Duality of translation association schemes coming from certain actions
aa r X i v : . [ m a t h . C O ] A ug Duality of translation association schemescoming from certain actions
Dae San Kim a , Hyun Kwang Kim b a Department of Mathematics, Sogang University, Seoul 121-742, Korea b Department of Mathematics, Pohang University of Science and Technology,Pohang 790-784, Korea
Abstract
Translation association schemes are constructed from actions of finite groupson finite abelian groups satisfying certain natural conditions. It is also shown thatthe mere existence of maps from finite groups to themselves sending each elementin their groups to its ‘adjoint’ entails the self-duality of the constructed associationschemes. Many examples of these, including Hamming scheme and sesquilinear formsschemes, are provided. This construction is further generalized to show the dualityof the association schemes coming from actions of two finite groups on the samefinite abelian group. An example of this is supplied with weak Hamming schemes.
Key words:
Action ; Orbit ; Translation association scheme ; Adjoint ;Self-duality ; Duality
Let G be a finite group acting on a finite additive abelian group X , and let O = { } , O , · · · , O d be the G -orbits. Assume that the action satisfies theconditions g ( x + x ′ ) = gx + gx ′ , f or all g ∈ G and x, x ′ ∈ X, and x ∈ O i ⇒ − x ∈ O i , f or ≤ i ≤ d. E-mail address : [email protected] (D.S.Kim)This work was supported by grant No. R01-2007-000-11176-0 from the Basic Re-search Program of the Korea Science and Engineering Foundation.
Preprint submitted to Elsevier 5 November 2018 hen it is easy to see that X G = ( X, { R i } di =0 ), with ( x, y ) ∈ R i ⇔ y − x ∈ O i ,is a translation association scheme (cf. Theorem 2).Assume now further that the following map sending each element to itsadjoint exists, i.e., there is a map ι : G → G such that h gx, y i = h x, ι ( g ) y i , f or all g ∈ G, x, y ∈ X. Here h , i : X × X → C × is an inner product on X . Then it is shownthat such a simple requirement is enough to guarantee the self-duality of X G (cf. Corollary 7). Many examples of such pairs ( G, X ) satisfying the abovethree conditions are provided. For instance, the Hamming scheme and mostof sesquilinear forms schemes fall within this category (cf. Section 4).Suppose now that ˇ G is another finite group acting on the same finite abeliangroup X , with ˇ O = { } , ˇ O , · · · , ˇ O d the ˇ G -orbits, and that the correspondingconditions to (1) and (2) for ˇ G and ˇ O i (0 ≤ i ≤ d ) are satisfied. Assume inaddition that there is a map ι : G → ˇ G such that h gx, y i = h x, ι ( g ) y i , f or all g ∈ G, x, y ∈ X. Then it is shown that the existence of the map sending each element to itsadjoint is strong enough to yield the duality of X G and X ˇ G . An example of thisis illustrated with what we call the weak Hamming scheme H ( n , · · · , n t , q ).It is the wreath product of the Hamming schemes H ( n , q ) , · · · , H ( n t , q ). Ourresult now says that H ( n , · · · , n t , q ) and H ( n t , · · · , n , q ) are dual to eachother. This would have been observed already in [14] or even in the earlierworks [10] and [13]. The weak Hamming scheme can be also constructed inconnection with weak order poset weight (cf. Section 6). Also, it is an exampleof weak metric schemes which are the subject of the recent paper [12]. A pair X = ( X, { R i } di =0 ) consisting of a finite set X (referred to as thevertex set of X ) and d + 1 nonempty subsets R i of X × X is called a d -class(symmetric) association scheme if(i) { R , R , · · · , R d } is a partition of X × X ,(ii) R = △ X = { ( x, x ) | x ∈ X } ,(iii) t R i = R i , for all i , where t R i = { ( x, y ) | ( y, x ) ∈ R i } ,(iv) for any i, j, k (0 ≤ i, j, k ≤ d ), there are numbers, called intersectionnumbers, p kij such that, for any ( x, y ) ∈ R k , p kij = { z ∈ X | ( x, z ) ∈ R i , ( z, y ) ∈ R j } . X = ( X, { R i } di =0 ) be an association scheme. Let A i be the adjacencymatrix of R i , for 0 ≤ i ≤ d . Then A , A , · · · , A d generate a ( d +1)-dimensionalcommutative subalgebra A of symmetric matrices in M n ( C ) ( n = | X | ), calledthe Bose-Mesner algebra of X . A has another nice basis E , E , · · · , E d , calledthe irreducible idempotents of A . They are determined (up to permutation ofthe indices 1 , · · · , d ) by :(i) E i E j = δ ij E i , for all i, j, (ii) P di =0 E i = I, (iii) E = | X | − J ( J all-one matrix) , (iv) E , E , · · · , E d are linearly independent over C . (1)The C -space generated by A , A , · · · , A d is also closed under the Hadamardmultiplication ◦ , with J as the multiplicative identity. Write E i ◦ E j = | X | − d X k =0 q kij E k (0 ≤ i, j ≤ d ) . Then q kij ’s are actually nonnegative real numbers, called the Krein parameters.Let A j = d X i =0 p ij E i , E j = | X | − d X i =0 q ij A i . The p ij ’s and q ij ’s are respectively called p - and q -numbers. In particular, v i = p i = { z | ( x, z ) ∈ R i } , for any fixed x ∈ X , and m i = q i = rank E i are respectively called the valencies and the multiplicities. Also, P = ( p ij ) and Q = ( q ij ) are respectively called the first and the second eigenmatrix of X .Let X be a finite additive abelian group, and let X = ( X, { R i } di =0 ) be a d -class association scheme. Then X is called a translation association schemeif ( x, y ) ∈ R i ⇒ ( x + z, y + z ) ∈ R i , f or all z ∈ X and i.
Let X i = { x ∈ X | (0 , x ) ∈ R i } , f or ≤ i ≤ d. (2)Then X , X , · · · , X d give a partition of X , and( x, y ) ∈ R i ⇔ y − x ∈ X i (0 ≤ i ≤ d ) . The dual association scheme X ∗ = ( X ∗ , { R ∗ i } di =0 ) of X consists of the group X ∗ of characters on X together with d + 1 nonempty subsets R ∗ i of X ∗ × X ∗ determined by : ( χ, ψ ) ∈ R ∗ i ⇔ ψχ − ∈ X ∗ i , X ∗ i = { χ ∈ X ∗ | E i χ = χ } . Here χ is viewed as the column vectorwith the x -component ( x ∈ X ) given by χ ( x ). For the proof of the followingtheorem, see [3, Theorem 2.10.10]. Theorem 1
Let X = ( X, { R i } di =0 ) be a translation association scheme withparameters p kij , q kij , v i , m i , P, Q . Then we have the following : (a) The dual scheme X ∗ = ( X, { R ∗ i } di =0 ) is also a translation associationscheme with parameters p ∗ kij = q kij , q ∗ kij = p kij , v ∗ i = m i , m ∗ i = v i ,P ∗ = Q, Q ∗ = P . (b) (i) p ij = P x ∈ X j χ ( x ) , f or χ ∈ X ∗ i , (ii) q ij = P χ ∈ X ∗ j χ ( x ) , f or x ∈ X i , (iii) E j = | X | − P χ ∈ X ∗ j χ t χ , (iv) v j = | X j | , (v) m j = | X ∗ j | . Two association schemes X = ( X, { R X,i } di =0 ) and Y = ( Y, { R Y,i } di =0 ) aresaid to be isomorphic if there are a bijection f : X → Y and a permutation σ of { , , · · · , d } such that( x, y ) ∈ R X,i ⇔ ( f ( x ) , f ( y )) ∈ R Y,σ ( i ) , f or ≤ i ≤ d. Here we always assume that R X, = △ X , R Y, = △ Y , so that ( x, y ) ∈ R X, ⇔ ( f ( x ) , f ( y )) ∈ R Y, . In particular, two d -class translation association schemes X = ( X, { R X,i } di =0 ) and Y = ( Y, { R Y,i } di =0 ) are isomorphic if there is a groupisomorphism f : X → Y and a permutation σ of { , , · · · , d } such that x ∈ X i ⇔ f ( x ) ∈ Y σ ( i ) , f or ≤ i ≤ d. Here X i = { x ∈ X | (0 , x ) ∈ R X,i } , Y i = { y ∈ Y | (0 , y ) ∈ R Y,i } . If two associ-ation schemes are isomorphic, we may assume that all the parameters of theschemes are the same.For further facts about association schemes, one is referred to [2] and [3]. Let G be a finite group acting on a finite additive abelian group X , with O = { } , O , · · · , O d the G -orbits. We assume that this action satisfies theconditions g ( x + x ′ ) = gx + gx ′ , f or all g ∈ G, x, x ′ ∈ X, (3)and x ∈ O i ⇒ − x ∈ O i , f or ≤ i ≤ d. (4)4ote here that g g ∈ G , as O = { } . This together with (3)implies that g ( − x ) = − gx, f or all g ∈ G, x ∈ X. (5)A special case of the following theorem appeared in [11]. Theorem 2 X G = ( X, { R i } di =0 ) , given by ( x, y ) ∈ R i ⇔ y − x ∈ O i (0 ≤ i ≤ d ) , is a translation association scheme. PROOF.
Here the conditions (3) and (4) are needed. All are easy to check,perhaps except for the condition on the intersection numbers. Let x, y, x ′ , y ′ ∈ X , with u = y − x, v = y ′ − x ′ ∈ O k . Then we must see : { z ∈ X | z − x ∈ O i , y − z ∈ O j } = { z ∈ X | z − x ′ ∈ O i , y ′ − z ∈ O j } . Observe that { z ∈ X | z − x ∈ O i , y − z ∈ O j } → { z ∈ X | u − z ∈ O i , z ∈ O j } ( z y − z )is a bijection. So it is enough to show : { z ∈ X | u − z ∈ O i , z ∈ O j } = { z ∈ X | v − z ∈ O i , z ∈ O j } . As G acts transitively on each O i , and u, v ∈ O k , there is an h ∈ G such that hu = v , and hence { z ∈ X | u − z ∈ O i , z ∈ O j } → { z ∈ X | v − z ∈ O i , z ∈ O j } ( z hz )is a bijection. Notice that (3), (4) and (5) are used here. ✷ Remark 3 (1)
For the association scheme X G = ( X, { R i } di =0 ) , X i = O i , f or ≤ i ≤ d ( cf. (2)) . (2) Let the condition (4) be replaced by : f or each i (0 ≤ i ≤ d ) , x ∈ O i ⇒ − x ∈ O j , f or some j. (6)If (3) is satisfied along with (6), then in fact we have x ∈ O i ⇔ − x ∈ O j ,and hence the association scheme X = ( X, { R i } di =0 ), given by ( x, y ) ∈ R i ⇔ y − x ∈ O i (0 ≤ i ≤ d ), is not a symmetric and yet commutative associationscheme. However, in this paper we will consider only the association schemes5oming from the actions satisfying (3) and (4).Let h , i : X × X → C × be an inner product on the finite additive abeliangroup X , i.e.,(i) h x, y i = h y, x i , for all x, y ∈ X ,(ii) h x, y + z i = h x, y ih x, z i , for all x, y, z ∈ X ,(iii) h x, y i = h x, z i , for all x ∈ X ⇒ y = z . Remark 4
It is well-known that such an inner product always exists on afinite abelian group X . h , x i will denote the character on X given by y y, x i , so that X ∗ = {h , x i| x ∈ X } . Also, h , x i will indicate the column vector of size | X | whose y -component is h y, x i ( y ∈ X ) . Theorem 5
The following are equivalent. (a) X G = ( X, { R i } di =0 ) → X ∗ G = ( X ∗ , { R ∗ i } di =0 ) , given by x
7→ h , x i , is anisomorphism of translation association schemes. (b) There is a permutation σ of { , , · · · , d } such that x ∈ X j ⇔ h , x i ∈ X ∗ σ ( j ) , f or j = 1 , · · · , d. (c) E j = | X | − P x ∈ X j h , x i t h , x i (0 ≤ j ≤ d ) are the irreducible idempotentsof the association scheme X G = ( X, { R i } di =0 ) . (d) f j : X → C , given by f j ( y ) = P x ∈ X j h y, x i , is constant on each X i (0 ≤ i ≤ d ) , for all j = 0 , · · · , d . PROOF. ( a ) ⇔ ( b ) This is just the definition.( a ) ⇒ ( d ) As ( a ) ⇔ ( b ), q iσ ( j ) = P x ∈ X j h y, x i , for y ∈ X i , is the q -number of X , and hence is constant on each X i (0 ≤ i ≤ d ), for all j = 0 , , · · · , d .( d ) ⇒ ( c ) Let E j = | X | − P x ∈ X j h , x i t h , x i , for all j . Then E j = | X | − P di =0 q ij A i ,with q ij = P x ∈ X j h y, x i ( y ∈ X i ). So E j belongs to the Bose-Mesner algebra of X G . ( E i E j ) ab = | X | − X x ∈ Xiy ∈ Xj h a, x ih b, y i t h , x ih , y i . Note that t h , x ih , y i = X z ∈ X h z, y − x i = , y = x, | X | , y = x.
6o ( E i E j ) ab = , i = j, | X | − P x ∈ X i h a − b, x i = ( E i ) ab , i = j, and hence E i E j = δ ij E i . Similarly, P di =0 E i = I . Assume that P di =0 α i E i = 0,with α i ∈ C . Then α i E i = 0, for all i . To show independence of E , · · · , E d ,it is enough to see that E i = 0, for all i . This is indeed the case, since E i h , x i = h , x i , if x ∈ X i , , if x / ∈ X i . (7)Thus E , E , · · · , E d are the irreducible idempotents of X (cf. (1)). Note herethat E = | X | − J .( c ) ⇒ ( b ) Recall that e E , e E , · · · , e E d , with e E j = | X | − P χ ∈ X ∗ j χ t χ , is alsothe irreducible idempotents of X G (cf. Theorem 1, (b)). Observe also that e E = | X | − J . As the irreducible idempotents are unique up to permutation,there is a permutation σ of { , , · · · , d } such that E j = e E σ ( j ) , for j = 1 , · · · , d .Now, using (7) we have : x ∈ X j ⇔ E j h , x i = h , x i⇔ e E σ ( j ) h , x i = h , x i⇔ h , x i ∈ X ∗ σ ( j ) . ✷ Assume now further that there is a map ι : G → G such that h gx, y i = h x, ι ( g ) y i , f or all g ∈ G, x, y ∈ X, (8)where h , i : X × X → C × is an inner product. Lemma 6
Under the assumption of (8), the sum X x ∈ X j h y, x i ( y ∈ X i ) depends only on i , for all i, j with ≤ i, j ≤ d . PROOF.
Let y , y ∈ X i . Then y = hy , for some h ∈ G . So X x ∈ X j h y , x i = X x ∈ X j h hy , x i = X x ∈ X j h y , ι ( h ) x i = X x ∈ X j h y , x i . ✷ X G is self-dual. Corollary 7
Let X G = ( X, { R i } di =0 ) be the translation association schemeobtained from the action of the finite group G on the finite abelian group X ,satisfying (3) and (4). Assume that there is a map ι : G → G such that h gx, y i = h x, ι ( g ) y i , f or all g ∈ G, x, y ∈ X, where h , i : X × X → C × is an inner product. Then (a) X G = ( X, { R i } di =0 ) → X ∗ G = ( X ∗ , { R ∗ i } di =0 ) , given by x
7→ h , x i , is anisomorphism, i.e., X G is self-dual, (b) E j = | X | − P x ∈ X j h , x i t h , x i (0 ≤ j ≤ d ) are the irreducible idempotentsfor X G , (c) q ij = P x ∈ X j h y, x i ( y ∈ X i ) are the q -numbers for X G , (d) p kij = q kij , p ij = q ij , v i = m i . Here we will demonstrate that there are abundant examples of actions offinite groups G on finite abelian groups X satisfying (3) and (4), and (8) forsuitable inner products on X . So the schemes X G = ( X, { R i } di =0 ) constructedfrom these actions are, in particular, self-dual. In below, λ will always denotea fixed nontrivial additive character on F q . The examples (a) and (b) beloware adopted from [6].(a) Let X be a finite abelian group with period ν . Then ( Z / ( ν )) × acts on X via ( Z / ( ν )) × × X → X (( m, x ) mx ) . This action satisfies (3) and (4), and the orbits O = { } , O , · · · , O d arecalled the central classes of X . Further, given any inner product on X , (8) issatisfied with ι the identity map.(b) Let ω be a primitive element in F q ( q an odd prime power), and let d bea positive integer such that 2 d | q −
1. Let G = h ω d i be the cyclic subgroup of F × q of order r = ( q − /d . Then G acts on X = ( F q , +) by left multiplication.Here the orbits are O = { } , O , · · · , O d , where, for 0 ≤ i ≤ d − O i +1 = { ω i , ω d + i , · · · , ω ( r − d + i } . , O , · · · , O d are called the cyclotomic classes of F q . The condition (3) isobviously satisfied, and the condition (4) is also valid, as we assume 2 d | q − h x, y i = λ ( xy ) is an inner product on X , and (8) is satisfied with ι theidentity map.The examples (c)-(f) will be about sesquilinear forms association schemes.There are many articles about this topic. Here we are content with just men-tioning [3, Sections 9.5-6] and [5,7,8,16].(c) Let X = ( F m × nq , +), with m ≤ n , and let G = GL ( m, q ) × GL ( n, q ) be thedirect product of general linear groups. G now acts on X via G × X → X ((( α, β ) , A ) φ ( α,β ) A := t αAβ ) . Then O i = { A ∈ X | rank ( A ) = i } ( i = 0 , , · · · , m ) are the G -orbits, and theconditions (3) and (4) hold. The associated scheme X G is called the bilinearforms scheme, which is usually denoted by Bil ( m × n, q ). Moreover, for A · B = tr ( A t B ) = P i,j A ij B ij ( A, B ∈ X ), and ( α, β ) ∈ G , φ ( α,β ) A · B = A · φ ( t α, t β ) B. So, for the inner product h A, B i = λ ( A · B ) and ι : G → G (( α, β ) ( t α, t β )),(8) is valid.(d) Let X be the group of all alternating matrices of order m over F q . Recallhere that ( A ij ) is alternating ⇔ A ii = 0, for 1 ≤ i ≤ m , and A ji = − A ij , for1 ≤ i < j ≤ m . G = GL ( m, q ) acts on X via G × X → X (( α, A ) φ ( α ) A := t αAα ) . Then O i = { A ∈ X | rank ( A ) = 2 i } ( i = 0 , , · · · , n = ⌊ m ⌋ ) are the G -orbits, and the conditions (3) and (4) are satisfied. The associated scheme X G is called the alternating forms scheme which is denoted by Alt ( m, q ). For A · B = P i The following are equivalent. (a) X ˇ G = ( X, { ˇ R i } di =0 ) → X ∗ G = ( X ∗ , { R ∗ i } di =0 ) , given by x 7→ h , x i , is anisomorphism of translation association schemes. (b) There is a permutation σ of { , , · · · , d } such that x ∈ ˇ X j ⇔ h , x i ∈ X ∗ σ ( j ) , f or j = 1 , · · · , d. (c) E j = | X | − P x ∈ ˇ X j h , x i t h , x i (0 ≤ j ≤ d ) are the irreducible idempotents f the association scheme X G = ( X, { R i } di =0 ) . (d) f j : X → C , given by f j ( y ) = P x ∈ ˇ X j h y, x i is constant on each X i (0 ≤ i ≤ d ) , for all j = 0 , , · · · , d . Assume now further that there is a map ι : G → ˇ G such that h gx, y i = h x, ι ( g ) y i , f or all g ∈ G, x, y ∈ X. (9)Then, as in Lemma 6, we have the following. Lemma 9 Under the assumption of (9), the sum X x ∈ ˇ X j h y, x i ( y ∈ X i ) depends only on i , for all i, j with ≤ i, j ≤ d . So we obtain the following corollary from Theorem 8 and Theorem 1. Corollary 10 Let X G = ( X, { R i } di =0 ) and X ˇ G = ( X, { ˇ R i } di =0 ) be the twotranslation association schemes obtained from the actions of the finite groups G and ˇ G on the same finite abelian group X , and satisfying (3) and (4).Assume that there is a map ι : G → ˇ G such that h gx, y i = h x, ι ( g ) y i , f or all g ∈ G, x, y ∈ X, where h , i : X × X → C × is an inner product. Then (a) X ˇ G = ( X, { ˇ R i } di =0 ) → X ∗ G = ( X ∗ , { R ∗ i } di =0 ) ( x 7→ h , x i ) is an isomorphism,i.e., X G and X ˇ G are dual to each other, (b) E j = | X | − P x ∈ ˇ X j h , x i t h , x i (0 ≤ j ≤ d ) are the irreducible idempotentsfor X G , (c) q ij = P x ∈ ˇ X j h y, x i ( y ∈ X i ) are the q -numbers for X G , (d) p ij = P x ∈ X j h y, x i ( y ∈ ˇ X i ) are the p -numbers for X G , (e) p kij = ˇ q kij , p ij = ˇ q ij , m i = ˇ v i , q kij = ˇ p kij , q ij = ˇ p ij , v i = ˇ m i . Here we are content with giving only one example for Section 5 which iswhat we call the weak Hamming scheme. Let H ( m, q ) denote, as usual, theHamming scheme whose vertex set is F mq and i -th relation is given by( x, y ) ∈ R i ⇔ d H ( x, y ) = w H ( x − y ) = i ( i = 0 , , · · · , m ) . w H and d H are respectively the Hamming weight and the Hamming met-ric. Then the weak Hamming scheme H ( n , · · · , n t , q ) is given as the wreathproduct (cf. [1]) H ( n , · · · , n t , q ) = H ( n , q ) ≀ · · · ≀ H ( n t , q ) , so that the vertex set of that is F n q × · · · × F n t q , and( x, y ) ∈ R n + ··· + n i − + i ⇔ x i +1 = y i +1 , · · · , x t = y t and d H ( x i , y i ) = w H ( x i − y i ) = i , for i = 1 , · · · , t, ≤ i ≤ n i , or i = 1 , i = 0.There is another description of H ( n , · · · , n t , q ) which has to do with poset-weight (poset-metric). This notion of poset-weight (poset-metric) was firstintroduced in [4]. Let P = ([ n ] , ≤ ) be a poset, with [ n ] = { , , · · · , n } . The P -weight w P is the function on F nq , which is given by : w P ( x ) = { i ∈ [ n ] | i ≤ j, for some j ∈ Supp ( x ) } . Here Supp ( x ) = { j ∈ [ n ] | x j = 0 } , for x = ( x , · · · , x n ) ∈ F nq . Then d P ( x, y ) = w P ( x − y ) is a distance function on F nq , called P -metric. We now specialize P as the weak order poset P . P = n L · · · L n t is given as the ordinal sumof the antichains n i on the set { n + · · · + n i − + 1 , · · · , n + · · · + n i − + n i } ,for i = 1 , · · · , t , i.e., the underlying set is [ n ] ( n = n + · · · + n t ) and the orderrelation is given by : k < l ⇔ k ∈ n i , l ∈ n j , for some i < j. Let G = Aut ( F nq , w P ) be the group of all linear automorphisms of F nq pre-serving w P -weight, i.e., w P ( φu ) = w P ( u ), for all u ∈ F nq . Then G actson F nq in a natural way and the G -orbits are O i = { x ∈ F nq | w P ( x ) = i } , i = 0 , , · · · , n . Clearly, the conditions (3) and (4) are valid. Moreover, for i = 1 , · · · , t, ≤ i ≤ n i , or i = 1 , i = 0, y − x ∈ O n + ··· + n i − + i ⇔ w P ( y − x ) = n + · · · + n i − + i ⇔ x i +1 = y i +1 , · · · , x t = y t and d H ( x i , y i ) = i ⇔ ( x, y ) ∈ R n + ··· + n i − + i . Here we identified F nq with F n q × · · · × F n t q by writing the elements x ∈ F nq asthe blocks of coordinates x = ( x , · · · , x t ) ∈ F n q × · · · × F n t q . So the associatedscheme X G is nothing but the weak Hamming scheme H ( n , · · · , n t , q ). Simi-larly, the weak Hamming scheme H ( n t , · · · , n , q ) = H ( n t , q ) ≀ H ( n t − , q ) ≀ · · · ≀ H ( n , q ) is also obtained from the action of the group ˇ G = Aut ( F nq , w ˇ P ) on F nq . Here ˇ P = n t L n t − L · · · L n is the dual poset of P .Let { e = (1 , , · · · , , e = (0 , , · · · , , · · · , e n = (0 , · · · , , } be thestandard basis of F nq . Let g ∈ G = Aut ( F nq , w P ). Then, for each 1 ≤ s ≤ t ,13nd each i ∈ n s , g ( e i ) = a i e ρ s ( i ) + n + ··· + n s − X l =1 b li e l , (10)where a i ∈ F × q , b li ∈ F q , ρ s ( i ) ∈ n s , and the assignment i ρ s ( i ) is apermutation on n s . Conversely, if g is the linear map given by (10), for1 ≤ s ≤ t , i ∈ n s , then g ∈ G = Aut ( F nq , w P ). Let x · y = P ni =1 x i y i , for x = ( x , · · · , x n ), y = ( y , · · · , y n ) ∈ F nq . Then h x, y i = λ ( x · y ) is an innerproduct on F nq , where λ is a fixed nontrivial additive character on F q . Now, wewill show the existence of a map ι : G = Aut ( F nq , w P ) → ˇ G = Aut ( F nq , w ˇ P )such that h gx, y i = h x, ι ( g ) y i , for all g ∈ G , x, y ∈ F nq . For this, it is enoughto show that there is a map ι : G → ˇ G satisfying ge i · e k = e i · ι ( g ) e k , f or all g ∈ G, i, k = 1 , · · · , n. (11)Let g ∈ G be the map given by (10), for each 1 ≤ s ≤ t , and each i ∈ n s . If(11) is to be satisfied, we must have :for 1 ≤ j ≤ t , k ∈ n j , ι ( g ) e k · e i = ge i · e k = ( a i e ρ s ( i ) + n + ··· + n s − X l =1 b li e l ) · e k = , if s < j,a i , if s = j and k = ρ s ( i ) , , if s = j and k = ρ s ( i ) ,b ki , if s > j. This yields that, for 1 ≤ j ≤ t , k ∈ n j , ι ( g ) e k = a ρ − j ( k ) e ρ − j ( k ) + n + ··· + n t X l = n + ··· + n j +1 b kl e l . (12)Now, ι ( g ) given by (12) belongs to ˇ G = Aut ( F nq , w ˇ P ), so that a map ι : G → ˇ G is defined and (11) is satisfied. Thus the results stated in Corollary 10 holdtrue. In particular, H ( n , n , · · · , n t , q ) and H ( n t , n t − , · · · , n , q ) are dual toeach other. Remark 11 (1) As we have seen in the above, the weak Hamming scheme H ( n , · · · , n t , q ) is the associated scheme X G when G = Aut ( F nq , w P ) ( n = n + · · · + n t ) acts on F nq , with P = n L · · · L n t the weak order poset. Theweak order poset is unique in many respects. Indeed, the following has beenshown. Let P be a poset on [ n ] . Then (1) P is a weak order poset on [ n ] ⇔ (2) ( P , ˇ P ) is a weak dual MacWilliams pair ( wdM p ) ⇔ (3) The group Aut ( F nq , w P ) acts transitively on each P -sphere S P ( i ) = { x ∈ F nq | w P ( x ) = i } (0 ≤ i ≤ n ) ⇔ (4) X = ( F nq , { R i } ni =0 ) , with ( x, y ) ∈ R i ⇔ x − y ∈ S P ( i ) (0 ≤ i ≤ n ) , is n association scheme. Here ˇ P is the dual poset of P , a pair of posets ( P , ˇ P ) on [ n ] is called a wdM p if the P -weight distribution of C uniquely determines ˇ P -weight distribution of C ⊥ , for every linear code C ⊆ F nq . For details aboutthese, one is referred to [9-11,13,14].(2) The structure of Aut ( F nq , w P ) was explicitly determined in [10]. So the map ι : G = Aut ( F nq , w P ) → ˇ G = Aut ( F nq , w ˇ P ) would have been more explicitlydetermined, just as we did in the example (g) of Section 4. References [1] S. Bang, S. Y. Song, On generalized semidirect product of association schemes, Discrete Math. 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