Linear Groups of Isometries with Poset Structures

Linear Groups of Isometries with Poset Structures Luciano Panek∗ Marcelo Firer† Hyun Kwang Kim‡ Jong Yoon Hyun§ Abstract Let V be an n-dimensional vector space over a finite field Fq and P = {1, 2, . . . , n} a poset. We consider on V the poset-metric dP . In this paper, we give a complete description of groups of linear isometries of the metric space (V, dP ), for any poset-metric dP . We show that a linear isometry induces an automorphism of order in poset P , and consequently we show the existence of a pair of ordered bases of V relative to which every linear isometry is represented by an n × n upper triangular matrix. Key words: Poset codes, poset metrics, linear isometries. Coding theory takes place in finite dimensional linear spaces over finite fields. One of the main questions of the theory (classical problem) asks to find a k-dimensional subspace in Fnq , the space of n-tuples over the finite field Fq , with the largest minimum distance possible. There are many possible metrics that can be defined in Fnq , the most common ones are the Hamming and Lee metrics. In 1987 Harald Niederreiter generalized the classical problem of coding theory (see [11]). Brualdi, Graves and Lawrence (see [3]) also provided in ∗ Centro de Ciências Exatas, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900 - Maringá - PR, Brazil. Email: [email protected] † IMECC - UNICAMP, Universidade Estadual de Campinas, Cx. Postal 6065, 13081-970 - Campinas - SP, Brazil. Email: [email protected] ‡ Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, South Korea. Email: [email protected] § Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, South Korea. Email: [email protected] 1 1995 a wider situation for the above problem: using partially ordered sets and defining the concept of poset-codes, they started to study codes with a posetmetric. This has been a fruitful approach, since many new perfect codes have been found with such poset metrics (see [1], [3], [5], [8] and [9]). We let P be a partially ordered set (abbreviated as poset) of cardinality n with order relation denoted, as usual, by ≤. An ideal of P is a subset I ⊆ P with the property that x ∈ I and y ≤ x implies that y ∈ I. Given A ⊆ P , we denote by hAi the smallest ideal of P containing A. Without loss of generality, we assume that P = {1, 2, . . . , n} and that the coordinates of vectors in Fnq are in one-to-one correspondence with the elements of P . Given x = (x1 , x2 , . . . , xn ) ∈ Fnq , the support of x is the set supp (x) := {i ∈ P : xi 6= 0} , and we define the P -weight of x to be the cardinality of the smallest ideal containing supp(x): wP (x) = |hsupp (x)i| . The function dP : Fnq × Fnq → N defined by dP (x, y) = wP (x − y) is a metric in Fnq ([3, Lemma 1.1]), called a poset-metric or a P -poset-metric, when it is important ¡to stress ¢ the order n taken in consideration. We denote such a metric space by Fq , dP . An [n, k, δP ]q poset-code is a k-dimensional subspace C ⊂ Fnq , where Fnq is endowed with a poset-metric dP and δP (C) = min {wP (x) : 0 6= x ∈ C} is the P -minimum distance of the code C. If P is an antichain order, that is, an order with no comparable elements, P -weight, P -poset-metric and P minimum distance become the Hamming weight, Hamming metric and minimum distance of classical coding theory. Further notice that the RosenbloomTsfasman metric, introduced in [12], can be viewed as a P -poset-metric which corresponds to the poset consisting of finite disjoint union of chains of equal lengths. ¡ ¢ A linear isometry T of the metric space Fnq , dP is a linear transformation T : Fnq → Fnq that preserves P -poset-metric, dP (T (x) , T (y)) = dP (x, y) , for every x, y ∈ Fnq . Equivalently, a linear transformation T is an isometry if ¢ ¡ wP (T (x)) = wP (x) for every x ∈ Fnq . A linear isometry of Fnq , dP is said 2 ¡ ¢ to be a P -isometry. We denote by GLP Fnq the group of linear isometries of ¢ ¡ n Fq , dP . In [4], [6], [10] some authours determined the group of linear isometries of the Rosenbloom-Tsfasman space, generalized Rosenbloom-Tsfasman space and crown space. In this work, we give a complete description of those groups, for any given poset-metric P . The property of permuting chains of same length, showed in [10], corresponds, in the case of a general poset P , to Theorem 1.1 of the first section, which assures that every linear isometry T induces an automorphism of the poset P . The key-point for these proof is Proposition 1.1, which assures that hsupp (T (u))i ⊆ hsupp (T (v))i if hsupp (u)i ⊆ hsupp (v)i, u, v ∈ Fnq . The characterization of linear isometries is given in Theorem ¡ ¢ 1.2: there is an ordered base β of Fnq relative to which every T ∈ GLP Fnq , is represented by the product A · U of matrices, where U is a monomial matrix corresponding to an isomorphism of the poset P and A is an upper-triangular matrix. The second is devoted to some examples, with a complete descrip¡ nsection ¢ tion of GLP Fq where we give a detailed description of with some of the most commonly used poset-metrics: when the posets are disjoint union of chains, weak-metric and crown-metric. 1 Linear Isometries for a General Poset Structures We will present only the concepts of the theory of partially ordered sets that are strictly necessary for this work, refereing the reader to [13] for more details. A totally ordered set (or linearly ordered set) is a poset P in which any two elements are comparable. A subset C of a poset P is called a chain if C is a totally ordered set when regarded as a subposet of P . Two posets P and Q are isomorphic if there exists an order-preserving bijection φ : P → Q, called of isomorphism, whose inverse is order preserving; that is, x ≤ y in P if and only if φ(x) ≤ φ(y) in Q. An isomorphism φ : P → P is called an automorphism. Given x, y ∈ P , we say that y covers x if x < y and if no element z ∈ P satisfies x < z < y. A chain x1 < x2 < . . . < xk in a finite poset P is called saturated if xi covers xi−1 for i ∈ {1, 2, . . . , k}. From here on, we denote by {e1 , e2 , . . . , en } the canonical base of Fnq . Given an order automorphism φ P: P → P , we Pdefine the canonical linear P -isometry Tφ induced by φ as Tφ ( ni=1 ai ei ) := ni=1 ai eφ(i) . 3 ¡ ¢ We will show that a linear isometry T ∈ GLP Fnq induces an automorphism of the poset P in the following way: given i ∈ {1, 2, . . . , n} we consider any saturated chain i1 < i2 < . . . < ik containing i. Then there are ej1 , ej2 , . . . , ejk , with js+1 covering js for all s ∈ {1, 2, . . . , k − 1}, such that hsupp (ejl )i = hsupp (T (eil ))i for any l ∈ {1, 2, . . . , k}. So, if i = il , we can define the order automorphism φ by φ (il ) = jl . The key to prove this is to show that ¡hsupp ¢ (T (u))i ⊆ hsupp (T (v))i if n hsupp (u)i ⊆ hsupp (v)i, for every T ∈ GLP Fq . We will start with some preliminary lemmas. Lemma 1.1 Let P = {1, 2, . . . , n} be a poset, T ∈ GLP (Fnq ) and {e1 , e2 , . . . , en } the canonical base of Fnq . If hsupp (ei )i ⊆ hsupp (ej )i, then hsupp (T (ei ))i ⊆ hsupp (T (ej ))i . Proof. We observe that, for any vectors u, v ∈ Fnq , if supp (u) ⊆ supp (v) then wP (u) ≤ wP (v). Moreover, the inequality is strict if and only if hsupp (u)i ( hsupp (v)i. We remember that T is a linear isometry, so that wP (v) = wP (T (v)), for every vector v. We prove the lemma by contradiction, assuming that hsupp (T (ei ))i * hsupp (T (ej ))i. Suppose hsupp (T (ei ))i ∩ hsupp (T (ej ))i = ∅. Since T is linear, wP (T (ei + ej )) = wP (T (ei ) + T (ej )) and since the ideals do not intersect, we have that wP (T (ei ) + T (ej )) = wP (T (ei )) + wP (T (ej )) . Since T is an isometry, we find that wP (T (ei )) + wP (T (ej )) = wP (ei ) + wP (ej ) > wP (ej ) wP (T (ei + ej )) = wP (ei + ej ) . However, we are assuming that hsupp (ei )i ⊆ hsupp (ej )i, so that wP (ei + ej ) = wP (ej ), a contradiction. Now we can assume that hsupp (T (ei ))i ∩ hsupp (T (ej ))i 6= ∅. If we put supp (T (ei )) ∩ supp (T (ej )) = {k1 , . . . , kr }, we have two cases to consider. Case 1: {k1 , . . . , kr } 6= ∅. In this case, we can write supp (T (ei )) = {k1 , . . . , kr } ∪ {i1 , . . . , is } 4 and T (ei ) = αk1 ek1 + . . . + αkr ekr + βi1 ei1 + . . . + βis eis . Let y = ei − βi1 T −1 (ei1 ) − . . . − βis T −1 (eis ) . Then wP (y) ≥ wP (ei ) , unless ei = βi1 T −1 (ei1 ) + . . . + βis T −1 (eis ) = T −1 (βi1 ei1 + ... + βis eis ) , contradicting the hypothesis that {k1 , . . . , kr } 6= ∅. But T (y) = αk1 ek1 + . . . + αkr ekr , and since there is il ∈ {i1 , . . . , is } ⊆ supp (T (ei )) such that il ∈ / supp (T (ej )), we find that wP (T (y)) < wP (T (ei )) = wP (ei ). So wP (T (y)) < wP (y) , a contradiction. Case 2: {k1 , . . . , kr } = ∅. This means that supp (T (ei )) ∩ supp (T (ej )) = ∅. Put T (ei ) = αi1 ei1 + . . . + αit eit . Then there is an l ∈ hsupp (T (ei ))i \supp (T (ei )) . (1) Let y = ei − αi1 T −1 (ei1 ) − . . . − αit T −1 (eit ) + T −1 (el ) . Then wP (y) ≥ wP (ei ) , unless ei = T −1 (el ), and this contradicts (1). But, T (y) = el and hence wP (T (y)) = wP (el ) < wP (ei ) ≤ wP (y) , again a contradiction. ¤ Lemma 1.2 Let P = {1, 2, . . . , n} be a poset, T ∈ GLP (Fnq ) and {e1 , e2 , . . . , en } the canonical base of Fnq . Then, * à s !+ s [ X hsupp (T (eji ))i = supp T (eji ) , i=1 i=1 for every s ∈ {1, 2, . . . , n} and j1 , . . . , js ∈ {1, . . . , n}. 5 Ps T (eji ))i, there is an i such that j ∈ hsupp (T (eji ))i, !+ à s s X [ ⊆ T (eji ) hsupp (T (eji ))i . supp Proof. If j ∈ hsupp ( so that * i=1 i=1 i=1 We will prove the other inclusion by induction on s. The case s = 1 is trivial and we can assume, as the induction hypothesis that * à s−1 !+ s−1 X [ supp T (eji ) = hsupp (T (eji ))i , i=1 i=1 for every subset {j1 , . . . , js−1 } ⊆ {1, . . . , n}. Given J = {j1 , . . . , js } ⊆ {1, . . . , n} and t ∈ {1, 2, . . . , s}, we can define à s ! [ hsupp (T (eji ))i . ΘJ,t = hsupp (T (ejt ))i \ i=1,i6=t But ΘJ,t = ∅ means that every j ∈ hsupp (T (ejt ))i we have j∈ s [ hsupp (T (eji ))i i=1,i6=t so that s [ s [ hsupp (T (eji ))i = hsupp (T (eji ))i i=1,i6=t i=1 and by the induction hypothesis we have that !+ * à s s X [ . T (eji ) hsupp (T (eji ))i = supp Since * supp we have that * (2) i=1,i6=t i=1 supp à s X !+ T (eji ) i=1 à s X i=1 !+ T (eji ) ⊆ ⊆ s [ * supp 6 hsupp (T (eji ))i i=1 à s X i=1,i6=t !+ T (eji ) . (3) Since T is a linear isometry, we have that ! à à s à s !! à s ! X X X eji wP T (eji ) = wP T = wP eji , i=1 i=1 wP à s X i=1,i6=t But ! T (eji ) = wP T i=1 s X eji i=1,i6=t à s X wP à à i=1 eji ! ≥ wP à !! s X i=1,i6=t = wP eji à s X i=1,i6=t ! eji ! . (4) and since by definition, we have that wP (v) = |hsupp (v)i|, considering inequality (4) in (3) we find that !+ !+ * à s * à s X X T (eji ) = supp T (eji ) supp i=1,i6=t i=1 and from (2) we get that * à s !+ s X [ supp T (eji ) = hsupp (T (eji ))i , i=1 i=1 so that the lemma holds if for every s ≥ 2, there is J = {j1 , . . . , js } and t ∈ {1, 2, . . . , s} such that ΘJ,t = ∅. The case of an antichain P is trivial, so we can assume that the poset P is not an antichain order, and hence there are l1 , l2 ∈ {1, 2, . . . , n} such that l2 covers l1 . So, given s ≥ 2, for every J = {l1 , l2 , j3 , . . . , js } we have that ΘJ,l1 = ∅, since hsupp (el1 )i = hl1 i ⊆ hl2 i = hsupp (el2 )i . ¤ Now we can state and prove the proposition that extends Lemma 1.1 to general vectors. Proposition 1.1 Let P = {1, 2, . . . , n} be a poset, T ∈ GLP (Fnq ). Then, for every u, v ∈ Fnq , hsupp (T (u))i ⊆ hsupp (T (v))i , if hsupp (u)i ⊆ hsupp (v)i. 7 Proof. Let {e1 , e2 , . . . , en } be the canonical base of Fnq and express u and v as a linear combination of this base: u = α1 eu1 + α2 eu2 + . . . + αr eur v = β1 ev1 + β2 ev2 + . . . + βs evs with supp (u) = {u1 , . . . , ur } and supp (v) = {v1 , . . . , vs }. Since hsupp (u)i ⊆ hsupp (v)i we have that hsupp (eui )i ⊆ hsupp (v)i for­every ¡i ∈ ¢® {1, 2, . . . , r}, so ⊆ supp evj . But Lemma there is an j ∈ {1, 2, . . . , s} such that ­ hsupp ¡ (e ¡ ui )i¢¢® . It follows that 1.1 assures that hsupp (T (eui ))i ⊆ supp T evj !+ * à r X T (eui ) hsupp (T (u))i = supp i=1 ⊆ ⊆ r [ i=1 s [ j=1 hsupp (T (eui ))i ­ ¡ ¡ ¢¢® supp T evj and by Lemma 1.2 we have that hsupp (T (v))i = = = * supp j=1 s [ ­ j=1 s [ j=1 à s X ­ !+ ¢ ¡ T βj evj ¢¢® ¡ ¡ supp T βj evj ¡ ¡ ¢¢® supp T evj and we find hsupp (T (u))i ⊆ hsupp (T (v))i . ¤ An ideal I of a poset P is said to be a prime ideal if it contains a unique maximal element. Lemma 1.3 Let P = {1, 2, ..., ¢ be a poset, β = {e1 , e2 , . . . , en } be the canon¡ n} n n ical base of Fq and T ∈ GLP Fq . Then, for every r ∈ {1, 2, . . . , n}, we have that hsupp (T (er ))i is a prime ideal. 8 Proof. We want to prove that the ideal hsupp (T (er ))i is generated by a single greatest element (greater than every other element), or alternatively, it has only one maximal element (no one greater than it). Let {j1 , j2 , . . . , jk } be a set of maximal elements in hsupp (T (er ))i. Then we have that hsupp (T (er ))i = = k [ i=1 k [ hji i hsupp (eji )i i=1 = * supp à r X i=1 eji !+ . But Proposition 1.1 assures that we can apply T −1 to both sides of the equation above preserving the equality, so that à r !!+ * à X ­ ¡ −1 ¢® eji . (5) hsupp (er )i = supp T T (er ) = supp T −1 i=1 Since T −1 is linear, we have that !!+ * à r à r * à !+ X X = supp supp T −1 T −1 (eji ) eji i=1 i=1 and by Lemma 1.2, we have that !+ * à r k X [ ­ ¡ ¢® T −1 (eji ) = supp supp T −1 (eji ) . i=1 (6) i=1 S But looking at equations (5) and (6) we find that ki=1 hsupp (T −1 (eji ))i is the prime ideal hsupp (er )i. Since we are expressing a prime ideal as the union of ideals, one of them, let us say hsupp (T −1 (ejs ))i for some s ∈ {1, 2, . . . , r}, must contain the maximal element r and hence hsupp (T −1 (ejs ))i = hsupp (er )i. Using again Proposition 1.1, we find that hsupp (ejs )i = hsupp (T (er ))i so that hsuppT (er )i is a prime ideal and consequently {j1 , j2 , . . . , jk } = {js }. ¤ Now we can state and prove the proposition that extends Lemma 1.3 to the general case. 9 ¡ ¢ Proposition 1.2 Let P = {1, 2, . . . , n} be a poset and T ∈ GLP Fnq . Then, for every v ∈ Fnq such that hsupp (v)i is a prime ideal, we have that hsupp (T (v))i is also a prime ideal. Proof. Let {e1 , e2 , . . . , en } the canonical base of Fnq and v ∈ Fnq . Suppose that v = α1 ei1 + . . . + αs eis . Then hsupp (v)i = hsupp (α1 ei1 + . . . + αs eis )i = hsupp (ei1 )i ∪ . . . ∪ hsupp (eis )i , and since hsupp (v)i is a prime ideal, it follows there is an k ∈ {1, 2, . . . , s} such that hsupp (ei1 )i ∪ . . . ∪ hsupp (eis )i = hsupp (eik )i so that hsupp (v)i = hsupp (eik )i. Lemma 1.1 assures that hsupp (T (v))i = hsupp (T (eik ))i , and as hsupp (T (eik ))i is a prime ideal (by Lemma 1.3), and we conclude that hsupp (T (v))i is a prime ideal. ¤ Lemma 1.4 If k covers i and J is an ideal such that hii ⊆ J ⊆ hki, then J = hii or J = hki. Proof. If hii = J, there is nothing to be proved. So, we assume that hii J ⊆ hki. Then, there is an j ∈ J such that j i. Since J ⊆ hki it follows that j ≤ k. So i  j ≤ k, and since k covers i, we have that j = k and hence J = hki. ¤ Theorem 1.1 Let P = {1, 2,¡. . . ,¢n} be a poset, {e1 , e2 , . . . , en } be the canonical base of Fnq and T ∈ GLP Fnq linear isometry. Then, for every saturated chain with a minimal element ii < i2 < . . . < ir there is a unique saturated sequence of prime ideals hsupp (ej1 )i ⊂ hsupp (ej2 )i ⊂ . . . ⊂ hsupp (ejr )i . such that hsupp (T (eik ))i = hsupp (ejk )i for every k ∈ {1, 2, . . . , r} and φ: P ik −→ P 7−→ φ (ik ) := jk is a well defined poset automorphism. 10 Proof. Proposition 1.2 assures us that hsupp (T (eik ))i is a prime for all k ∈ {1, 2, . . . , r}, since hsupp (eik )i is a prime ideal. Then for each k ∈ {1, 2, . . . , r} there is just one maximal element jk ∈ hsupp (T (eik ))i. So hsupp (T (eik ))i = hsupp (ejk )i for all k ∈ {1, 2, . . . , r}. Since hsupp (ei1 )i ⊂ hsupp (ei2 )i ⊂ . . . ⊂ hsupp (eir )i , it follows, from Proposition 1.1, that hsupp (ej1 )i ⊂ hsupp (ej2 )i ⊂ . . . ⊂ hsupp (ejr )i . We affirm now that the sequence above is saturated. Suppose that for some k ∈ {1, 2, . . . , r} there is j ′ such that hjk i & hj ′ i & hjk+1 i . Since hjk i = hsupp hsupp­ (T (e¡ik ))i¡ , ¢¢® ­ (ej¡k )i = ¢® , hjk+1 i = supp ejk+1 = supp T eik+1 it follows, applying Proposition 1.1) to the linear P -isometry T −1 , that ­ ¡ ¢® hik i = supp T −1 T (eik ) ¢® ­ ¡ & supp T −1 (ej ′ ) ¢¢® ­ ¡ ¡ = hik+1 i , & supp T −1 T eik+1 what contradicts, by Lemma 1.4, the hypothesis that i1 < . . . < ir is a saturated chain. Let us now define φ : P → P by φ (il ) = jl . Since jl is uniquely defined and does not depends on the choice of the saturated chain containing il (but only on T (eil )), we have that φ is well defined. Moreover, let us suppose that x < y in P , and let i1 < . . . < ik−1 < x < ik+1 < . . . < il−1 < y < il+1 < . . . < ir be a saturated chain containing x and y. Then there is only one saturated chain j1 < . . . < jk−1 < jk < jk+1 < . . . < jl−1 < jl < jl+1 < . . . < jr such that φ (x) = jk and φ (y) = jl . Since jk < jl we get that φ (x) < φ (y). Therefore φ is an application that preserves the order on P . 11 Finally, we affirm that φ is one-to-one. In fact, suppose that φ (x) = φ (y). As φ (x) = max hsupp (T (ex ))i and φ (y) = max hsupp (T (ey ))i then hsupp (T (ex ))i = hsupp (T (ey ))i , and from Proposition 1.1 follows that ­ ¡ ¢® ­ ¡ ¢® hsupp (ex )i = supp T −1 T (ex ) = supp T −1 T (ey ) = hsupp (ey )i . As both ideals hsupp (ex )i and hsupp (ey )i are primes, we must have x = y. Being φ one-to-one and P finite, we find that φ is a bijection that preserves the order and we conclude that φ is an automorphism of P . ¤ The m-th level Γ(m) (P ) is the set of elements of P that generates a prime ideal with cardinality m: Γ(m) (P ) = {i ∈ P : |hii| = m} = {i ∈ P : wP (ei ) = m} . We now describe the main result of this work: Theorem 1.2 Let P = {1, 2, . . . , n} be a poset and {e1 , e2 , . . . , en } be the canonical base of Fnq . Then T ∈ GLP (Fnq ) if and only if X xij eφ(i) T (ej ) = i∈hji where φ : P → P is an order automorphism and xjj 6= 0, for any j ∈ {1, 2, . . . , n}. Moreover, there is a pair of ordered bases β and β ′ of Fnq relative ¡ ¢ to which every linear isometry T ∈ GLP Fnq is represented by an n × n upper triangular matrix (aij )1≤i,j≤n with aii 6= 0 for every i ∈ {1, 2, . . . , n}. Proof. Since hsupp (ej )i is a prime ideal, it follows from Proposition 1.2 that hsupp (T (ej ))i is also a prime ideal, for every j ∈ {1, 2, . . . , n}. Given j ∈ {1, 2, . . . , n}, let j ′ = φ (j) be the unique maximal element of the ideal hsupp (T (ej ))i, where φ : P → P is the order automorphism induced by the isometry T (see Theorem 1.1). Then ¢® ­ ¡ hsupp (T (ej ))i = hsupp (ej ′ )i = supp eφ(j) , and since φ is a automorphism of order we have that ¢® ­ ¡ supp eφ(j) = {φ (i) : i ∈ hji} . 12 Therefore hsupp (T (ej ))i = {φ (i) : i ∈ hji}. Being φ (j) = max {φ (i) : i ∈ hji}, we conclude that X T (ej ) = xij eφ(i) (7) i∈hji with xjj 6= 0. It is straightforward to verify that for a given order automorphism φ : P → © P , any linear map ª defined as in (7) is a P -isometry. Let βm = ei : i ∈ Γ(m) (P ) and β = β1 ∪ β2 ∪ . . . ∪ βk . be a decomposition of the canonical base of Fnq as a disjoint union, where k = max {wP (ei ) : i = 1, 2, . . . , n}. We order this base β = {ei1 , ei2 , . . . , ein } in the following way (and denoted this total order by ≤β ): if eir ∈ βjr and eis ∈ βjs with r 6= s then, eir ≤β eis if and only jr ≤ js . In other words, we begin enumerating the the vectors of β1 and after exhausting them, we enumerate the vectors of β2 and so on. We define another ordered base β ′ as the base induced by the order automorphism φ, ª © β ′ := eφ(i1 ) , eφ(i2 ) , . . . , eφ(in ) and let A be the matrix of T relative to the basis β and β ′ : [T ]β,β ′ = A = (akl )1≤k,l≤n . We find by the construction of the bases β and β ′ that akl 6= 0 implies il ∈ hφ (ik )i. But il ∈ hφ (ik )i and hil i 6= hφ (ik )i implies that l < k so that A is upper triangular. Since A is invertible and upper triangular, we must have Qn ¤ det (A) = i=1 aii 6= 0 so that aii 6= 0, for every i ∈ {1, 2, . . . , n}. The upper triangular matrix obtained in the previous theorem is called a canonical form of T . We note that the ordered bases chosen in the theorem is unique up to re-ordination within the linearly independent sets βi , i = 1, 2, . . . , k. As in [14], a monomial matrix is a matrix with exactly one nonzero entry in each row and column. Thus a monomial matrix over F2 is a permutation matrix, and a monomial matrix over an arbitrary finite field is a permutation matrix times an invertible diagonal matrix. ¡ ¢ Corollary 1.1 Given T ∈ GLP Fnq there is an ordering β = {ei1 , ei2 , . . . , ein } of the canonical base such that [T ]β,β is given by the product A · U where A 13 is an invertible upper triangular matrix and U is a monomial matrix obtained from the identity matrix by permutation of the columns, corresponding to the automorphism of order induced by T . Proof. Let φ be the automorphism of order induced by T . Let Tφ−1 be the linear isometry defined as Tφ−1 (ej ) = eφ−1 (j) , for j ∈ {1, 2, . . . , n}. As we saw in Theorem 1.2, X T (ej ) = xij eφ(i) . i∈hji So, ¢ ¡ T ◦ Tφ−1 (ej ) = T eφ−1 (j) X = xiφ−1 (j) eφ(i) i∈hφ−1 (j)i = xiφ−1 (j) ej + X xiφ−1 (j) eφ(i) . i∈hφ−1 (j)i,i6=φ−1 (j) It follows that the automorphism of order induced by T ◦ Tφ−1 is the identity, so, when taking the base β ′ as in the Theorem 1.2, we find that β ′ = β and the matrix of T ◦ Tφ−1 relative to this base is an upper triangular matrix A = [T ◦ Tφ−1 ]β . But Tφ−1 acts on Fnq as a permutation of the vectors in β, so that in any ordered base containing those vectors, U −1 = [Tφ−1 ] is obtained from the identity matrix by permutation of the columns. We note that Tφ = (Tφ−1 )−1 and it follows that [T ]β = [T ◦ Tφ−1 ◦ Tφ ]β = [T ◦ Tφ−1 ]β [Tφ ]β = A · U. ¤ Given a poset P = {1, 2, . . . , n}, we denote by Aut (P ) the group of the order-automorphisms of P . © ª Corollary 1.2 Let P = {1, . . . , n} by a poset and k = max m : Γ(m) (P ) 6= ∅ . Then ! à k Y ¯ ¡ n ¢¯ (i) ¯GLP Fq ¯ = (q − 1)n · q (i−1)|Γ (P )| · |Aut (P )| . i=1 14 ¡ ¢ Proof. From Corollary 1.1, if T ∈ GLP Fnq there is an ordered base β = {ei1 , ei2 , . . . , ein } of the canonical base of Fnq such that |hil i| ≤ l for all l ∈ {1, 2, . . . , n} and [T ]β = A · U , being A = (akl )1≤k,l≤n an upper triangular matrix with akl = 0 if ik ∈ / hil i and U = [Tφ ]β the matrix representing the automorphism φ induced by linear isometry T (see Theorem 1.2). Moreover, such base β depends only on φ and for every φ ∈ Aut (P ), any matrix A as in the previous Corollary defines a linear P -isometry. Given l ∈ {1, 2, . . . , n}, there are (q − 1) possible different entries for all (since all 6= 0). But A is upper triangular, given 1 ≤ i < j ≤ n we have that aij 6= 0 only if i ∈ hji, so there are at most |hji| − 1 possible nonzero indices (i, j) with 1 ≤ i < j ≤ n, and for there are q possible different ¯ each of those ¯ entries. Since there are exactly ¯Γ(|hji|) (P )¯ such indices, we find that, up to considering the order automorphism induced by the isometry, there are ! à k Y (i) n q (i−1)|Γ (P )| (q − 1) · i=1 linear P -isometries and we conclude counting the elements of Aut(P ). ¤ Let Mn×n (Fq ) be the set of all n × n matrices over Fq and   ¯ ¯ Fq if i <P j   ¯ i=j GP = (aij ) ∈ Mn×n (Fq ) : aij ∈ ¯¯ F∗q if .   ¯ 0 otherwise ¡ ¢ As we have seen, this is the set of elements in GLP Fnq that corresponds to isometries that induces the trivial automorphism of order. So, we have the following characterization: ¡ n ¢ Corollary 1.3 With the definitions above, the group of isometries of F , d P q ¡ ¢ is the semi-direct product GLP Fnq ≃ GP ⋊Aut (P ). Proof. Let A = (aij ) and B = (bij ) be elements in GP . Since (AB)ij = n X aik bkj = k=1 X aik bkj i≤P k≤P j we have that AB ∈ GP . We note that every element in GP is an upper triangular matrix with nonzero diagonal entries. Hence, such elements are invertible. Since the inverse of an element in GP is a polynomial in that 15 element, ¡ n ¢such an element is in GP . So, we see¡ that ¢ GP is a subgroup of n GLP Fq . Since we already proved that GLP Fq = GP · Aut (P ), all is ¡ ¢ left to show is that GP is a normal subgroup of GLP Fnq . Given φ ∈ Sn , it acts on n × n matrices by permuting columns or rows. We denote by Aφ and φ A respectively the column of the matrix A. It is ¡φ and ¢−1 row permutation φ straightforward to show that Id = Id ([4]). It follows that ¡φ ¢ ¡φ ¢−1 φ φ Id A Id = A for every n × n matrix A. If A = (aij ) ∈ GP , for each i = 1, 2, ..., n we have that n X ¡φ ¢ ¡φ ¢−1 φ φ Id A Id (ei ) = A (ei ) = aφ(k)φ(i) ek k=1 X = aφ(k)φ(i) ek φ(k)≤P φ(i) = X aφ(k)φ(i) ek k≤P i ¡ ¢ and aφ(i)φ(i) 6= 0 for every i. Thus, we find that GP is normal in GLP Fnq and the proposition follows. ¤ a Corollary 1.4 Let P and Q be order posets. Then we have 1. GP ×Q = GP ⊗ GQ ; ◦ 2. GP ∪Q ≃ GP × GQ ; 3. If Q is a disjoint union of m’s posets P on {1, 2, ..., n}, then we have Aut (Q) ≃ Aut (P ) Sn . Proof. All the claims follow straight from the definitions. 2 ¤ Examples In this section, we illustrate the results of this paper with three examples, the main classes of poset-metrics: the posets that are disjoint union of chains, the weak order and the crown order. 16 ◦ ◦ ◦ Example 2.1 Let D = P1 ∪P2 ∪. . .∪Ps be a poset consisting of a disjoint union of r chains. Denoted by µi the cardinality of the i-th chain, i ∈ {1, 2, . . . , s}. For every j ∈ {1, 2, . . . , n} let νj = |{Pi : |Pi | = j}|. From Corollary 1.1 follows that there is an ordered base β of Fnq relative to which every linear isometry ¡ ¢ T ∈ GLP Fnq is represented by the product A · U of n × n matrices, where U is a monomial matrix that acts exchanging coordinate subspaces with isomorphic supports and   A1 0 0 · · · 0  0 A2 0 · · · 0      A =  0 0 A3 · · · 0  ,  .. .. .. . . . ..   . . . .  0 0 0 · · · As where each Ai is a µi × µi upper triangular matrix with non zero diagonal entries. If P = {1, 2, . . . , n} be a totally ordered set, then there¡ is ¢an ordered base β of Fnq relative to which every linear isometry T ∈ GLP Fnq is represented by the n × n upper triangular matrix with xii 6= 0 for every i ∈ {1, 2, . . . , n}. If R consisting of finite disjoint union of chains of equal lengths, then wR become the Rosenbloom-Tsfasman weight defined on the linear space Mn×m (Fq ) of all n × m matrices over Fq : if (aij ) ∈ Mn×m (Fq ), then wR ((aij )) = m X |hsupp (a1j , a2j , . . . , anj )i| . j=1 From Corrollary 1.3 ([10, Theorem 1]) it follows that GLP (Mn×m (Fq )) ≃ (Tn )m ⋊Sm , where (Tn )m denotes the direct product of m copies of the group Tn of all upper triangular matrices of size n over Fq with nonzero diagonal elements. Remark 2.1 For the case of modular rings Zn , we observed that if n 6= 2, there is no partial order P = {1, 2, . . . , m} such that the poset-weight wP coincide with the Lee weight wLee : if x = (x1 , . . . , xm ) ∈ Zm n then wLee (x) = m X min {|xi | , m − |xi |} , i=1 with 0 ≤ xi ≤ n the representative integer of the class xi . If n = 2 then wLee = wH . Therefore, if P is antichain and n = 2, then wP = wLee . Now, if 17 ³¥ ¦ ¥ n ¦´ n ∈ Zm , . . . , n , where ⌊x⌋ denotes the greatest integer 2 2 ¥ ¦ less than or equal to x, follows that wP (x) = m and wLee (x) = m · n2 > m. Hence wP (x) 6= wLee (x) (wP (x) < wLee (x)). In summary: if n 6= 2 is a positive integer, then there is no partial order P such that wP = wLee over Zm n. n 6= 2, taking y = Example 2.2 Let n1 , . . . , nt be positive integers with n1 + . . . + nt = n. Then W = n1 1⊕ . . . ⊕ nt 1 will denote the weak order given by the ordinal sum of the antichains ni 1 with ni elements (see [7]). Explicitly, W = n1 1⊕ . . . ⊕ nt 1 is the poset whose underlying set and order relation are given by {1, 2, . . . , n} = n1 1∪n2 1∪ . . . ∪ nt 1, ni 1 = {n1 + . . . + ni−1 + 1, n1 + . . . + ni−1 + 2, . . . , n1 + . . . + ni−1 + ni } and x < y if and only if x ∈ ni 1, y ∈ nj 1 for some i, j with i < j. Notice that if n1 = . . . = nt = 1, then W = 11⊕ . . . ⊕ 11 is totally ordered with 1 < 2 < . . . < t and if t = 1 then W = n1 is antichain. For a weak order W = n1 1⊕ . . . ⊕ nt 1 we have that Γ(m) (W ) = ns 1 if m = n1 +n2 + . . . + ns−1 + 1, for any s ∈ {1, 2, . . . , t} and Γ(m) (W ) = ∅ otherwise. The group of the automorphism of order Aut (W ) is isomorphic to the cartesian product Sn1 × Sn2 × . . . × Snt (Aut (W ) is just the group of the applications φ that permutes only the elements of each m-th level). Corollary 1.2 assures us then that à t ! Y ¯ ¡ n ¢¯ ¯GLW Fq ¯ = (q − 1)n · q ni (n1 +n2 +...+ni−1 +1) · n1 ! · n2 ! · . . . · nt !. i=2 From Theorem 1.2 follows that there are bases β and β ′ of Fnq such that the matrix [T ]β,β ′ is equal   Dn1 ×n1 ∗ ∗ ··· ∗   0 Dn2 ×n2 ∗ ··· ∗     0 0 Dn3 ×n3 · · · ∗ ,    .. .. .. . . . .   . . . . . 0 0 0 · · · Dnt ×nt where ¡ ¢ Dns ×ns = diag aΣns−1 +1,Σns−1 +1 , aΣns−1 +2,Σns−1 +2 , . . . , aΣns−1 +ns ,Σns−1 +ns 18 is a diagonal matrix for each s = 1, 2, . . . , t, and Σnj−1 := n1 + n2 + . . . + nj−1 . Considering the particular weak order W = 41⊕41⊕41 (Hasse diagram illustrated in Figure 1), the matrix of a linear P -isometry [T ]β,β ′ of T ∈ ¡ ¢ is an upper triangular matrix as bellow: GLW F12 q 9 10 11 12 5 6 7 8 1 2 3 4 Figure 1: Weak order W = 41⊕41⊕41.                     a1,1 0 0 0 0 0 0 0 0 0 0 0 0 a2,2 0 0 0 0 0 0 0 0 0 0 0 0 a3,3 0 0 0 0 0 0 0 0 0 0 0 0 a4,4 0 0 0 0 0 0 0 0 a1,5 a2,5 a3,5 a4,5 a5,5 0 0 0 0 0 0 0 a1,6 a2,6 a3,6 a4,6 0 a6,6 0 0 0 0 0 0 a1,7 a2,7 a3,7 a4,7 0 0 a7,7 0 0 0 0 0 a1,8 a2,8 a3,8 a4,8 0 0 0 a8,8 0 0 0 0 a1,9 a2,9 a3,9 a4,9 a5,9 a6,9 a7,9 a8,9 a9,9 0 0 0 a1,10 a2,10 a3,10 a4,10 a5,10 a6,10 a7,10 a8,10 0 a10,10 0 0 a1,11 a2,11 a3,11 a4,11 a5,11 a6,11 a7,11 a8,11 0 0 a11,11 0 a1,12 a2,12 a3,12 a4,12 a5,12 a6,12 a7,12 a8,12 0 0 0 a12,12                     Example 2.3 The crown is a poset with elements C = {1, 2, . . . , 2n}, n > 1, in which i < n + i, i + 1 < n + i for each i ∈ {1, 2, . . . , n − 1}, and 1 < 2n, n < 2n and these are the only strict comparabilities ([1]). The Hasse diagram of crown poset P with n = 4 is illustrated in Figure 2. Given a crown C = {1, 2, . . . , 2n}, we have that Aut (C) is isomorphic to the dihedral group Dn , consisting of the orthogonal transformations which preserve a regular n-sided polygon centered at the origin of the euclidian plane. Considering the usual inclusion ι : Dn → Sn , the action of Dn on C is defined 19 5 6 7 8 1 2 3 4 Figure 2: Crown poset P = {1, 2, 3, 4, 5, 6, 7, 8}. by g (k) = ½ ι ◦ g (k) for k = 1, 2, . . . , n ι ◦ g (k − n) for k = n + 1, . . . , 2n We note that Γ(1) (C) = {1, 2, . . . , n}, Γ(3) (C) = {n + 1, . . . , 2n}, and Γ(k) (C) = ∅, for k 6= 1, 3. So, it follows from Corollary 1.2 that ¯ ¡ ¢¯ ¯GLC F2n ¯ = (q − 1)2n · q 2n · 2n. q Theorem 1.2 assures there is a pair of ordered bases β and β ′ of Fnq relative to ¡ ¢ which every linear isometry T ∈ GLP Fnq is represented by the [T ]β,β ′ n × n upper triangular matrix                  a1,1 0 0 .. . 0 a2,2 0 .. . 0 0 a3,3 .. . ··· ··· ··· .. . 0 0 0 .. . a1,n+1 a2,n+1 0 .. . 0 0 0 .. . 0 0 0 .. . 0 0 0 .. . ··· ··· ··· ... an,n 0 0 .. . 0 an+1,n+1 0 .. . 0 0 0 0 0 0 ··· ··· 0 0 0 0 ··· ··· ··· .. . 0 0 0 .. . a1,2n 0 0 .. . an+2,n+2 .. . ··· ··· ··· ... an,2n−1 0 0 .. . an,2n 0 0 .. . 0 0 ··· ··· a2n−1,2n−1 0 0 a2n,2n 0 a2,n+2 a3,n+2 .. . 0 0          .        In the particular case when W = {1, 2, 3, 4, 5, 6, 7, 8} (see Figure 2), the 20 canonical form of a linear P -isometry  a1,1 0 0 0  0 a2,2 0 0   0 0 a3,3 0   0 0 0 a 4,4   0 0 0 0   0 0 0 0   0 0 0 0 0 0 0 0 is a1,5 a2,5 0 0 a5,5 0 0 0 0 a2,6 a3,6 0 0 a6,6 0 0 0 0 a3,7 a4,7 0 0 a7,7 0 a1,8 0 0 a4,8 0 0 0 a8,8       .      Example 2.4 The Boolean n-cube B n is the product of n chains of cardinality 2, that is, B n = 2 × 2 × · · · × 2 (n times) where 2 is a chain of cardinality 2. It is well known ([2]) that Aut (B n ) ≃ Sn . The Boolean cube may also be described as the Boolean order (defined by the set inclusion order) in the set P (n) of all subsets of {1, 2, ..., n}. So, we find that the order of subset µ ¶ n k {i1 , i2 , ..., ik } is 2 , and there are exactly subsets of cardinality k, that is, k ( ³n´ ¯ (m) ¯ if m = 2k ¯Γ (P )¯ = . k 0 otherwise It follows, from Corollary 1.2 that ¯ ¡ ¢¯ ¯GLB n Fnq ¯ = (q − 1)2n From Theorem 1.2, we know we the matrix [T ]β,β ′ is like  D1 A2  0 D2   0 0   0 0   .. ..  . . 0 0 µ ¶  n n (2i −1) Y  i  · q  n!. i=0 n can find ordered bases β and β ′ of F2q such A3 A4 C2,3 C2,4 D3 C3,4 0 D4 .. .. . . 0 0 21  · · · An · · · B2   · · · B3   · · · B4   . . . ..  .  · · · Dn ¡ ¢ ¡ ¢ where Di ¡is¢ an¡ ¢ni × ni diagonal matrix with determinant, Ai (Bi ) ¡n¢non¡nzero ¢ n n is an 1 × ( i × 1) matrix, and Ci,j is an i × j matrix, having (at least) ¡n¢ ¡ i ¢ i ¡ ¢ ¡ ¢ − j zero entries in each column and (at least) nj − n−i zero entries in j j−i each row. The computations done in all the examples of this work is summarize in the tables bellow. We recall we are denoting by T , D, A, W , C and B total, disjoint union of chains, antichain, weak, crowns and Boolean orders. We recall that νj is the number of the components in D with cardinality equal to j (see Exemple 2.1). Table 1: Aut (P ) and |Aut (P )|. P T D A W C B Aut (P ) {id} Sν1 × Sν2 × . . . × Sνn Sn Sn1 × Sn2 × . . . × Snt Dn Sn |Aut (P )| 1 ν1 ! · ν2 ! · . . . · νt ! n! n1 ! · n2 ! · . . . · n t ! 2n n! ¯ ¯ Table 2: Γ(m) (P ) 6= ∅ and ¯Γ(m) (P )¯. P T D A W C B Γ(m) (P ) 6= ∅ Γ(m) © (T ) = {1, 2, . . . , m} ª (m) Γ (D) = im , iΣµ1 +m , . . . , iΣµs−1 +m Γ(1) (A) = A Γ(Σns−1 +1) (W ) = ns 1 Γ(1) (C) = {1, 2, . . . , n} Γ(3) (C) = {n + 1, n + 2, . . . , 2n} Subsets of cardinality m if m = 2k ∅ otherwise ¯ ¡ ¢¯ Table 3: ¯GLP Fnq ¯. 22 ¯ (m) ¯ ¯Γ (P )¯ ¯ (m) m ¯ ¯Γ (D)¯ ≤ s n ns n ³n´ k 0 if m = 2k otherwise P T D A W C B ¯ ¡ ¢¯ ¯GLP Fnq ¯ Q i−1 (q − 1)n · (´ ni=2 ³Q ³Qq ) µ (µ −1) ´ k k s s 2 (q − 1)n · j=1 νj ! · k=1 q (q − 1)n · n! (q − 1)n · ´ ¢ ³Qt ni (Σni−1 +1) · q n ! j i=2 j=1 ¡Qt (q − 1)n · q n · n if n isµeven ¶  n i (2 −1) n Q i  (q − 1)2 ·  ni=0 q  n! ¯ ¡ ¢¯ In the table bellow we explicity compute ¯GLP Fnq ¯ for T , D, A, W , C and B with q = 2 and n = 2, 3, . . . , 10: Table 4: Numbers of linear isometries of |GLP (Fn2 )|. n |GLT (Fn2 )| 2 2 3 8 4 64 5 1024 6 32768 7 2097152 8 268435456 9 ∼ 6.8719 × 1010 10 ∼ 3.5184 × 1013 |GLA (Fn2 )| |GLC (Fn2 )| 2 8 6 ∗ 24 64 120 ∗ 720 384 5040 ∗ 40320 2048 362880 ∗ 3628800 10240 ¯ ¡ ¢¯ ¯GLB F22n ¯ 64 3145 728 ∼ 8. 854 4 × 1020 ∼ 3. 949 2 × 1065 ∼ 1. 102 2 × 10203 ∼ 3. 335 7 × 10623 ∼ 3. 977 8 × 101902 ∼ 4. 034 7 × 105776 ∼ 6. 687 5 × 1017473 References [1] J. Ahn, H. K. Kim, J. S. Kim and M. Kim, Classification of perfect linear codes with crown poset structure, Discrete Mathematics 268 (2003) 21-30. [2] R. Belding, Structures Characterizing Partially Odered Sets and Tehir Automorphism Groups, Discrete Mathematics 27 (1979) 117-126. [3] R. Brualdi, J. S. Graves and M. Lawrence, Codes with a poset metric, Discrete Mathematics 147 (1995) 57-72. 23 [4] S. H. Cho and D. S. Kim, Automorphism group of the crown-weight space, submitted [5] Y. Jang and J. Park, On a MacWilliams Type Identity and a Perfecteness for a Binary Linear (n, n − 1, j)-poset code, Discrete Mathematics 265 (2003) 85-104. [6] D. S. Kim, Association schemes and MacWilliams dualities for generalized Rosenbloom-Tsfasman poset, submittied. [7] D. S. Kim and J. G. Lee, A MacWilliams-Type Identity for Linear Codes on Weak Order, Discrete Mathematics 262 (2003) 181-194. [8] Jong Yoon Hyun and Hyun Kwang Kim, The poset strutures admitting the extended binary Hamming code to be a perfect code, Discrete Mathematics 288 (2004) 37-47. [9] Yongnam Lee, Projective systems and perfect codes with a poset metric, Finite Fields and Their Applications 10 (2004) 105-112. [10] K. Lee, Automorphism group of the Rosenbloom-Tsfasman space, Eur. J. Combin. 24 (2003) 607-612. [11] H. Niederreiter, A combinatorial problem for vector spaces over finite fields, Discrete Mathematics 96 (1991) 221-228. [12] M. Yu Rosenbloom and M. A. Tsfasman, Codes for the m-metric, Probl. Inf. Transm. 33 (1997) 45-52. [13] R. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole, Monterey, CA, 1986. [14] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. 24