + 1)-dimensional com- m u tative sub algebra M = h A 0 , A 1 , . . . , A d i of the full matrix algebra M X ( C ) o ver the field of complex n u m b ers C . The algebra M is kn o wn as th e Bose-Mesner algebra of th e scheme. T h e Bose-Mesner algebra for a comm utativ e asso ciation sc heme, b eing semi-simple, admits a second basis E 0 , E 1 , . . . , E d of primitive idemp oten ts. Note that M is also closed u n der the Hadamard (en trywise) multiplicatio n “ ◦ ” of matrices. So there are nonnegativ e r eal num b ers q h ij called the Krein parameters, suc h that (4) E i ◦ E j = | X | − 1 d X h =0 q h ij E h . There exist t wo sets of the ( d + 1) 2 complex num b ers p j ( i ) and q j ( i ) according to the d + 1 expressions A j = d X i =0 p j ( i ) E i , and E j = | X | − 1 d X i =0 q j ( i ) A i . The n umb er p j ( i ) is c haracterized b y the relation A j E i = p j ( i ) E i . That is p j ( i ) is the eigen v alue of A j , asso ciated with the eigenspace spann ed b y the columns of E i , o ccurring with the m ultiplicit y m i = rank( E i ). W e define P to b e the ( d + 1) × ( d + 1) m atrix whose ( i, j )-en try is p j ( i ). This P is referred to as the charact er table or fir st eigenmatrix of the scheme X . Giv en an n -elemen t set X , and th e C -algebra M X ( C ) (or M n ( C )), by the stand ard m o dule of X , w e mean the n -dimensional ve ctor space V = C X = L x ∈ X C ˆ x of column vecto r s whose co ordinates are indexed by X . Here for eac h x ∈ X , w e d en ote b y ˆ x th e column v ector with 1 in the x th p osition, and 0 elsewhere. Observe that M X ( C ) acts on V by left m u ltiplication. W e endow V with the Hermitian inner pr o duct defin ed by h u, v i = u t v ( u, v ∈ V ). F or a giv en asso ciation sc h eme X = ( X, { R i } 0 ≤ i ≤ d ), V can b e written as the direct sum of V i = E i V where V i are the maximal common eigenspaces of A 0 , A 1 , . . . , A d . Giv en an element x ∈ X , let R i ( x ) = { y ∈ X : ( x, y ) ∈ R i } . The v alencies k 0 , k 1 , . . . , k d of X are denoted by k i = | R i ( x ) | = p 0 ii ′ . Let V ∗ i = V ∗ i ( x ) = L y ∈ R i ( x ) C ˆ y . Both R i ( x ) and V ∗ i are referred to as the i th sub c onstituent of X with resp ect to x . Let E ∗ i = E ∗ i ( x ) b e th e orthogonal pro jection map from V = d L i =0 V ∗ i to the i th sub constituent V ∗ i . So, E ∗ i can b e r epresen ted by the diagonal matrix giv en b y ( E ∗ i ) y y =  1 if ( x, y ) ∈ R i 0 if ( x, y ) / ∈ R i . The m atrices E ∗ 0 , E ∗ 1 , . . . , E ∗ d are linearly indep enden t, and they form a basis for a su balgebra M ∗ = M ∗ ( x ) = h E ∗ 0 , E ∗ 1 , . . . , E ∗ d i of M X ( C ). Let A ∗ i = A ∗ i ( x ) b e the diagonal m atrix in M X ( C ) with y y en try ( A ∗ i ) y y = n · ( E i ) y x . Then w e ha ve (1) A ∗ 0 = I , (2) A ∗ 0 + A ∗ 1 + · · · + A ∗ d = nE ∗ 0 , (3) A ∗ i A ∗ j = d P h =0 q h ij A ∗ h = A j A i for all i, j . 3 F urthermore, we also hav e A ∗ j = d X i =0 q j ( i ) E ∗ i and E ∗ j = 1 n d X i =0 p j ( i ) A ∗ i . Th us, A ∗ 0 , A ∗ 1 , . . . , A ∗ d form a second basis for M ∗ . T he algebra M ∗ is s ho wn to b e comm utativ e, semi-simple s ubalgebra of M X ( C ). This algebra is called the d ual Bose-Mesner algebra of X with resp ect to x . Let X = ( X , { R i } 0 ≤ i ≤ d ) den ote a s c heme and fix a vertex x ∈ X , and let T = T ( x ) denote the subalgebra of M X ( C ) generated by the Bose-Mesner algebra M and the d ual Bose-Mesner algebra M ∗ . W e call T the T erwil liger algebr a of X with r esp e ct to x . The facts d iscussed in th e rest of the section will b e u seful when we describ e the irreducible T -mo du les for our sc hemes in the su bsequent s ections. Lemma 2.1 [19] L et X = ( X, { R i } 0 ≤ i ≤ d ) denote a d -class asso ciation scheme. F or an arbitr ary fixe d vertex x ∈ X , let T = T ( x ) . Ther e exists a set Φ = Φ( x ) and a b asis { e λ : λ ∈ Φ } of the c enter of T such that (i) I = P λ ∈ Φ e λ , (ii) e λ e µ = δ λµ e λ (for al l λ, µ ∈ Φ ). W e r efer to e λ as the c entr al primitive idemp otents of T . Let V = C X denote the standard mo dule. Lemma 2.2 [19] L et X = ( X , { R i } 0 ≤ i ≤ d ) denote a scheme and fix a vertex x ∈ X , and let T = T ( x ) . L et { e λ : λ ∈ Φ } b e the c entr al primitive idemp otents of T . (i) V = P λ ∈ Φ e λ V (Ortho gonal dir e ct sum). M or e over, e λ : V → e λ V is an ortho gonal pr oje c- tion for al l λ ∈ Φ . (ii) F or e ach irr e ducible T -mo dule W , ther e is a uni q ue λ ∈ Φ such that W ⊆ e λ V . We r efer to λ as the typ e of W . (iii) L et W and W ′ denote irr e ducible T - mo dules. Then W and W ′ ar e T -i somorph ic if and only if W and W ′ have the same typ e. (iv) F or al l λ ∈ Φ , e λ V c an b e de c omp ose d as an ortho gonal dir e ct sum of irr e ducible T -mo dules of typ e λ . (v) R eferring to (iv), the numb er of irr e ducible T -mo dules in the de c omp osition is indep endent of the de c omp osition. We shal l denote this numb er b y mult ( e λ ) (or simply mult ( λ ) ) and r efer to it as the multiplicity (in V ) of the irr e duci ble T -mo dule of typ e λ . The s et of tr ip le p ro du cts E ∗ i A j E ∗ h in T ( x ) pla ys a sp ecial role in our stud y , so we lo ok at them little closely . W e can view E ∗ i A j E ∗ h as a linear map from V ∗ h → V ∗ i suc h that E ∗ i A j E ∗ h ˆ y = X z ∈ R i ( x ) ∩ R j ( y ) ˆ z for eac h ˆ y ∈ V ∗ h . T erw illiger prov ed the follo wing k ey fact in [19, Lemma 3.2]. 4 Prop osition 2.3 F or 0 ≤ h, i, j ≤ d , E ∗ i A j E ∗ h = 0 if and only if p h ij = 0 . Note that for ev ery i ∈ { 0 , 1 , . . . , d } , A i and E ∗ i can b e w r itten in terms of the triple p ro ducts E ∗ i A j E ∗ h . Thus, the triple pro du cts E ∗ i A j E ∗ h generate the T erwilliger algebra. It is often easier to find the irr educible mo du les if we w ork with the trip le pro ducts E ∗ i A j E ∗ h instead of A i and E ∗ i . 2.2 The W reath pro duct of asso ciation sc hemes W e briefly recall the n otion of the wreath pro duct. Let X = ( X, { R i } 0 ≤ i ≤ d ) and Y = ( Y , { S j } 0 ≤ j ≤ e ) b e asso ciation schemes of order | X | = m and | Y | = n . The w reath pro d uct X ≀ Y of X and Y is d efi ned on the set X × Y ; but w e tak e Y = { y 1 , y 2 , . . . , y n } , and regard X × Y as the disjoin t union of n copies X 1 , X 2 , . . . , X n of X , where X j = X × { y j } . The relations on X 1 ∪ X 2 ∪ · · · ∪ X n is defined b y the follo wing rule: F or any j , the r elations b et we en the elements of X j are d etermined by the asso ciation relations b et ween the fir s t co ordinates in X . F or an y i and j , the relations b et w een X i and X j are determined by the association relation of th e second co ordinates y i and y j in Y and the relation is indep endent from the fi rst co ordinates. W e may arrange the r elations W 0 , W 1 , . . . , W d + e of X ≀ Y as f ollo ws: • W 0 = { (( x, y ) , ( x, y )) : ( x, y ) ∈ X × Y } • W k = { (( x 1 , y ) , ( x 2 , y )) : ( x 1 , x 2 ) ∈ R k , y ∈ Y } , for 1 ≤ k ≤ d ; and • W k = { (( x 1 , y 1 ) , ( x 2 , y 2 )) : x 1 , x 2 ∈ X , ( y 1 , y 2 ) ∈ S k − d } for d + 1 ≤ k ≤ d + e . Then the wreath pro d uct X ≀ Y = ( X × Y , { W k } 0 ≤ k ≤ d + e ) is a ( d + e )-cla s s asso ciation scheme. It is clear that X ≀ Y is comm utativ e (resp. sym m etric) if and only if X and Y are. Let A 0 , A 1 , . . . , A d and C 0 , C 1 , . . . , C e b e the adjacency matrices of X and those of Y , r esp ectiv ely . Then the adja- cency matrices W k of X ≀ Y are giv en b y W 0 = C 0 ⊗ A 0 , W 1 = C 0 ⊗ A 1 , . . . , W d = C 0 ⊗ A d , W d +1 = C 1 ⊗ J m , . . . , W d + e = C e ⊗ J m , where “ ⊗ ” d enotes the Kr on eck er pro du ct: A ⊗ B = ( a ij B ) of tw o matrices A = ( a ij ) and B . With the ab o ve ordering of the asso ciation relations of X ≀ Y , the relation table of the w reath pro du ct is describ ed by R ( X ≀ Y ) = d + e X k =0 k · A k = I n ⊗ R ( X ) + { R ( Y ) + d ( J n − I n ) } ⊗ J m . 3 The Dimension of the T erwilliger Algebra of K n 1 ≀ K n 2 ≀ · · · ≀ K n d In this section, we first calcula te the dimension of th e T erwilliger algebra of the wreath pr o duct of one-class asso ciation sc hemes. W e also show that th ese wreath pr o duct sc hemes satisfy the triple regularity prop erty . 5 Throughout the pap er, for the n otational s im p licit y , by [ n ] we denote the set of int egers { 1 , 2 , . . . , n } , and by K n = H (1 , n ) we denote b oth the one-class asso ciation scheme ([ n ] , { R 0 , R 1 } ) with A 1 = J − I , and the complete graph on n v ertices. Let X = ( X , { R i } 0 ≤ i ≤ d ) denote the d -class asso ciation sc heme K n 1 ≀ K n 2 ≀ · · · ≀ K n d with X = [ n 1 ] × [ n 2 ] × · · · × [ n d ] = { ( a 1 , a 2 , . . . , a d ) : a i ∈ [ n i ] , f or i = 1 , 2 , . . . , d } . Let (1 , 1 , . . . , 1) ∈ X b e a fixed b ase v ertex x of X . Without loss of generalit y , we can arrange the asso ciation relations such th at • R 1 ( x ) = { ( a, 1 , 1 , . . . , 1) : a ∈ { 2 , 3 , . . . , n 1 }} , • f or i = 2 , 3 , . . . , d , R i ( x ) = { ( a 1 , a 2 , . . . , a i − 1 , b, 1 , 1 , . . . , 1) : a k ∈ [ n k ] for k ∈ { 1 , 2 , . . . , i − 1 } , b ∈ [ n i ] − { 1 }} . W e obs erv e that k i = | R i ( x ) | = ( n i − 1) i − 1 Q k =1 n k . W e can arrange ro ws and columns of th e relation table of X by the order of parts in the partition X = R 0 ( x ) ∪ R 1 ( x ) ∪ · · · ∪ R d ( x ). Example 3.1 The fol lowing is the r elation table for the wr e ath pr o duct of thr e e asso ciation schemes K 2 , K 2 and K 3 arr anging the elements in the or der describ e d ab ove. R ( K 2 ≀ K 2 ≀ K 3 ) =                      0 1 2 2 3 3 3 3 3 3 3 3 1 0 2 2 3 3 3 3 3 3 3 3 2 2 0 1 3 3 3 3 3 3 3 3 2 2 1 0 3 3 3 3 3 3 3 3 3 3 3 3 0 1 2 2 3 3 3 3 3 3 3 3 1 0 2 2 3 3 3 3 3 3 3 3 2 2 0 1 3 3 3 3 3 3 3 3 2 2 1 0 3 3 3 3 3 3 3 3 3 3 3 3 0 1 2 2 3 3 3 3 3 3 3 3 1 0 2 2 3 3 3 3 3 3 3 3 2 2 0 1 3 3 3 3 3 3 3 3 2 2 1 0                      We note that any table obtaine d fr om this table by p ermuting the or der of r ows and c orr esp onding c olumns simultane ously r epr esents the same asso ciation scheme. Lemma 3.2 L et X = ( X , { R i } 0 ≤ i ≤ d ) = K n 1 ≀ K n 2 ≀ · · · ≀ K n d . Then the c omplete list of nonzer o p h ij wher e h, i, j ∈ { 0 , 1 , 2 , . . . , d } is as fol lows. (1) F or h = 0 , k 0 = p 0 00 = 1 , k 1 = p 0 11 = n 1 − 1 , and k j = p 0 j j = ( n j − 1) j − 1 Q l =1 n l for j = 2 , 3 , . . . , d . (2) F or h = 1 , 2 , . . . , d , 6 ( a ) p h hh = ( n h − 2) h − 1 Q l =1 n l . ( b ) p h j j = ( n j − 1) j − 1 Q l =1 n l for h + 1 ≤ j ≤ d ( c ) p h j h = p h hj = ( n j − 1) j − 1 Q l =1 n l for 1 ≤ j ≤ h − 1 ( d ) p h 0 h = p h h 0 = 1 Pro of: It is straigh tforward to calculate the in tersection n u m b ers. Due to this lemma, we ha ve the follo wing list of non-zero triple pro du cts in T . Theorem 3.3 The c omp lete list of nonzer o triple pr o ducts E ∗ i A j E ∗ h among al l h, i, j ∈ { 0 , 1 , 2 · · · , d } in X = K n 1 ≀ K n 2 ≀ · · · ≀ K n d is given as fol lows. (1) E ∗ i A i E ∗ 0 for 0 ≤ i ≤ d (2) E ∗ h A h E ∗ h if and only if n h ≥ 3 for 1 ≤ h ≤ d , (3) E ∗ j A j E ∗ h for 2 ≤ h + 1 ≤ j ≤ d (4) E ∗ j A h E ∗ h for 1 ≤ j + 1 ≤ h ≤ d (5) E ∗ h A j E ∗ h for 1 ≤ j + 1 ≤ h ≤ d Pro of: Imm ediate from the ab o ve lemma by Prop osition 2.3. In order to calculate the dimension of the T erwilliger algebra for the d -class asso ciation sc heme X = ( X , { R i } 0 ≤ i ≤ d ) = K n 1 ≀ K n 2 ≀ · · · ≀ K n d . Let x ∈ X b e a fixed v ertex, and consider the subspace T 0 = T 0 ( x ) of T = T ( x ) spanned by { E ∗ i A j E ∗ h : 0 ≤ i, j, h ≤ d } . It is easy to see th at T is generated by T 0 as an algebra since T 0 con tains A i and E ∗ i for all i , but in general, T 0 ma y b e a prop er linear subspace of T . How ev er, w e will see that T 0 = T for K n 1 ≀ K n 2 ≀ · · · ≀ K n d shortly . First we ha ve the follo win g dimension form u la for T 0 . Theorem 3.4 L et X = K n 1 ≀ K n 2 ≀ · · · ≀ K n d . Then the dimension of T 0 is given by dim( T 0 ) = ( d + 1) 2 + 1 2 d ( d + 1) − b wher e b is the numb er of K 2 factors in the wr e ath pr o duct. In p articular, ( d + 1) 2 +  d 2  ≤ dim ( T 0 ) ≤ ( d + 1) 2 +  d + 1 2  Pro of: In the Th eorem 3.3, the n umb er of non-zero trip le pr o ducts can b e coun ted as d+1 from (1), d − b fr om (2), 1 2 d ( d − 1) from (3), and d ( d + 1) from (4) and (5). As th ey are indep end en t of eac h other we hav e the dim ( T 0 ( x )) = d + 1 + d − b + 1 2 d ( d − 1) + d ( d + 1) = ( d + 1) 2 + 1 2 d ( d + 1) − b as 7 desired. The case when all n i = 2 give s th e lo wer b ound as in this case b = d . The u pp er b ound is giv en b y the case wher e n i ≥ 3 for all i . In such a situ ation b = 0. This completes the pro of. W e n o w show that T = T 0 for the w r eath p ro duct sc heme K n 1 ≀ K n 2 ≀ · · · ≀ K n d ; so the scheme has the triple-regularit y prop er ty . The concept of triple-regularit y wa s fir st studied by T erwilliger. F or more information on it, we refer to [14, p.120]). W e use the follo w ing equiv alen t prop erties of triple-regularit y obs er ved b y Munemasa. Prop osition 3.5 ( [16]) L e t X b e a c ommutative asso ciation scheme. Then the fol lowing ar e e quivalent. 1 . X is triply r e gular; i.e. X has the pr op erty that the size of the set R i ( x ) ∩ R j ( y ) ∩ R h ( z ) dep ends only on the set { i, j, h, l , m, n } wher e ( x, y ) ∈ R l , ( x, z ) ∈ R m and ( y , z ) ∈ R n . 2 . A i E ∗ j A h ∈ T 0 for any h, i, j . 3 . T ( x ) = T 0 ( x ) for x ∈ X . According to this pr op osition, it suffices to v erify that all triple pro du cts A i E ∗ h A j b elong to T 0 in order to sho w that T = T 0 for K n 1 ≀ K n 2 ≀ · · · ≀ K n d . Lemma 3.6 F or the d -class scheme K n 1 ≀ K n 2 ≀ · · · ≀ K n d , we have the fol lowing. (1) A i E ∗ h A j = ( A i E ∗ h )( E ∗ h A j ) for al l h, i, j ∈ { 0 , 1 , . . . , d } . (2) A h E ∗ h = P h j =0 E ∗ j A h E ∗ h for al l h ∈ { 0 , 1 , . . . , d } . (3) F or 0 ≤ i < h ≤ d , A i E ∗ h = E ∗ h A i E ∗ h . (4) F or 0 ≤ h < i ≤ d , A i E ∗ h = E ∗ i A i E ∗ h . Pro of: (1) It is trivially true as E ∗ h are idemp oten ts. (2) The nonzero en tries of A h E ∗ h are the nonzero entries of the columns of A h indexed by v er tices in R h ( x ). The rest of the en tries are zero. Th e columns of A h indexed by v ertices in R h ( x ) ha v e 1 in ro w s indexed by the v ertices in R 0 ( x ) ∪ R 1 ( x ) ∪ · · · ∪ R h − 1 ( x ). In addition the rows indexed by the v ertices R h ( x ) ha ve zero in the diagonal blo cks of size ( Q h − 1 i =1 n i ) × ( Q h − 1 i =1 n i ), and 1 elsewhere. These are essen tially h P j =0 E ∗ j A h E ∗ h . (3) If i < h , the nonzero entries of A i E ∗ h are the nonzero entries of the columns of A i indexed b y vertice s in R h ( x ). Th e rest of the en tries are zero. The columns of A i indexed b y vertic es in R h ( x ) hav e 1 in ro ws indexed by the v ertices in R h ( x ). The rest of the ent ries are zero. (4) If i > h , the nonzero entries of A i E ∗ h are the nonzero entries of the columns of A i indexed b y vertice s in R h ( x ). Th e rest of the en tries are zero. The columns of A i indexed b y vertic es in R h ( x ) h a ve 1 in ro ws indexed by the v ertices in R i ( x ). Th e rest of the ent r ies are zero. Th is completes the p ro of. 8 Lemma 3.7 In K n 1 ≀ K n 2 ≀ · · · ≀ K n d , (1) E ∗ h A h = P h j =0 E ∗ h A h E ∗ j for h ∈ { 0 , 1 , . . . , d } ; (2) for 0 ≤ i < h ≤ d , E ∗ h A i = E ∗ h A i E ∗ h ; (3) for 0 ≤ h < i ≤ d , E ∗ h A i = E ∗ h A i E ∗ i . Pro of: It f ollo ws from the pr evious lemma and the fact that the trans p ose of A i E ∗ h is E ∗ h A i . Lemma 3.8 F or K n 1 ≀ K n 2 ≀ · · · ≀ K n d , we have the fol lowing line ar c ombinations for A i E ∗ h A j . (1) F or h ∈ { 2 , 3 , . . . , d } , A h E ∗ h A h = ( n h − 1) h − 1 Y k =1 n k ! d X m =0 h − 1 X l =0 h − 1 X n =0 E ∗ l A m E ∗ n + ( n h − 2) h − 1 Y k =1 n k ! ( d X m =0 h − 1 X n =0 E ∗ h A m E ∗ n + d X m =0 h X l =0 E ∗ l A m E ∗ h ) . (2) ( a ) F or d ≥ j > h ≥ 2 , A h E ∗ h A j = ( n h − 1) h − 1 Y k =1 n k ! d X m =0 h − 1 X l =0 E ∗ l A m E ∗ j + ( n h − 2) h − 1 Y k =1 n k ! d X m =0 E ∗ h A m E ∗ j ; ( b ) for 2 ≤ j < h ≤ d , A h E ∗ h A j = ( n j − 1) j − 1 Y k =1 n k ! d X m =0 h − 1 X l =0 E ∗ l A m E ∗ h . (3) ( a ) F or d ≥ i > h ≥ 2 , A i E ∗ h A h = ( n h − 1) h − 1 Y k =1 n k ! d X m =0 h − 1 X n =0 E ∗ i A m E ∗ n + ( n h − 2) h − 1 Y k =1 n k ! d X m =0 E ∗ i A m E ∗ h ; ( b ) for i < h , A i E ∗ h A h = n 1 · · · n i − 1 ( n i − 1) d X m =0 h − 1 X n =0 E ∗ h A m E ∗ n . (4) ( a ) F or d ≥ i > h ≥ 2 , A i E ∗ h A i = n 1 · · · n h − 1 ( n h − 1) d X m =0 E ∗ i A m E ∗ i ; 9 ( b ) for 2 ≤ i < h ≤ d , A i E ∗ h A i = n 1 · · · n i − 1 ( n i − 1) d X m =0 E ∗ h A m E ∗ h . Pro of: (1) Applying the iden tities in Lemmas 3.6 and 3.7 to A i E ∗ h A j = ( A i E ∗ h )( E ∗ h A j ) w e ha ve A i E ∗ h A j = k h d X m =0 h − 1 X l =0 h − 1 X n =0 E ∗ l A m E ∗ n +( k h − ( k 0 + · · · + k h − 1 )) ( d X m =0 h − 1 X n =0 E ∗ h A m E ∗ n + D X m =0 h X l =0 E ∗ l A m E ∗ h ) with k h = n 1 n 2 · · · n h − n 1 n 2 · · · n h − 1 as desired. (2) The pro of of part (a) follo ws from Lemma 3.6(1) and 3.7(3), while (b) follo ws from Lemma 3.6(1) and 3.7 (2). (3) The p ro of is a similar to part (2). (4) The pro of of (a) follo ws from Lemma 3.6(3) and 3.7(3), and (b) follo ws from Lemma 3.6 (2) and 3.7(2). Lemma 3.9 In K n 1 ≀ K n 2 ≀ · · · ≀ K n d , for i, j, h ∈ { 0 , 1 , . . . , d } , su pp ose that no two of i, j, h ar e e qual. ( i ) If i > h and j > h , then A i E ∗ h A j = ( n h − 1) h − 1 Y k =1 n k ! d X m =0 E ∗ i A m E ∗ j . ( ii ) If i < h < j , then A i E ∗ h A j = ( n i − 1) i − 1 Y k =1 n k ! d X m =0 E ∗ h A m E ∗ j . ( iii ) If i > h > j , then A i E ∗ h A j = ( n j − 1) j − 1 Y k =1 n k ! d X m =0 E ∗ i A m E ∗ h . ( iv ) If i < h and j < h , then ( a ) for i < j , A i E ∗ h A j = ( n i − 1) i − 1 Y k =1 n k ! E ∗ h A j E ∗ h ; 10 ( b ) for i > j , A i E ∗ h A j = ( n j − 1) j − 1 Y k =1 n k ! E ∗ h A i E ∗ h . Pro of:(i),(ii) and (iii) are similar to L emm a 3.2.8 (iv)(a) Lemmas 3.6(ii) and 3.7(ii) giv es us nonzero entries of A i E ∗ h A j o ccur in the ro ws and columns indexed by the ve rtices R h ( x ). Consider the diagonal blo c ks of size k j × k j inside A i E ∗ h indexed by the rows and columns of vertic es in R h ( x ). These diagonal blo c ks hav e k i 1’s in eac h ro w and column. T he off diagonal entries are all zero. I n a similar mann er consider diagonal blo c ks of size k j × k j inside A i E ∗ h A j indexed by the r o ws and columns of vertic es in R h ( x ). These diagonal blo c ks are all zero and the off diagonal en tries are 1. This observ ation giv es us A i E ∗ h A j = n 1 · · · n i − 1 ( n i − 1) E ∗ h A j E ∗ h (b) S im ilar to part (a). Theorem 3.10 The d -class asso ciation scheme K n 1 ≀ K n 2 ≀ · · · ≀ K n d is triply r e gular and T = T 0 . Pro of: It is straigh tforw ard to chec k that eac h of the rest of A i E ∗ h A j that are not co vered in Lemma 3.8 and Lemma 3.9, also can b e expressed as linear com bination of the generators of T 0 . Th us the conclusion follo ws from Prop osition 3.5. In sum mary , w e ha ve the follo w ing. Theorem 3.11 The dimension of the T erwil liger algebr a T of K n 1 ≀ K n 2 ≀ · · · ≀ K n d is dim ( T ) = ( d + 1) 2 + 1 2 d ( d + 1) − b wher e b denotes the numb er of factors K n i with n i = 2 . Pro of: Imm ediate consequence of Lemma 3.4, P rop osition 3.5, and Theorem 3.10 . 4 T erwilliger Algebras of W reath P o w ers of K m In this section we describ e the structure of the T erwilliger algebra of th e wr eath p ow er ( K m ) ≀ d = K m ≀ K m ≀ · · · ≀ K m of d copies of the one-class asso ciation scheme K m . W e b egin the section with the description of the T erw illiger algebra of th e one-class asso ciation sc h eme K m of order m . The n on trivial r elation graph of K m is the adjacency matrix of the complete graph K m whic h is also view ed as Hamming graph H (1 , m ). W e then describ e the T erwilliger algebra of ( K m ) ≀ 2 whose fi r st relation graph is th e complete m -partite strongly regular graph with parameters ( v , k , λ, µ ) = ( m 2 , m ( m − 1) , m ( m − 2) , m ( m − 1)). W e th en compare th e com b in atorial structures 11 of w reath s q u are ( K m ) ≀ 2 and cub e ( K m ) ≀ 3 of K m to describ e irreducible T -mo d ules of ( K m ) ≀ 3 b y extending those of ( K m ) ≀ 2 . Similarly , the structure of the irreducible T -mo dules of ( K m ) ≀ d for an y higher d will b e describ ed f r om that of ( K m ) ≀ ( d − 1) . It is sho wn that all non-p rimary irreducible T -mo d ules of wr eath p o we rs of K m are of d imension 1. W e conclude th e section by describing the T erwilliger algebra of the d -p o wer ( K m ) ≀ d for an arbitrary d ≥ 2. 4.1 The T erwilliger algebra of K m Let K m = ([ m ] , { R 0 , R 1 } ), and let x = 1. T hen R 1 ( x ) = { 2 , 3 , . . . , m } and, we m a y denote A 1 b y A 1 = J − I =   0 1 t 1 J m − 1 − I m − 1   where 1 is the ( m − 1)-dimensional column v ector all of w hose en tr ies are 1. Remark 4.1 By The or em 3.11, we know that the dimension of the T erwil lige r algebr a of K m is 5 if m > 2 and 4 if m = 2 . A lso by The or em 3.3, al l the matric es in the T erwil liger algebr a of K m is a line ar c ombination of the matric es E ∗ 0 A 0 E ∗ 0 , E ∗ 0 A 1 E ∗ 1 , E ∗ 1 A 1 E ∗ 0 , E ∗ 1 A 0 E ∗ 1 , and E ∗ 1 A 1 E ∗ 1 . (If m = 2 , then E ∗ 1 A 1 E ∗ 1 = 0 .) If we s et E 11 = E ∗ 0 A 0 E ∗ 0 , E 12 = E ∗ 0 A 1 E ∗ 1 , E 21 = 1 m − 1 E ∗ 1 A 1 E ∗ 0 , E 22 = 1 m − 1 ( E ∗ 1 A 0 E ∗ 1 + E ∗ 1 A 1 E ∗ 1 ) , then these matrices form a subalgebra U of T ( x ) as its multiplica tion table is giv en by E 11 E 12 E 21 E 22 E 11 E 11 E 12 0 0 E 12 0 0 E 11 E 12 E 21 E 21 E 22 0 0 E 22 0 0 E 21 E 22 Considering the isomorp hism b et wee n U and M 2 ( C ) that tak es E 11 7→  1 0 0 0  ; E 12 7→  0 1 0 0  ; E 21 7→  0 0 1 0  ; E 22 7→  0 0 0 1  . w e see that T ( x ) = M 2 ( C ) ⊕ M 1 ( C ) b y W edderburn -Artin’s Theorem (cf. [9, Sec. 2.4]). Sp ecif- ically , if we set F = E ∗ 1 A 0 E ∗ 1 − E 22 , then it tu rns out that F X = 0 for all X ∈ U . Th is gives us T ( x ) = C F ⊕ U . While F = 0 for m = 2, F 6 = 0 for all m > 2. T h erefore, we reasserted th e follo wing. Theorem 4.1 [15] The T e rwil lige r algebr a of K m c an b e describ e d as fol lows: T ( x ) ∼ =  M 2 ( C ) if m = 2 M 2 ( C ) ⊕ M 1 ( C ) if m > 2 . 12 4.2 The T erwilliger algebra of ( K m ) ≀ 2 and its ir reducible mo dules Let X = ( K m ) ≀ 2 = ( X , { R 0 , R 1 , R 2 } ) b e the wreath square of K m . Without loss of generalit y , w e arrange the relations so that the fi rst r elation graph ( X, R 1 ) is to b e the complete m -p artite strongly regular grap h with p arameters ( v , k , λ, µ ) = ( m 2 , m ( m − 1) , m ( m − 2) , m ( m − 1)), which is also the wreath squ are of the complete graph K m . Let X = [ m ] × [ m ] = { ( i, j ) : i, j ∈ [ m ] } , and let x = (1 , 1). W e will refer to (1 , 1) as the base v ertex x . Th en R 1 ( x ) = { ( i, j ) : i ∈ [ m ] , j ∈ { 2 , 3 , . . . , m }} , R 2 ( x ) = { ( i, 1) : i ∈ { 2 , 3 , . . . , m }} . Let the adjacency matrices A i and the relation table R of X b e decomp osed according to the partition X = R 0 ( x ) ∪ R 1 ( x ) ∪ R 2 ( x ). Then, A 1 =   0 1 t 2 0 t 1 1 2 B 2 L 0 1 L t B 1   , A 2 =   0 0 t 2 1 t 1 0 2 C 2 N 1 1 N t C 1   , R =   0 1 t 2 2 1 t 1 1 2 B 2 + 2 C 2 L 2 1 1 L t 2 C 1   . where 1 2 and 1 1 are all-ones column vec tors of size m ( m − 1) and ( m − 1), resp ectiv ely; 0 2 and 0 1 are all-zeros column v ectors of size m ( m − 1) and m − 1, r esp ectiv ely; L and N are m ( m − 1) × ( m − 1) all-ones and all-zeros matrices, resp ectiv ely; B 2 = J m ( m − 1) − ( I m − 1 ⊗ J m ), and B 1 is a ( m − 1) × ( m − 1) zero matrix, wh ile C 2 = I m − 1 ⊗ ( J m − I m ) and C 1 = J m − 1 − I m − 1 . W e n ote that A 2 = 1 m ( m − 1) A 2 1 − m − 2 m − 1 A 1 − I . W e also note that ( K m ) ≀ 2 is form ally self-dual P - and Q -p olynomial asso ciation sc heme with its first and second eigenmatrices P = Q =   1 m ( m − 1) m − 1 1 0 − 1 1 − m m − 1   . The c haracteristic p olynomial of A 1 is θ 2 +( µ − λ ) θ + ( µ − k ) = 0, and the eigen v alues of A 1 are giv en b y k = m ( m − 1) , r = 0 , and s = − m with multiplicities, 1 , m ( m − 1) , and m − 1, resp ectiv ely . The induced subgraphs of the graph ( K m ) ≀ 2 on the verte x sets R 1 ( x ) and R 2 ( x ) with adjacency matrices B 2 and B 1 resp ectiv ely , are called the sub constituents of the graph with resp ect to x . Set λ = p 1 11 = m ( m − 2), µ = p 2 11 = m ( m − 1), k 1 = m 1 = p 0 11 = m ( m − 1) , and k 2 = m 2 = m − 1. F urthermore, B 2 1 2 = m ( m − 1) 1 2 and C 1 1 1 = ( m − 1) 1 1 . The eigen v alues of B 2 are m ( m − 2), − m and 0 with multiplici ties 1, m − 2 and m 2 − 2 m + 1 resp ectiv ely . F or B 1 , 0 is the only eigen v alue w ith multiplic it y m − 1. Cameron, Go ethals and S eidel in tr o duced the concept of restricted eigen v alues an d eigen v ec- tors [6]. An eigen v alue of B 2 (resp. B 1 ) is called restricted if it h as an eigen v ector orthogonal to the all-ones v ector of size k 1 (resp. k 2 ). T omiyama and Y amaza k i u sed r estricted eigen ve ctors, the eigen v ectors asso ciated w ith the restricted eigen v alue of B i that are orthogonal to 1 i , to describ e the sub constituen t algebra of a 2-class asso ciation sc heme constru cted from a strongly regular graph. In order to describ e the irreducible T -mo dules of our scheme ( K m ) ≀ 2 , we will n eed the result of Theorem 5.1 in [6 ] for our sc h eme. Lemma 4.2 With the ab ove notations for ( K m ) ≀ 2 and its eige nvalues m ( m − 1) , 0 and − m , we have the fol lowing. 13 1. Su pp ose y is a r estricte d eige nve ctor of B 2 with an eigenvalue θ . Then y is the e i genve ctor of LL t with the eigenvalue − θ ( θ + m ) , and L t y i s the zer o ve ctor or the r estricte d ei genve ctor of B 1 with the eigenv alue − m − θ . In p articular, L t y is zer o if and only if θ ∈ { 0 , − m } . 2. Su pp ose z is a r estricte d eige nve ctor of B 1 with an eigenvalue θ ′ . Then z is the eigenve ctor of L t L with the eige nv alue − θ ′ ( θ ′ + m ) , and L z is the zer o ve ctor or the r estricte d eigenve ctor of B 2 with the eigenv alue − m − θ ′ . In p articular, L z i s zer o if and only if θ ′ ∈ { 0 , − m } . Pro of: It is immediate from the fact that B 2 is the adjacency matrix of th e str ongly regular graph with parameters ( m ( m − 1) , m ( m − 2) , m ( m − 3) , m ( m − 2)). The next three lemmas are results of T omiya m a and Y amazaki (rep orted in [22]) su ited to our asso ciation scheme ( K m ) ≀ 2 . Lemma 4.3 L et T denote the T erwil liger algebr a of ( K m ) ≀ 2 . Then with the ab ove notations in this subse ction, we have the f ol lowing. (1) Supp ose y is a r estricte d ei g enve ctor of B 2 with an eigenvalue θ . Then the ve ctor sp ac e W over C which is sp anne d by (0 , y t , 0 t 1 ) t is a thin irr e ducible T -mo dule over C and dim W = 1 if θ ∈ { 0 , − m } . (2) Supp ose z is a r estricte d eigenve ctor of B 1 with an eigenvalue θ ′ . Then the ve ctor sp ac e W ′ over C which is sp anne d by (0 , 0 t 2 , z t ) t is a thin irr e duci b le T - mo dule over C and dim W ′ = 1 if θ ′ ∈ { 0 , − m } . Pro of: (1) Since y is the restricted eigen v ector of B 2 asso ciated w ith eigen v alue θ , so ( B 2 y ) t = ( θ y ) t . Hence, (0 , ( B 2 y ) t , 0 t 1 ) t ∈ sp an { (0 , y t , 0 t 1 ) t } , and W is spanned by (0 , y t , 0 t 1 ) t . Observe that A 1 (0 , y t , 0 t 1 ) t = (0 , ( B 2 y ) t , 0 t 1 ) t , and also y is orthogonal to 1 2 with the asso ciated eigen v alue 0 or − m . By L emm a 4.2 L t y = 0 1 . Therefore, W is A 1 -in v arian t and thus M -in v ariant . W is also M ∗ -in v arian t. (2) The p ro of is similar to that of (1). Lemma 4.4 F or the asso ciation scheme ( K m ) ≀ 2 , let V denote the standar d T -mo dule. Ther e exist irr e ducible T - mo dules { W i } 1 ≤ i ≤ m ( m − 1) and { W ′ j } 2 ≤ j ≤ m − 2 such that V = ( ⊕ W i ) ⊕ ( ⊕ W ′ j ) . Pro of: Using Gram Sc hmidt pro cess we can fi nd eigen vect ors v 1 , v 2 , · · · , v m ( m − 1) of B 2 suc h that { v i } 1 ≤ i ≤ m ( m − 1) span E ∗ 1 V ∼ = C m ( m − 1) , h v i , v j i = 0 for i 6 = j with v 1 b eing the all-ones vect or 1 2 . Let θ i b e th e eigen v alue of B 2 with resp ect to the eigen ve ctor v i for 2 ≤ i ≤ m ( m − 1). Then θ i ∈ { 0 , − m } for 2 ≤ i ≤ m ( m − 1). Let W i denote the linear span of ( h v i , v 1 i , 0 t 2 , 0 t 1 ) t , (0 , v t i , 0 t 1 ) t , and (0 , 0 t 2 , ( L t v i ) t ) t o ve r C . Then W i is a thin irreducible T -mod ule and W i ∩ W j = { 0 } for i 6 = j . Also, dimW i =  3 if i = 1 1 if 2 ≤ i ≤ m ( m − 1) Note that W 1 is the primary mo dule generated b y ( h v 1 , v 1 i , 0 t 2 , 0 t 1 ) t , (0 , v t 1 , 0 t 1 ) t , and (0 , 0 t 2 , ( L t v 1 ) t ) t o ve r C . F or eac h i , 2 ≤ i ≤ m ( m − 1), W i is generated by (0 , v t i , 0 t 1 ) t . Note that w 1 = L t v 1 14 is an eigen vecto r of B 1 . Let w 2 , · · · , w m − 1 b e the eigen v ectors of B 1 suc h that w 1 , · · · , w m − 1 span E ∗ 2 V ∼ = C m − 1 and h w i , w j i = 0 for i 6 = j . Let W ′ i b e the linear sp an of (0 , 0 t 2 , w t i ) t o ve r C for 2 ≤ i ≤ m − 1. W ′ i is a thin irreducible T -mo dule of dimens ion 1. Thus, we ha ve V = ( ⊕ W i ) ⊕ ( ⊕ W ′ j ) as d esir ed . Lemma 4.5 F or ( K m ) ≀ 2 , let { θ i } 1 ≤ i ≤ m ( m − 1) , { θ ′ i } 2 ≤ i ≤ m − 1 , { W i } 1 ≤ i ≤ m ( m − 1) and { W ′ i } 2 ≤ i ≤ m − 1 . Then the fol lowing hold. 1. F or al l i with 2 ≤ i ≤ m ( m − 1) , W 1 and W i ar e not T -isomorph i c . 2. F or al l i and j with 1 ≤ i ≤ m ( m − 1) and 2 ≤ j ≤ m − 1 , W i and W ′ j ar e not T -isomorphic. 3. F or i and j with 2 ≤ i, j ≤ m ( m − 1) , W i and W j ar e T - isomorphic if and only if θ i = θ j . 4. F or i and j with 2 ≤ i, j ≤ m − 1 , W ′ i and W ′ j ar e T - isomorphic. Pro of: It is straigh tforward fr om the p revious lemma according to Lemma 3.4 in [22]. Remark 4.2 In or der to describ e the T erwil liger algebr a of ( K m ) ≀ 2 , let Λ denote the index set for the isomorphism classes of irr e ducible T - mo dules. F ol lowing the r esults of L emmas 4.4 and 4.5, we c an gr oup the isomorph ic irr e ducible mo dules to gether, say V λ for λ ∈ Λ , so that V = L λ ∈ Λ V λ . By L emma 2.2, we know that for e ach subsp ac e V λ ther e is a unique c entr al idemp otent e λ such that V λ = e λ V . L et W b e an irr e duci ble T -mo dule in the de c omp osition of V . Then the map taking A ∈ e λ T to the endomorphism w 7→ A w wher e w ∈ W is an isomorphism. Henc e we have e λ V ∼ = E nd C W . Thus T = L λ ∈ Λ e λ T is isomorph i c to a dir e ct sum of c omplex matrix algebr a M k ( C ) wher e k = dim ( W ) as in the fol lowing. In what fol lows we use the notation M 1 ( C ) ⊕ l for the dir e c t sum M 1 ( C ) ⊕ M 1 ( C ) ⊕ · · · ⊕ M 1 ( C ) of l c opies of M 1 ( C ) . Theorem 4.6 L et T b e the T e rwil lige r algebr a of ( K m ) ≀ 2 . Then dim T = 12 and T ∼ = M 3 ( C ) ⊕ M 1 ( C ) ⊕ 3 . Pro of: By Lemm a 4.5, the list of non-isomorphic irredu cible T -mo d u les of ( K m ) ≀ 2 consists of (i) the primary mo dule W 1 of d im en sion 3, (ii) tw o one-dimensional non-isomorphic irr educible mo dules from W i , 2 ≤ i ≤ m ( m − 1), that r ep resen t t wo isomorphism classes corresp ond ing to the eigen v alues 0 and − m , and (iii) one one-dimensional irreducible m o dule wh ic h represen ting the class of W ′ j for all 2 ≤ j ≤ m − 1. 15 4.3 The T erwilliger algebra of ( K m ) ≀ 3 and its ir reducible mo dules W e n o w extend the irredu cible mo dules of ( K m ) ≀ 2 to fi nd irred u cible mo dules of ( K m ) ≀ 3 in this section, and th en generalize it to describ e the T erwilliger algebra f or ( K m ) ≀ d for an arbitrary d ≥ 3 in the next section. W e note that the T erw illiger algebra of ( K m ) ≀ 2 is generated by A 1 , E ∗ 0 and E ∗ 1 . As we mo v e on to three or higher wreath p ow ers of K m , the sc heme is n o longer P -p olynomial, so, w e must consid er many more, but w ithin 2 d + 1 generators for a d -p o wer case. W e will see that a ‘concrete T erwilliger algebra’ for th e wreath p ow ers can b e describ ed as w ell. (Here the term ‘concrete’ is used in comparison with an ‘abstract’ T erwilliger algebra describ ed in terms of generators and relations as in [10].) W e describ e the irred ucible T -mo dules of ( K m ) ≀ d b y extending the irreducible T -mo du les of ( K m ) ≀ ( d − 1) for d ≥ 3. This can b e done b y inv estigating the structural relations b et ween the wr eath squ are and the wreath cub e to b egin th e iterativ e pro cess. Let X = ( K m ) ≀ 3 = ( X, { R i } 0 ≤ i ≤ 3 ) b e a 3-class asso ciation sc heme of order m 3 . Let X = { ( a, b, c ) : a, b, c ∈ [ m ] } . Cho ose x 1 = (1 , 1 , 1) and fix it as the b ase ve rtex and we will r efer to it as x h enceforth. Let us order th e association relations suc h that R 0 ( x ) = { (1 , 1 , 1) } , and R 1 ( x ) = { ( a, b, c ) : a, b ∈ [ m ] , c ∈ { 2 , 3 , . . . , m }} , R 2 ( x ) = { ( a, b, 1) : a ∈ [ m ] , b ∈ { 2 , 3 , . . . , m }} , R 3 ( x ) = { ( a, 1 , 1) : a ∈ { 2 , 3 , . . . , m }} . Let the relation table of X b e decomp osed into blo c k matrix form according to the p artition X = R 0 ( x ) ∪ R 1 ( x ) ∪ R 2 ( x ) ∪ R 3 ( x ). W e can see th at th e relation m atrix of ( K m ) ≀ 2 is em b edded in to th at of ( K m ) ≀ 3 b y r elab eling the asso ciation relations. T o s ee this, consider th e relation matrix R for ( K m ) ≀ 2 giv en in th e previous subsection. Denote the blo c k lab eled by B 2 + 2 C 2 of R by D . Th en D has en tries 0, 1, or 2. W e n o w form a n ew blo c k from D by replacing ent ries 2 with 3, 1 with 2 and 0 with 0, and call this D 2 . It is easy to see that D 2 is the blo ck of the relation table ( K m ) ≀ 3 that is ind exed by the vertice s in R 2 ( x ). The relation table of ( K m ) ≀ 3 is giv en b y     0 1 t 3 2 1 t 2 3 1 t 1 1 3 D 3 L 2 L 1 2 1 2 L t 2 D 2 2 L 3 1 1 L t 1 2 L t 3 C 1     where 1 3 , 1 2 , 1 1 are all-ones column v ectors of dimension m 2 ( m − 1) , m ( m − 1) , m − 1, resp ec- tiv ely; L 2 , L 1 , L are all-ones matrices of size m 2 ( m − 1) × m ( m − 1) , m 2 ( m − 1) × ( m − 1) , m ( m − 1) × ( m − 1), resp ectiv ely; D 3 = I m ⊗ D 2 + ( J m − I m ) ⊗ J m ( m − 1) and C 1 = J m − 1 − I m − 1 . W e are no w ready to see the descriptions of the irreducible T -mo d ules of the wreath cub e. Theorem 4.7 The primary irr e ducible T -mo dule of X = ( K m ) ≀ 3 is sp anne d by the four ve ctors (1 , 0 t 3 , 0 t 2 , 0 t 1 ) t , (0 , 1 t 3 , 0 t 2 , 0 t 1 ) t , (0 , 0 t 3 , 1 t 2 , 0 t 1 ) t , (0 , 0 t 3 , 0 t 2 , 1 t 1 ) t wher e 0 3 , 0 2 , 0 1 ar e al l-zer os ve ctors of dimensions m 2 ( m − 1) , m ( m − 1) , ( m − 1) , r esp e ctively; and 1 3 , 1 2 , 1 1 ar e al l-ones v e ctors of dimensions m 2 ( m − 1) , m ( m − 1) , ( m − 1) , r esp e ctively. 16 Pro of: It is straigh tforward. Theorem 4.8 L et V b e the standar d mo dule of ( K m ) ≀ 3 . Ther e exist irr e ducible T -mo dules { W 1 i } 2 ≤ i ≤ ( m − 1) such that (0 , 0 t 3 , 0 t 2 , 1 t 1 ) t and { W 1 i } 2 ≤ i ≤ m − 1 to g e ther c onstitute E ∗ 3 V . Pro of: W e ha v e seen in the p ro of of Lemm a 4.4 that there exist ve ctors w 1 , · · · , w m − 1 whic h span C m − 1 and h w i , w j i = 0 for 1 ≤ i 6 = j ≤ m − 1. Let W 1 i b e the linear span of (0 , 0 t 3 , 0 t 2 , w t i ) t o ve r C . F or ( K m ) ≀ 2 , the generators E ∗ i A h E ∗ i act on the mo d ules W ′ i and the s ignifi can t nonzero actions are E ∗ 2 A 0 E ∗ 2 and E ∗ 2 A 2 E ∗ 2 . The em b edded structure of ( K m ) ≀ 2 in ( K m ) ≀ 3 ensures that for ( K m ) ≀ 3 the generators E ∗ i A h E ∗ i act on the linear sp ace W 1 i , 2 ≤ i ≤ ( m − 1) and the only nonzero actions are E ∗ 3 A 0 E ∗ 3 and E ∗ 3 A 3 E ∗ 3 . It is clear th at (0 , 0 t 3 , 0 t 2 , w t i ) t are E ∗ 0 A 3 E ∗ 3 , E ∗ 1 A 1 E ∗ 3 , E ∗ 2 A 2 E ∗ 3 - in v ariant , and eac h W 1 i is an irreducible T -mo du le of dimension 1. Thus, the result follo ws. Theorem 4.9 In the standar d mo dule V of ( K m ) ≀ 3 , ther e exist irr e ducible mo dules { W 2 i } 1 ≤ i ≤ m ( m − 1) such that (0 , 0 t 3 , 1 t 2 , 0 t 1 ) t and { W 2 i } 2 ≤ i ≤ m ( m − 1) c onstitute E ∗ 2 V . Pro of: W e ha ve seen in Lemma 4.4 that th er e exist vect ors v 1 , . . . , v m ( m − 1) suc h th at v 1 , . . . , v m ( m − 1) span C m ( m − 1) and h v i , v j i = 0 f or 1 ≤ i 6 = j ≤ m ( m − 1). Let W 2 i b e th e linear sp an of (0 , 0 t 3 , v t i , 0 t 1 ) t o ve r C . F or ( K m ) ≀ 2 the generators E ∗ i A h E ∗ i act on the m o dules W i and the significan t nonzero actions are E ∗ 1 A 0 E ∗ 1 , E ∗ 1 A 1 E ∗ 1 and E ∗ 1 A 2 E ∗ 1 . Th e em b edded structure of ( K m ) ≀ 2 in ( K m ) ≀ 3 ensures th at f or ( K m ) ≀ 3 the generators E ∗ i A h E ∗ i act on the linear spaces W 2 i , 2 ≤ i ≤ m ( m − 1) and the nonzero actions are E ∗ 1 A 0 E ∗ 1 , E ∗ 2 A 2 E ∗ 2 and E ∗ 2 A 3 E ∗ 2 . Also, we see that (0 , 0 t 3 , v t 1 , 0 t 1 ) is inv ariant under the action of E ∗ 0 A 2 E ∗ 2 , E ∗ 1 A 1 E ∗ 2 and E ∗ 3 A 2 E ∗ 2 . Eac h of { W 2 i } 2 ≤ i ≤ m ( m − 1) is a irr educible T -mo d ule of dimension 1, and th e result follo ws. W e no w d escrib e the irr educible m o dules th at sp an E ∗ 1 V . W e kno w th at E ∗ 1 V ∼ = C m 2 ( m − 1) . W e observe th at an y m 2 ( m − 1) dimensional column vecto r can b e p artitioned into m equ al parts eac h with m ( m − 1) comp onent s. L et eac h part b e denoted by the index j where j ∈ { 1 , 2 , . . . , m } so that the m 2 ( m − 1) dimensional column v ector is of the form ( u t 1 , u t 2 , . . . , u t m ) t where eac h u j is an m ( m − 1) dimens ional column v ector for eac h j ∈ { 1 , 2 , . . . , m } . Let u j,i = ( u t 1 , u t 2 , . . . , u t l , . . . , u t m ) t denote the m 2 ( m − 1) dim en sional column vecto r su c h that u l = δ lj v i where v i are the ones giv en in Lemm a 4.4 and δ lj is the Kroneck er delta; i.e., δ lj is 1 if and only if l = j , and it is zero otherwise. Lemma 4.10 In the standar d mo dule V of ( K m ) ≀ 3 , f or e ach p air j, i , 1 ≤ j ≤ m, 2 ≤ i ≤ m ( m − 1) , the line ar subsp ac e W j,i of sp anne d by (0 , u t j,i , 0 t 2 , 0 t 1 ) t is an irr e ducible T -mo dule c ontaine d in E ∗ 1 V . Pro of: In the T erw illiger algebra of ( K m ) ≀ 2 , the generators E ∗ i A h E ∗ i act on the mo d ules W i and the only nonzero actions are d ue to E ∗ 1 A 0 E ∗ 1 , E ∗ 1 A 1 E ∗ 1 and E ∗ 1 A 2 E ∗ 1 . F or ( K m ) ≀ 3 , the sub matrix D 3 = I m ⊗ D 2 + ( J m − I m ) ⊗ J m ( m − 1) indicates that in th e T erwilliger algebra of ( K m ) ≀ 3 the generators E ∗ i A h E ∗ i act on the linear sp aces in { W j,i : 1 ≤ j ≤ m, 2 ≤ i ≤ m ( m − 1) } w here W j,i 17 is the linear span of (0 , u t j,i , 0 t 2 , 0 t 1 ) t and the only n onzero actions are d ue to E ∗ 1 A 0 E ∗ 1 , E ∗ 1 A 2 E ∗ 1 , E ∗ 1 A 3 E ∗ 1 , E ∗ 0 A 1 E ∗ 1 , E ∗ 2 A 1 E ∗ 1 and E ∗ 3 A 1 E ∗ 1 . It follo ws that W j,i is an irred u cible mo du le of dimen- sion 1 for eac h j, i . Remark 4.3 We note that i f W 0 denotes the line ar sp an of (1 , 0 t 3 , 0 t 2 , 0 t 1 ) t over C , then W 0 is an irr e ducib le mo dule that sp ans E ∗ 0 V . W e r e c al l that, by L emma 4.5, among the irr e ducible mo dules { W i } 2 ≤ i ≤ m ( m − 1) of ( K m ) ≀ 2 ther e ar e two non-isomorphic T - mo dules of dimension 1 . In sum, so far, for ( K m ) ≀ 3 we have the fol lowing non-isomorph i c T -mo dules 1. The primary mo dule of dimension 4 . 2. Two non-isomorphic T - mo dules of dimension 1 in E ∗ 1 V . 3. Two non-isomorphic T - mo dules of dimension 1 in E ∗ 2 V . 4. O ne non-isomorph i c T -mo dules of dimension 1 in E ∗ 3 V . It lea v es u s w ith one non -isomorp h ic irred u cible T -mo dule of dimension 1 whic h is not ac- coun ted for as the total dimension of the T erwilliger algebra must b e 22 b y the Theorem 3.11. It is in the sub constituen t E ∗ 1 V as in the follo wing. Lemma 4.11 Pick any nonzer o ve ctor u ∈ E ∗ 1 V which is ortho gonal to al l the irr e ducible mo dules { W j,i : 1 ≤ j ≤ m, 2 ≤ i ≤ m ( m − 1) } . The u sp ans a one-dimensional irr e ducible T -mo dule. The actions of E ∗ 1 A 0 E ∗ 1 , E ∗ 1 A 1 E ∗ 1 , E ∗ 1 A 2 E ∗ 1 , E ∗ 1 A 3 E ∗ 1 , E ∗ 0 A 1 E ∗ 1 , E ∗ 2 A 1 E ∗ 1 and E ∗ 3 A 1 E ∗ 1 on this T - mo dule ar e nonzer o and r est ar e al l zer o. Pro of: It is straigh tforward. Theorem 4.12 F or ( K m ) ≀ 3 , T ∼ = M 4 ( C ) ⊕ M 1 ( C ) ⊕ 6 . Pro of: It follo ws from the ab o ve remark and lemmas. 4.4 The T erwilliger algebra of ( K m ) ≀ d for m ≥ 3 The d escription of th e concrete T erw illiger algebra of a sc h eme inv olv es describing the irr educible mo dules that constitute different sub constituen ts of the algebra. Earlie r in this section, w e ha ve already studied all the sub constituen ts of th e 2-class and 3-class wreath p ow er asso ciation sc hemes. F r om the ir reducible mo du les of ( K m ) ≀ 3 w e can d escrib e all irreducible mo du les for ( K m ) ≀ 4 , from ( K m ) ≀ 4 w e can describ e all irreducible mo d ules for ( K m ) ≀ 5 , and so on. W e no w dev elop a r ecursiv e metho d to describ e the T erwilliger algebra of a d -class asso ciation sc h eme ( K m ) ≀ d from a ( d − 1)-class scheme ( K m ) ≀ ( d − 1) . Let X = ( X, { R i } 0 ≤ i ≤ d ) denote the d -class sc heme ( K m ) ≀ d with X = [ m ] × [ m ] × · · · × [ m ] = { ( a 1 , a 2 , · · · , a d ) : a i ∈ [ m ] , for i = 1 , 2 , . . . , d } 18 Let (1 , 1 , . . . , 1) ∈ X b e a fixed b ase v ertex x of X . Without loss of generalit y , we can arrange the lab el of asso ciate relations and the v ertices so th at for i = 1 , 2 , . . . , d − 1, R i ( x ) = { ( a 1 , a 2 , . . . , a d − i − 1 , a, 1 , 1 , . . . , 1) : a k ∈ [ m ] f or 1 ≤ k ≤ d − i − 1 , a ∈ [ m ] − { 1 }} ; R 0 ( x ) = { x } ; R d ( x ) = { ( a, 1 , 1 , . . . , 1) : a ∈ [ m ] − { 1 }} . In the same manner, we can get the relation table for d − 1 wreath p o wer as we ll. First let us lo ok at the relation table of ( K m ) ≀ ( d − 1) .          0 1 t d − 1 2 1 t d − 2 · · · ( d − 2) 1 t 2 ( d − 1) 1 t 1 1 d − 1 T d − 1 J d − 1 ,d − 2 · · · J d − 1 , 2 J d − 1 , 1 2 1 d − 2 J t d − 1 ,d − 2 T d − 2 · · · 2 J d − 2 , 2 2 J d − 2 , 1 . . . . . . . . . . . . . . . . . . ( d − 2) 1 2 J t d − 1 , 2 2 J t d − 2 , 2 · · · T 2 ( d − 2) J 1 , 1 ( d − 1) 1 1 J t d − 1 , 1 2 J t d − 2 , 1 · · · ( d − 2) J t 1 , 1 T 1          where 1 i are all-ones column vecto r s of size m i − 1 ( m − 1), J j,k are all-ones matrices of size m j − 1 ( m − 1) × m k − 1 ( m − 1), T 1 = ( d − 1)( J m − 1 − I m − 1 ) and T i = I m ⊗ T i − 1 + ( d − i )( J m − I m ) ⊗ J m i − 2 ( m − 1) for i ∈ { 2 , 3 , . . . , d − 1 } . No w using this table, w e can describ e the relation table for ( K m ) ≀ d as f ollo ws. In the d iagonal blo c ks T i mak e the follo wing c h anges. Th e ent r y 0 is k ept same, and the en tries i are replaced with i + 1, for all i = 1 , 2 , . . . , d − 1. Let us name the resulting new blo c ks U i for i ∈ { 1 , 2 , . . . , d − 1 } . It is n ot hard to s ee that U i are the diagonal blo cks of the relation table of ( K m ) ≀ d . Let U d = I m ⊗ U d − 1 + ( J m − I m ) ⊗ J m d − 2 ( m − 1) , 1 d is the all-ones column v ector of size m d − 1 ( m − 1). Abusin g notation and denoting all the all-ones matrices in the relation table as J for all d im en sions th e relation table of the d -class asso ciation scheme is             0 1 t d 2 1 t d − 1 3 1 t d − 2 · · · ( d − 1) 1 t 2 d 1 t 1 1 d U d J d,d − 1 J d,d − 2 · · · J d, 2 J d, 1 2 1 d − 1 J t d,d − 1 U d − 1 2 J d − 1 ,d − 2 · · · 2 J d − 1 , 2 2 J d − 1 , 1 3 1 d − 2 J t d,d − 2 2 J t d − 1 ,d − 2 U d − 2 · · · 3 J d − 2 , 2 3 J d − 2 , 1 . . . . . . . . . . . . . . . . . . . . . ( d − 1) 1 2 J t d, 2 2 J t d − 1 , 2 3 J t d − 2 , 2 · · · U 2 ( d − 1) J 1 , 1 d 1 1 J t d, 1 2 J t d − 1 , 1 3 J t d − 2 , 1 · · · ( d − 1) J t 1 , 1 U 1             In the next couple of paragraph s w e will d iscuss the sub constituen ts of the d -class asso ciation sc heme ( K m ) ≀ d . Let 0 i denote zero column v ectors of size m i − 1 ( m − 1) for 1 ≤ i ≤ d . Any m d dimensional column v ectors can b e divided in to subp arts 1, m d − 1 ( m − 1), m d − 2 ( m − 1), . . . , m ( m − 1) and ( m − 1) resp ectiv ely so that an y v ector lo oks like ( p, p t d , · · · , p t 1 ) t where p i is a m i − 1 ( m − 1) d imensional column vect or for 1 ≤ i ≤ d . With these notations, the primary m o dule of the d -class asso ciation scheme ( K m ) ≀ d ma y b e describ ed as f ollo w s. Theorem 4.13 L et V b e the standar d mo dule of ( K m ) ≀ d . The ve ctor (1 , 0 t d , . . . , 0 t 1 ) t and ve ctors q i = (0 , p t d , . . . , p t j , . . . , p t 1 ) t for 1 ≤ i ≤ d such that p j =  1 j if i = j 0 j if i 6 = j 19 gener ates the primary T -mo dule. Pro of: S tr aigh tforwa r d. Let us consider the d -class asso ciation scheme ( K m ) ≀ d of order m d and let V denote its standard mo dule. Finding th e irreducible mo dules of the sub constituents E ∗ i V for 2 ≤ i ≤ d and E ∗ 0 V is more routine. E ∗ 1 V needs to b e tr eated differently than th e other and we will come to that as we go along. Supp ose that th e m d − 1 dimensional column v ectors (0 , h t j ) t for 1 ≤ j ≤ m d − i ( m − 1) − 1 generate the one dimensional mo dules of th e sub constituent E ∗ i − 1 V f or th e ( d − 1)-class asso ciation sc heme ( K m ) ≀ ( d − 1) . I f we add th e m d − 1 ( m − 1) dimens ional column ve ctor 0 t d righ t after 0 in the ab o v e v ectors we land up with m d dimensional vect ors. F or 1 ≤ j ≤ m d − i ( m − 1) − 1 the v ectors (0 , 0 t d , h t j ) t and (0 , 0 t d , . . . , 1 t d − i − 1 , . . . , 0 t 1 ) t span C m d − i ( m − 1) . Also, h (0 , h t j ) , (0 , h t k ) i = 0 for j, k ∈ { 1 , 2 , . . . , m d − i ( m − 1) − 1 } . F or the sc heme ( K m ) ≀ ( d − 1) since (0 , h t j ) t generates a one dimensional mo d ule it is E ∗ i A j E ∗ h in v ariant for all i, j, h ∈ { 0 , 1 , . . . , d − 1 } . F or 1 ≤ j ≤ m d − i ( m − 1) − 1, let W ≀ d j b e the linear span of the ve ctor (0 , 0 t d , h t j ) t . Th e em b edded stru cture of the ( d − 1)-class asso ciation sc heme ( K m ) ≀ ( d − 1) in the d -class sc h eme ( K m ) ≀ d ensures that for 1 ≤ j ≤ m d − i ( m − 1) − 1, (0 , 0 t d , h t j ) t are E ∗ i A j E ∗ h in v ariant for all i, j, h ∈ { 0 , 1 , . . . , d } . No te that no w we are talking ab out the triple pro du cts of the d -class sc h eme ( K m ) ≀ d . The vecto r s (0 , 0 t d , h t j ) t for 1 ≤ j ≤ m d − i ( m − 1) − 1 generate all the one dimensional n on-primary irred ucible mo dules of the sub constituen t E ∗ i V for the D -class asso ciation sc h eme. W e ha ve so far describ ed the su b constituent s E ∗ 0 V , E ∗ 2 V , . . . , E ∗ d V for th e s c heme ( K m ) ≀ d . No w, E ∗ 1 V ∼ = C m d − 1 ( m − 1) . Obs er ve that any m d − 1 ( m − 1) dimensional column v ector can b e partitioned in to m equal parts eac h of dimension m d − 2 ( m − 1). Let eac h part b e denoted by the index j wher e j ∈ { 1 , 2 , . . . , m } so that the m d − 1 ( m − 1) dim en sional v ector is of the f orm ( r t 1 , r t 2 , . . . , r t m ) t where eac h r j , j ∈ { 1 , 2 , . . . , m } is a m d − 2 ( m − 1) dimens ional column ve ctor. F or eac h i ∈ { 2 , 3 , . . . , m d − 2 ( m − 1) } and j ∈ { 1 , 2 , . . . , m } let r j,i = ( r t 1 , r t 2 , . . . , r t l , . . . , r t m ) t denote the m d − 1 ( m − 1) d imensional column vec tor su c h that r l = δ lj h j where (0 , h t j ) generates the mo dules of E ∗ 1 V for the ( d − 1)-class scheme ( K m ) ≀ ( d − 1) . It is easy to see that eac h of the v ectors (0 , r t j,i , 0 t d − 1 , . . . , 0 t 1 ) t generates one dimensional irredu cible mo du le in the su b constituent E ∗ 1 V for the sc heme ( K m ) ≀ d . Th ese do n ot constitute all the irr educible m o dules in E ∗ 1 V . So far we ha ve accoun ted for th e follo wing non-isomorphic T -mo dules for ( K m ) ≀ d . 1. Th e primary mo dule of d im en sion d + 1. 2. d − 1 non -isomorp h ic T -mo dules of dimension 1 in E ∗ 1 V . 3. d − 1 non -isomorp h ic T -mo dules of dimension 1 in E ∗ 2 V . 4. d − 2 non -isomorp h ic T -mo dules of dimension 1 in E ∗ 3 V . . . 5. 2 non-isomorphic T -mo dules of d imension 1 in E ∗ d − 1 V . 6. 1 non-isomorphic T -mo dules of d imension 1 in E ∗ d V . 20 F rom Theorem 3.11 we ha ve the dimension of T (( K m ) ≀ d is ( d + 1) 2 + 1 2 d ( d + 1). F rom our ab o ve discussion w e hav e ( d + 1) 2 + ( d − 1) + ( d − 1) + ( d − 2) + · · · + 2 + 1 of the dimen s ion. That lea ves us with one non-isomorph ic T mod ule of dimen sion 1 in th e su b constituent E ∗ 1 V of m u ltiplicit y m − 1. Pic k any nonzero v ector r ∈ E ∗ 1 V w h ic h is orth ogonal to all the irreducible mo dules. Then r spans a one-dimensional irreducible T -mo dule. In th e d iscussion ab ov e w e s a w how w e could bu ild the irredu cible mo du les of the sc heme ( K m ) ≀ d . W e will conclude th e section b y collecti ng all our non-isomorphic T -mo dules and de- scribing th e T erwilliger algebra of ( K m ) ≀ d . (F urther detailed explanation of the conte n t of th is section can b e found in [4].) Theorem 4.14 L et X = ( K m ) ≀ d b e a d -class asso ciation scheme of or der m d , m ≥ 3 . Then the dimension of T is ( d + 1) 2 + 1 2 d ( d + 1) and T ∼ = M d +1 ( C ) ⊕ M 1 ( C ) ⊕ 1 2 d ( d +1) . 4.5 The Irreducible T -mo dules of ( K 2 ) ≀ d W e note that for the case ( K 2 ) ≀ d the n u m b er of nonzero triple pro ducts E ∗ i A j E ∗ h are f ew er in n u m b er than in the general case ( K m ) ≀ d with m ≥ 3. What is nice for ( K 2 ) ≀ d is that instead of just kno w in g the existence of vect ors that generate the ir r educible mo dules, we are actually able to get sp ecific ve ctors that generate the irred ucible T -mod ules. Let X = ( K 2 ) ≀ d b e a d -class asso ciation scheme of order 2 d . Without loss of generalit y w e lab el the 2 d elemen ts of X b y x 1 , x 2 , . . . , x 2 d . W e fix x 1 as the b ase v ertex and we will refer to it as x henceforth. Th en, without loss of generalit y , w e can arr ange the elemen ts suc h that X is partitioned w ith R 0 ( x ) = { x } , R 1 ( x ) = { x 2 , . . . , x 2 d − 1 +1 } , consisting of the 2 d − 1 elemen ts, R 2 ( x ) = { x 2 d − 1 +2 , . . . , x 2 d − 1 +2 d − 2 +1 } consisting of th e next 2 d − 2 elemen ts, and so on, with the last part R d ( x ) = { x 2 d } . Remark 4.4 L et 1 denote the al l-ones ve ctor in the standar d mo dule. Then the ve ctor sp ac e over C sp anne d by { E ∗ i 1 : 0 ≤ i ≤ d } is a thin irr e ducible T - mo dule of dimension d + 1 and is the primary mo dule denote d as P ; so, by setting d i = E ∗ i 1 , the set { d i : 0 ≤ i ≤ d } gener ates P . In order to describ e the irr educible T -mod ules, whose orthogonal direct su m forms the stan- dard mo du le V , we employ a particular set of v ectors. Let ˆ x denote the column ve ctor with 1 in the x -th p osition and 0 elsewhere. Lemma 4.15 F or l ∈ { 1 , 2 , . . . , d − 1 } define set of ve ctors { d l i } 1 ≤ i ≤ 2 d − l − 1 by d l i = 2 l − 1 − 1 X k =0 ˆ x i + j + k − 2 l − 1 X k =2 l − 1 ˆ x i + j + k . F or e ach i , the c orr esp ond ing values of j ar e suc c essively j = 1 , 1 + 2 l − 1 , 1 + 2(2 l − 1) , 1 + 3(2 l − 1) , . . . , 1 + (2 d − l − 2)(2 l − 1) . Then { d 1 i } 1 ≤ i ≤ 2 d − 1 − 1 ; d 1 i = ˆ x 2 i − ˆ x 2 i +1 21 { d 2 i } 1 ≤ i ≤ 2 d − 2 − 1 ; d 2 i = ˆ x i + j + ˆ x i + j +1 − ˆ x i + j +2 − ˆ x i + j +3 ; j = 1 , 4 , 7 , . . . { d 3 i } 1 ≤ i ≤ 2 d − 3 − 1 ; d 3 i = P 2 3 − 1 − 1 k =0 ˆ x i + j + k − P 2 3 − 1 k =2 3 − 1 ˆ x i + j + k ; j = 1 , 8 , 15 , . . . . . . d d − 1 1 = P 2 d − 2 +1 i =2 ˆ x i − P 2 d − 1 +1 i =2 d − 2 +2 ˆ x i . In p articular, h d l i , d k h i = 0 unless h = i and k = l . Pro of: Pr o of follo ws from the construction of the v ectors. Lemma 4.16 L et W d l i b e the line ar sp an of d l i for al l the ve ctors describ e d in L emma 4.15. Then W d l i is an irr e ducible T -mo dule of dimension 1 f or ( K 2 ) ≀ d . Pro of: T o pro ve that W d l i is an irred ucible T -mo d ule, we lo ok at the action of the n onzero generators E ∗ i A j E ∗ h of the T erw illiger algebra on the vec tor d l i . W e will consider th r ee cases. 1. Th e case when l = 1: (a) F or { d 1 i } 1 ≤ i ≤ 2 d − 2 b y the construction of d 1 i , the action of the triple pr o ducts E ∗ i A j E ∗ h on d 1 i can b e nonzero only for E ∗ 1 A 0 E ∗ 1 , E ∗ 1 A 1 E ∗ 1 , . . . , E ∗ 1 A d E ∗ 1 and E ∗ i A 1 E ∗ 1 for 2 ≤ i ≤ d . Among them E ∗ 1 A 1 E ∗ 1 = 0. The generators E ∗ i A j E ∗ h of T that act on d 1 i in a n onzero manner are E ∗ 1 A 0 E ∗ 1 and E ∗ 1 A d E ∗ 1 with E ∗ 1 A 0 E ∗ 1 d 1 i = d 1 i , E ∗ 1 A d E ∗ 1 d 1 i = − d 1 i . (b) F or { d 1 i } 2 d − 2 +1 ≤ i ≤ 2 d − 2 +2 d − 3 b y the construction of d 1 i , th e action of the trip le pro d ucts E ∗ i A h E ∗ h on d 1 i are nonzero only for E ∗ 2 A 0 E ∗ 2 and E ∗ 2 A d E ∗ 2 , and act in the follo w ing manner: E ∗ 2 A 0 E ∗ 2 d 1 i = d 1 i , E ∗ 2 A d E ∗ 1 d 1 i = − d 1 i . (c) F or every s et of v ectors corresp ondin g to the d ifferen t sub constituents will follo w the same pattern. Th e last case is the f ollo wing. F or d 1 2 d − 1 − 1 the generators E ∗ i A j E ∗ h of T that act on d 1 2 d − 1 − 1 in a nonzero manner for E ∗ d − 1 A 0 E ∗ d − 1 d 1 i = d 1 i , E ∗ d − 1 A d E ∗ d − 1 d 1 i = d 1 i . 2. Th e case when l ∈ { 2 , 3 , . . . , d − 2 } : (a) F or { d l i } 1 ≤ i ≤ 2 d − ( l +1) b y th e constru ction of d l i , the action of the trip le pro ducts E ∗ i A j E ∗ h on d 1 i can b e nonzero only for E ∗ 1 A 0 E ∗ 1 , E ∗ 1 A 1 E ∗ 1 , . . . , E ∗ 1 A d E ∗ 1 . Among them E ∗ 1 A 1 E ∗ 1 = 0. The generators E ∗ i A j E ∗ h of T that act on d l i in a nonzero manner are E ∗ 1 A 0 E ∗ 1 , E ∗ 1 A d E ∗ 1 , E ∗ 1 A d − 1 E ∗ 1 , . . . , E ∗ 1 A d − ( l − 1) E ∗ 1 in the follo wing man n er: E ∗ 1 A 0 E ∗ 1 d l i = d l i , E ∗ 1 A d E ∗ 1 d l i = d l i , E ∗ 1 A d − j E ∗ 1 d l i = 2 j d l i where 1 ≤ j ≤ l − 1 . 22 (b) F or { d l i } 2 d − ( l +1) +1 ≤ i ≤ 2 d − ( l +1) +2 d − ( l +2) b y the constru ction of d 1 i , the action of the triple pro du cts E ∗ i A j E ∗ h on d 1 i can b e n onzero only for E ∗ 2 A 0 E ∗ 2 , E ∗ 2 A 1 E ∗ 2 , . . . , E ∗ 2 A d E ∗ 2 . Among them E ∗ 2 A 1 E ∗ 2 = 0. The generators E ∗ i A j E ∗ h of T that act on d l i in a n onzero mann er are E ∗ 2 A 0 E ∗ 2 , E ∗ 2 A d E ∗ 2 , E ∗ 2 A d − 1 E ∗ 2 , . . . , E ∗ 2 A d − ( l − 1) E ∗ 2 in the follo wing manner: E ∗ 2 A 0 E ∗ 2 d l i = d l i , E ∗ 2 A d E ∗ 2 d l i = d l i , E ∗ 2 A d − j E ∗ 2 d l i = 2 j d l i for 1 ≤ j ≤ l − 2 , and E ∗ 2 A d − ( l − 1) E ∗ 2 d l i = − 2 l − 1 d l i . (c) F ollo wing a similar reasoning the last case will b e as follo ws. F or d l 2 d − l − 1 the gen- erators E ∗ i A j E ∗ h of T that act on d l 2 d − l − 1 in a nonzero manner are E ∗ d − 1 A 0 E ∗ d − 1 and E ∗ d − 1 A d E ∗ d − 1 in the follo win g manner: E ∗ d − l A 0 E ∗ d − l d l 2 d − l − 1 = d l 2 d − l − 1 , E ∗ d − l A d E ∗ d − l d l 2 d − l − 1 = d l 2 d − l − 1 , E ∗ d − l A d − j E ∗ d − l d l 2 d − l − 1 = 2 j d l 2 d − l − 1 for 1 ≤ j ≤ l − 2 , and E ∗ d − l A d − ( l − 1) E ∗ d − l d l 2 d − l − 1 = − 2 l − 1 d l 2 d − l − 1 . 3. Th e case when l = d − 1: By the constru ction of d d − 1 1 , the action of the triple pr o ducts E ∗ i A j E ∗ h on d d − 1 1 can b e nonzero only for E ∗ 1 A 0 E ∗ 1 , E ∗ 1 A 1 E ∗ 1 , . . . , E ∗ 1 A d E ∗ 1 . Among them E ∗ 1 A 1 E ∗ 1 = 0. Th e genera- tors E ∗ i A j E ∗ h of T that act on d d − 1 1 in a nonzero manner are E ∗ 1 A 0 E ∗ 1 , E ∗ 1 A d − 1 E ∗ 1 , . . . , E ∗ 1 A 2 E ∗ 1 in the follo wing manner: E ∗ 1 A 0 E ∗ 1 d d − 1 1 = d d − 1 1 , E ∗ 1 A d − j E ∗ 1 d d − 1 1 = 2 j d d − 1 1 for 1 ≤ j ≤ d − 3 , and E ∗ 1 A 2 E ∗ 1 d d − 1 1 = − 2 d − 2 d d − 1 1 . It is clear from th e ab ov e cases that eac h of W d l i is a T -mo dule. They are irreducible since they are vec tor sp aces generated by a single ve ctor. T his completes the pro of. As a consequence, w e ha ve the follo win g decomp osition of the standard mo dule. Theorem 4.17 L et ( K 2 ) ≀ d b e a d -class asso ciation scheme of or der 2 d . F or l ∈ { 1 , 2 , . . . , d − 1 } define set of ve ctors { d l i } 1 ≤ i ≤ 2 d − l − 1 by d l i = 2 l − 1 − 1 X k =0 ˆ x i + j + k − 2 l − 1 X k =2 l − 1 ˆ x i + j + k . F or e ach i the c orr esp onding values of j ar e suc c essively j = 1 , 1 + 2 l − 1 , 1 + 2(2 l − 1) , 1 + 3(2 l − 1) , . . . , 1 + (2 d − l − 2)(2 l − 1) . L et W d l i b e the line ar sp an of d l i . 23 (1) L et V b e the standar d mo dule and P b e the primary mo dule. Then V = P ⊕ X W v wher e v over al l d l i define d ab ove. (2) F or l ∈ { 1 , 2 , . . . , d − 1 } ( a ) W v wher e v ∈ { d l i : 1 ≤ i ≤ 2 d − ( l +1) } ar e T -i somorph ic . ( b ) W v wher e v ∈ { d l i : 2 d − ( l +1) + 1 ≤ i ≤ 2 d − ( l +1) + 2 d − ( l +2) } ar e T - isomorphic. F ol lowing ab ove we final ly have ( c ) W d l 2 d − l − 3 and W d l 2 d − l − 2 ar e T - isomorphic. ( d ) R est of W v ar e not T -isomorphic. Pro of: (1) is straigh tforw ard by the construction of the mo dules w h ic h are m utu ally orthogonal. (2) The p r o of is similar when w e c h o ose T -mo dules from same groups as d escrib ed in cases (a)-(c). W e sh all sho w that W d l 1 and W d l 2 are T -isomorphic. Defin e an isomorp h ism σ : W d l 1 → W d l 2 b y σ ( d l 1 ) = d l 2 . W e need to show that ( σ B − B σ ) W = 0 for all B ∈ T . Let us consider the action of nonzero E ∗ i A j E ∗ h on σ ; i.e, E ∗ 1 A 0 E ∗ 1 , E ∗ 1 A d E ∗ 1 , E ∗ 1 A d − 1 E ∗ 1 , . . . , E ∗ 1 A d − ( l − 1) E ∗ 1 . Now ( σ E ∗ 1 A 0 E ∗ 1 − E ∗ 1 A 0 E ∗ 1 σ ) W d l 1 = ( σ − E ∗ 1 A 0 E ∗ 1 σ ) W d l 1 = d l 2 − E ∗ 1 A 0 E ∗ 1 ( W d l 2 ) = 0 . Similar kind of reasoning shows that f or all E ∗ 1 A 0 E ∗ 1 , E ∗ 1 A d E ∗ 1 , E ∗ 1 A d − 1 E ∗ 1 , . . . , E ∗ 1 A d − ( l − 1) E ∗ 1 , ( σ E ∗ i A j E ∗ k − E ∗ 1 A 0 E ∗ 1 σ ) W d l 1 = 0. T h u s, W d l 1 and W d l 2 are T -isomorphic. Next w e shall sh o w that mo du les selected fr om differen t groups are not T -isomorphic. In particular, let us show that W d l 1 and W d l 2 d − ( l +1) +1 are not T -isomorphic. Supp ose we assume that there exists an isomorphism σ : W d l 1 → W d l 2 d − ( l +1) +1 suc h that ( σ B − B σ ) W d l 1 = 0 for all B ∈ T . Then for E ∗ 1 A 0 E ∗ 1 ∈ T , ( E ∗ 1 A 0 E ∗ 1 ) W d l 1 = W d l 1 and ( E ∗ 1 A 0 E ∗ 1 ) W d l 2 d − ( l +1) +1 = 0 . No w, [ σ ( E ∗ 1 A 0 E ∗ 1 ) − ( E ∗ 1 A 0 E ∗ 1 ) σ ] W d l 1 = [ σ − ( E ∗ 1 A 0 E ∗ 1 ) σ ] W d l 1 = W d l 2 d − ( l +1) +1 − 0 6 = 0 whic h is a con tradiction to our assumption. T hus, W d l 1 and W d l 2 d − ( l +1) +1 are not T -isomorphic. The other cases can b e pro ved with a similar approac h . Theorem 4.18 L et X = ( K 2 ) ≀ d b e a d -class asso ciation scheme of or der 2 d . T ∼ = M d +1 ( C ) ⊕ M 1 ( C ) ⊕ 1 2 d ( d − 1) . Pro of: It is straigh tforward. 24 5 Concluding Remarks There is furth er w ork th at is needed on th e theme related to our wo r k. Here we state a few problems that are of our in terest. 1. In ou r attempt to d escrib e the T erw illiger algebra of the d -class asso ciation scheme ( K m ) ≀ d our base w as the T erwilliger algebra describ ed by T omiy ama and Y amazaki [22] for a 2- class asso ciation sc heme constructed from a strongly regular graph. Although our 3-class asso ciation scheme w as neither strongly regular nor a P -p olynomial sc heme, we w ere able to describ e the T erwilliger algebra concrete mann er largely b ecause of the f act that d -class asso ciation scheme ( K m ) ≀ d turned out to b e triply regular, and the structur e of the ( d − 1)- class was b eautifu lly em b edded in the d -class asso ciation scheme. W e demonstr ated furth er ho w we could extend the same metho d us ed to describ e the T er w illiger algebra of th e 3-class asso ciation scheme to the d -cla ss asso ciation scheme ( K m ) ≀ d for d > 3. The general d -class wreath pro d uct sc heme K n 1 ≀ K n 2 ≀ · · · ≀ K n d is also trip ly regular, and its T erwilliger algebra h as the same dimension as in the case of th e wreath p ow er ( K m ) ≀ d . Ho we ver, it seems to b e muc h more inv olv ed to d escrib e the irreducible T -mo dules f or the general w r eath pr o duct scheme. The m etho d used for the case ( K m ) ≀ d do es not wo r k w ith differen t n i ’s. W e do not kno w how to find v ectors that generate the ir reducible T -mod ules. Ho we ver, Pa u l T erwilliger b eliev es that all non-pr imary mo dules still hav e dimension 1. If that is the case, then the structure of the T erwilliger algebra of K n 1 ≀ K n 2 ≀ · · · ≀ K n d is also the same as that of ( K m ) ≀ d . 2. In a slight ly differen t d irection, it will b e in teresting to lo ok at some sp ecific sc hemes obtained by taking the wreath p ow er of tw o asso ciation sc hemes, such as the Hamming H (2 , q ) in stead of H (1 , q ). W e know that the T erwilliger algebra of a Hammin g scheme can b e d escrib ed as symmetric d -tensors on the T erwilliger algebra of H (1 , q ) [15 ], although in general H ( d, q ) is not realized as a pro duct of H (1 , q ). It would b e interesti ng to see ho w the T erwilliger algebra changes wh en we tak e the wreath p o wer of a Hamming scheme H ( d, q ) for an arbitrary d > 1. 3. Th ere are also other pro du cts b esides the w r eath pro d uct. It is also an in teresting problem to lo ok at the direct p o wer of H (1 , q ). W e stud y the wreath p o wer fi rst b ecause the direct pro du ct of t w o asso ciation sc hemes has a lot more classes than the wreath p r o duct. Namely , the direct pro du ct of a d -class asso ciation sc heme and a e -class association scheme is of class de + d + e w h ile the wreath pro duct b ecomes ( d + e )-class asso ciation sc heme. S o a stu d y of direct p ow er requires a lot more work th an th at of wreath p o we r . How ev er, it ma y b e w orthy to lo ok at it n o w as w e know m ore ab out the sc h emes related to H (1 , q ). 4. In [18], the irr educible T -mo dules and T erwilliger algebra has b een inv estigate d for the Do ob s c hemes. The Doob sc hemes are the asso ciation sc hemes obtained by taking the direct pro du ct of copies of H (2 , 4) and copies of sc h emes coming from the Sh rikhande graph. In this case, the d irect p ro duct of th ese sc hemes preserves many p r op erties of th e original factor sc hemes. One im p ortan t prop ert y that is remained as the same is P -p olynomial p rop erty . In terms of graphs, the Hamming H (2 , 4) and the S hrikhande graph s are the only distance- regular graphs whose direct pro du ct is also distance-regular. Th is pr op ert y no longer holds for the direct p ro duct of other Hamming Schemes. Neverthele ss, the description of the 25 T erwilliger algebra of Do ob sc h emes in terms of those of H (2 , 4) an d S hrikhand e sc heme ma y shed a light in und erstanding ho w th e T erwilliger algebra of the pr o duct b eha ve s when w e study the T erwilliger algebra of the direct pro du ct of H ( d, q )s with v arious d . 5. As explained by Eric Egge [10] and introd uction of [19] it is p ossible to define an “abstract v ersion” of the T erwilliger algebra using generators and relations. In all cases the concrete T erwilliger algebra is a homomorp hic image of the abstract T erwilliger algebra, and in some cases they are isomorph ic. In the case of ( K m ) ≀ d the entire str u cture of the T erwilliger algebra is determined by th e in tersection num b ers and Kr ein parameters, so it ma y b e easy to see wh at is going on. 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