We compute hyperdeterminants of hypermatrices whose indices belongs in a
meet-semilattice and whose entries depend only of the greatest lower bound of
the indices. One shows that an elementary expansion of such a polynomial allows
to generalize a theorem of Lindstr\"om to higher-dimensional determinants. And
we gave as an application generalizations of some results due to Lehmer, Li and
Haukkanen.Comment: New version of "A remark about factorizing GCD-type
  Hyperdeterminants". Title changed. Results, examples and remarks adde

Luque, Jean-Gabriel

English

arXiv

New version of "A remark about factorizing GCD-type Hyperdeterminants". Title changed. Results, examples and remarks added.International audienceWe compute hyperdeterminants of hypermatrices whose indices belongs in a meet-semilattice and whose entries depend only of the greatest lower bound of the indices. One shows that an elementary expansion of such a polynomial allows to generalize a theorem of Lindström to higher-dimensional determinants. And we gave as an application generalizations of some results due to Lehmer, Li and Haukkanen

HAL - UPEC / UPEM

Hyperdeterminants on semilattices

HAL - Normandie Université

HAL-Ecole des Ponts ParisTech

Hyperdeterminants on semilatticesJean-Gabriel LuqueTo cite this version:Jean-Gabriel Luque. Hyperdeterminants on semilattices. Linear and Multilinear Algebra,Taylor & Francis, 2007, 56 (3), pp.333-344. <hal-00085204v3>HAL Id: hal-00085204https://hal.archives-ouvertes.fr/hal-00085204v3Submitted on 22 Nov 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.hal-00085204, version 3 - 22 Nov 2006Hyperdeterminants on semilatticesJean-Gabriel Luque ∗November 22, 2006AbstractWe compute hyperdeterminants of hypermatrices whose indicesbelongs in a meet-semilattice and whose entries depend only of thegreatest lower bound of the indices. One shows that an elementaryexpansion of such a polynomial allows to generalize a theorem of Lind-stro¨m to higher-dimensional determinants. And we gave as an applica-tion generalizations of some results due to Lehmer, Li and Haukkanen.keywords:Hyperdeterminants, meet semilattices, GCD-matrices, multiplicative functions.AMS: 15A15, 15A69, 06A12.1 IntroductionSince the end of the nineteen century, it is known that some determinants,with entries depending only of the gcd of the indices, factorize. Readers in-terested in the story of the problem can refer to [13] and [2]. In 1876, Smith[15] evaluated the determinant of a GCD matrix whose entries belong to afactor closed set (i.e., a set which contains all the factors of its elements) asa product of Euler’s totient. The interest of this computation lies in its linkswith arithmetic functions [1] and in particular multiplicative functions (see[7, 8] for interesting remarks about the last notion). During the last century,many generalizations of Smith’s theorem have been investigated. One of the∗Institut Gaspard Monge, Universite´ de Marne-la-Valle´e,77454 Marne-la-Valle´e cedex 2Franceemail:Jean-Gabriel.Luque@univ-mlv.fr1ways to extend his result consists in changing the set of the indices of thematrices. In 1990, Li [10] gave the value of GCD determinant for an arbitraryset of indices. Beslin and Ligh [3] shown that such a determinant factorizeswhen the indices belongs to a gcd-closed set (i.e., a set which contains thegcd of any pairs of its elements) as a product of certain functions evaluatedin terms of Euler’s totient. The fact that these determinants factorize canbe seen as a corollary of a very elegant theorem due to Lindstro¨m [11] whichevaluated the determinant of the GCD-matrix whose indices are chosen in ameet semilattice, i.e., a poset such that each pair admits a greatest lowerbound. Another way to generalize Smith’s result consists in computing mul-tidimensional analogous. In 1930 Lehmer gave [9] the first multi-indexedversion of Smith’s determinant. Other related computation are collected in[16, 17]. More recently, Haukkanen [12] generalized the results of Beslin andLigh [3] and Li [10]) to hyperdeterminants.We will see in Section 2 that the main trick for computing these multidi-mensional determinants consists in expanding it as a sum of (classical 2-way)determinants. In the aim to highlight this method, we apply it to a moregeneral object DetF.In Section 3, we recall shortly a classical technic and give a slight generaliza-tion of Lindstro¨m’s Theorem. As a consequence, we give a multidimensionalanalogue of Lindstro¨m’s Theorem.In Section 4, one investigates minors of meet hypermatrices and generalizestwo theorems due to Haukkanen [12].2 Hyperdeterminants and F-determinantsThe question of extending the notion of determinant to higher dimensionalarrays has been raised by Cayley [5, 6] few after he introduced the modernnotation as square arrays [4]. The simplest generalization is defined for akth order tensor on an n-dimensional space M = (Mi1,··· ,ik)1≤i1,··· ,ik≤n by thealternated sumDetM =1n!∑σ=(σ1,··· ,σk)∈Sknsign(σ)Mσ,where sign(σ) = sign(σ1) · · · sign(σk), Mσ = Mσ1(1)...σk(1) · · ·Mσ1(n)...σk(n) andSn is the symmetric group. A straightforward computation gives DetM = 02if k is odd.For any k (even if k is odd), one defines the polynomialDet1M =∑σ=(Id,σ2,··· ,σk)∈Sknsign(σ)Mσ.When k is even the two notions coincide but for k odd, only Det1 does notvanish. This is a special case of the ”less-than-full-sign” determinant theorydue to Rice [14].Let us denote by F a map from Sk−2n to a commutative ring. One defines amore general object, which will be called F-determinant of M byDetF(M) =∑σ=(σ2,··· ,σk)∈Sk−1nsign(σ2)F(σ3, . . . , σk)∏iMiσ2(i)...σk(i).It exists an elementary identity which consists in expanding the F-determinantas a sum of (n!)k−2 classical (2-way) determinants.Lemma 2.1 (Determinantal expansion)One hasDetFM =∑σ3,...,σkF(σ3, . . . , σk) det(Mσ3,...,σk),where Mσ3,...,σk denotes the n×n matrix such that Mσ3,...,σki,j =Mi,j,σ3(i),...,σk(i).Proof – It suffices to remark thatDetFM =∑σ3,...,σkF(σ3, . . . , σk)∑σ2sign(σ2)∏iMiσ2(i)σ3(i)···σk(i)One of the most important property of hyperdeterminants is the invarianceunder the action of k copies of the special linear group. It is a very classicalresult which can be recover as a straightforward consequence of the followingproposition.Proposition 2.2 The polynomial DetFM is invariant under the action oflinear group on M in the following senseDetFg.M = det gDetFM, (1)whereg.M =( ∑1≤j2≤ngi2j2Mi1,j2,i3··· ,ik)1≤i1,··· ,ik≤n.3Proof – By applying Lemma 2.1 to g.M , one getsDetFg.M =∑σ3,...,σkF(σ3, . . . , σk) det(g.Mσ3,...,σk)=∑σ3,...,σkF(σ3, . . . , σk) det g det(Mσ3,...,σk)= det gDetFM.3 Hyperdeterminants of meet hypermatrices3.1 Meet semilatticeConsider a partially ordered finite set L so that every pairs (x, y) ∈ L2 hasa greatest lower bound denoted by x ∧ y. Such a poset is called a meetsemilattice. One defines classically its ζ function byζ(x, y) ={1 if x ≤ y,0 otherwise.Its Mo¨bius function is the inverse of the zeta function and can be computedby the inductionµ(x, y) =1 if x = y,−∑x≤z<y µ(z, y) if x < y,0 in the other cases.If F and f verify the equalityF (x) =∑y≤xf(y) =∑y∈Lζ(y, x)f(y), (2)then, one hasf(x) =∑y∈Lµ(y, x)F (y) = (F ⋆ µ)(x) (3)where the symbol ⋆ means the convolution product.43.2 Lindstro¨m TheoremThe factorization properties of the GCD determinants are the consequenceof the semilattice structure of the integers with respect to divisibility andcan be stated in a more general way. The manipulations of the identities (2)and (3) are the keys of the proof of Lindstro¨m’s Theorem [11]. We recall itsproof in a very slightly more general version.For each x ∈ L, one considers a fixed element zx ≤ x. Let Fx be afunction from L to C (or more generally to a commutative ring). Let M bethe matrix defined byM = (Fx(zx ∧ y))x,y∈L .Remark that it suffices to define Fx(z) only when z ≤ x. In particular, onecan suppose that Fx(z) = F (z, x) is an incidence function, i.e., F (x, z) = 0unless x ≤ z. One hasFx(zx ∧ y) =∑z∈Lζ(z, zx)ζ(z, y)fx(z),where fx(z) =∑y∈L µ(y, z)Fx(y). Hence, detM factorizes as the productdetM = detΦ. detZ,where Φ = (ζ(y, zx)fx(y))x,y∈L and Z = (ζ(x, y))x,y∈L. As Φ and Z aretriangular, detZ = 1 anddetΦ =∏xζ(x, zx)fx(x) =∏xfx(y) if zx = x for each x,0 otherwise.(4)Then one obtains Lindstro¨m’s Theorem.Theorem 3.1 (Lindstro¨m [11])det (Fx(zx ∧ y))x,y∈L ={ ∏x fx(x) if zx = x for each x,0 otherwise.(5)Note that, the original Lindsto¨m ’s Theorem deals with the case where zx = xfor each x. Furthermore, equality (4) generalizes a lemma of Cesaro.Lemma 3.2 (Cesaro)Denote by gcdm(n) = gcd(m,n). One has,(µ ∗ (f ◦ gcdm))(n) ={(f ∗ µ)(n) if m = n,0 otherwise,(6)where ∗ is the Dirichlet convolution and ◦ is the composition of functions.53.3 Linstro¨m’s Theorem for F-determinantsLindsto¨m’s Theorem can be extended to F-determinants.Theorem 3.3 (Lindstro¨m’s theorem for F-determinants)If L = {x1, . . . , xn} denotes a meet semilattice, one hasDetF(Fxi1 (zxi1 ∧ · · · ∧ xik))=F(Id, · · · , Id)∏xfx(x) if zx = x for each x0 otherwiseProof – Lemma 2.1 givesDetF (Fx1(zx1 ∧ · · · ∧ xk))x1,··· ,xk∈L =∑σ3,...,σkF(σ3, . . . , σk) det(Fxi(zxi ∧ xj ∧ xσ3(i) ∧ · · · ∧ xσk(i))i,jFrom Linstro¨m’s Theorem (Theorem 3.1), one hasdet(Fxi(zxi ∧ xj ∧ xσ3(i) ∧ · · · ∧ xσk(i))6= 0if and only if for each xi one haszxi ∧ xσ3(i) ∧ · · · ∧ xσk(i) = xi.Equivalently,zxi ∧ xσ3(i) ∧ · · · ∧ xσk(i) ≥ xi,for eacu i. Hence, σ3 = · · · = σk = Id and zxi = xi for each i. The resultfollows.Example 3.4 Consider the semilattice constituted with two elements 2 ≥ 1.The expansion of the F-determinant givesDetF(Fi(i ∧ j ∧ k ∧ l)) = F(12, 12)∣∣∣∣ F1(1) F1(1)F2(1) F2(2)∣∣∣∣+ (F(12, 21) + F(21, 12) + F(21, 21))∣∣∣∣ F1(1) F1(1)F2(1) F2(1)∣∣∣∣= F(12, 12)F1(1)(F2(2) − F2(1))= F(12, 12)f1(2)f2(2)64 Minors of meet Hypermatrices4.1 Meet closed subsetsConsider a meet closed subset S of L (i.e., a subset closed under the opera-tion ∧) and fix a linear extension l = y1 . . . yn of S. Following the notationsof [2], we denote by x E yi the relation x ≤ yi and x 6≤ yj for each j < i.Consider a pair of functions f and F verifyingF (yi) =∑x≤yix∈Lf(x) (7)and set fˆ(yi) =∑xEyif(x). One has the following lemma.Lemma 4.1F (yi) =∑yk≤yifˆ(yk). (8)Proof – Remarking that for each x ∈ L, it exists i such that x E yi (it sufficesto set yi = min{j|x ≤ yj} and that x E yi and x E yj implies i = j, we haveF (yi) =∑x≤yix∈Lf(x) =∑yk≤yi∑xEykf(x) =∑yk≤yifˆ(yk).Note that this identity appears in [2] (Theorem 4.1, p 7). Hence, usingTheorem 3.3 and Lemma 4.1, one generalizes a result by Altinisik, Saganand Tuglu ([2], Theorem 4.1 p 7).Corollary 4.2DetF(Fyi1 (yi1 ∧ · · · ∧ yik)) = F(Id, · · · , Id)n∏i=1(∑x1Eyi∑x2∈Lµ(x1, x2)Fyi(x2)).Example 4.3 Consider the semilattice L given by its Hasse diagramL =4 5↑ ր ↑2 3↑ ր17where i→ j means i ≤ j. The sublattice generated by 2, 4 and 5,S =4 5↑ ր2is meet closed andfˆ2(2) = f2(2) + f2(1)fˆ4(4) = f4(4)fˆ5(5) = f5(5) + f5(3).Hence,DetF (Fi(i ∧ j ∧ k))i,j,k∈S = F(123)∣∣∣∣∣∣F2(2) F2(2) F2(2)F4(2) F4(4) F4(2)F5(2) F5(2) F5(5)∣∣∣∣∣∣+F(213)∣∣∣∣∣∣F2(2) F2(2) F2(2)F4(2) F4(2) F4(2)F5(2) F5(2) F5(5)∣∣∣∣∣∣+F(321)∣∣∣∣∣∣F2(2) F2(2) F2(2)F4(2) F4(2) F4(2)F5(2) F5(2) F5(2)∣∣∣∣∣∣+(F(132) + F(231) + F(312))∣∣∣∣∣∣F2(2) F2(2) F2(2)F4(2) F4(2) F4(2)F5(2) F5(2) F4(2)∣∣∣∣∣∣The permutation 123 is the only one having a non zero contribution inthis sum. Hence,DetF (Fi(i ∧ j ∧ k))i,j,k∈S = F(123)F2(2)(F4(4)− F4(2))(F5(5)− F5(2))= F(123)(f2(2) + f2(1))f4(4)(f5(5) + f5(3))= F(123)fˆ2(2)fˆ4(4)fˆ5(5)Remark 4.4 If L is the semilattice structure of the integers with respect todivisibility. By the specialization F(σ3, . . . , σk) = sign(σ3) . . . sign(σk), onerecovers the computations of Lehmer [9] as a special case of Theorem 3.3and the result of Haukkanen ([12] Theorem 1. p 56) from Corollary 4.2.4.2 Factor closed subsetsLet S be a factor closed subset of L. Then, f = fˆ andDetF (Fx1(x1 ∧ · · · ∧ xk))x1,...,xk∈S = F(Id, · · · , Id)∏xfx(x). (9)8As special cases of equality (9), one recovers Lehmer’s identities [9] andthe original result of Smith [15].4.3 General caseLet X = {x1, . . . , xn} be a subposet of a meet semi-lattice L. We will de-note by X = {x1, . . . , xn, xn+1, . . . , xm} the smallest factor-closed subset ofL containing X. The aim of this section consists in investigating the F -determinantDF (X) := DetF(Fxi1 (zxi1 ∧ xi2 ∧ · · · ∧ xik))1≤i1,...,ik≤n(10)where for each x ∈ X, zx denotes a fixed element of X such that z ≤ x. As inthe previous section, the result follows from the case k = 2 and Proposition2.1.Let us consider first the determinant det(Fxi(zxi ∧ xj))1≤i,j≤n. The functionsFxi can be chosen such that F (x, y) is an incidence function. The set Xbeing closed by factors, the functions fx and fˆx are equal. Hence,Fxi(zxi ∧ xj) =n+m∑k=1Cxi,xkζ(xk, xj) (11)whereCx,y = fx(y)ζ(y, zx) ={fx(y) if y ≤ zx0 otherwise.(12)One hasProposition 4.5det(Fxi(xi ∧ xj)) =∑1≤k1<···<kn≤n+mdet(Cxi,xkj)1≤i,j≤ndet (ζ(xki, xj))1≤i,j≤n .(13)9Proof – By multi-linearity, one obtainsdet(Fxi(xi ∧ xj)) = det(n+m∑k=1Cxi,xkζ(xk, xj))=∑1≤k1,...,kn≤n+mdet(Cxi,xkj ζ(xkj , xj))=∑1≤k1,...,kn≤n+mdet(Cxi,xkj)∏iζ(xki , xi)=∑1≤k1<···<kn≤n+mdet(Cxi,xkj)1≤i,j≤ndet (ζ(xki , xj))1≤i,j≤nExample 4.6 Let us consider the semilattice L whose Hasse diagram is4 5 61OO88rrrrrrrrrrrrrr2OO]]::::::::3]]::::::::OO0AA~~~~~~~~]]::::::::OOand X = {4, 5, 6}. We set z4 = 1, z5 = 2 and z6 = 6 . Consider thedeterminantdet(Fi(zi ∧ j ∧ k))i,j,k∈X =∣∣∣∣∣∣f4(1) + f4(0) f4(0) f4(1) + f4(0)f5(2) + f5(0) f5(2) + f5(0) f5(0)f6(1) + f6(0) f6(3) + f6(0) f6(6) + f6(3) + f6(1) + f6(0)∣∣∣∣∣∣= −f4(0)f6(1)f5 (2) + f4 (1)f6 0f5 (2) + f4 (1)f5 (0)f6 (3)+f4 1f5 (0)f6 (6) + f4 (0)f5 (2)f6 3 + 2 f4 (1)f5 (2)f6 (3)+f4 (1)f5 (2)f6 (6)Using the multilinearity of det, one recovers the expression given by Propo-10sition 4.5det(Fi(zi ∧ j)) =∣∣∣∣∣∣f4(0) f4(1) 0f5(0) 0 f5(2)f6(0) f6(1) 0∣∣∣∣∣∣ ·∣∣∣∣∣∣1 1 01 0 11 1 0∣∣∣∣∣∣+∣∣∣∣∣∣f4 (0) f4 (1 ) 0f5 (0) 0 0f6(0) f6(1) f6 (3)∣∣∣∣∣∣ ·∣∣∣∣∣∣1 1 01 0 01 1 1∣∣∣∣∣∣+∣∣∣∣∣∣f4 (0) f4 (1) 0f5(0) 0 0f6(0) f6(1) f6 (6)∣∣∣∣∣∣ ·∣∣∣∣∣∣1 1 01 0 01 1 1∣∣∣∣∣∣+∣∣∣∣∣∣f4 (0) 0 0f5(0) f5 (2) 0f6(0) 0 f6 (3)∣∣∣∣∣∣ ·∣∣∣∣∣∣1 0 01 1 01 0 1∣∣∣∣∣∣+∣∣∣∣∣∣f4 (1) 0 00 f5 (2) 0f6(1) 0 f6 (3)∣∣∣∣∣∣ ·∣∣∣∣∣∣1 0 00 1 01 0 1∣∣∣∣∣∣+∣∣∣∣∣∣f4 (1) 0 00 f5 (2) 0f6(1) 0 f6 (6)∣∣∣∣∣∣ ·∣∣∣∣∣∣1 0 00 1 01 0 1∣∣∣∣∣∣More generally, one has a multi-indexed version of Proposition 4.5.Theorem 4.7DF (X) =∑1≤k1<···<kn≤n+mDetF(fxi1 (xki2 )ζ(xki2 , zxi1 ∧ xi3 ∧ · · · ∧ xik))×× det (ζ(xki, xj))(14)Proof – We use Lemma 2.1 to expand DF (X) and we obtainDF (X) =∑σ1,...,σkF(σ1, . . . , σk) det(Fxi(zxi ∧ xj ∧ xσ3(i) ∧ · · · ∧ xσk(i))).Now, by Proposition 4.5, one getsDF (X) =∑σ1,...,σkF(σ1, . . . , σk)∑1≤k1<···<kn≤n+mdet (ζ(xki, xj))×× det(fxi(xkjζ(xkj , zxi ∧ xσ3(i) ∧ · · · ∧ xσk(i))))=∑1≤k1<···<kn≤n+mdet (ζ(xki, xj))∑σ1,...,σkF(σ1, . . . , σk)×× det(fxi(xkjζ(xkj , zxi ∧ xσ3(i) ∧ · · · ∧ xσk(i))))=∑1≤k1<···<kn≤n+mDetF(fxi1 (xki2 )ζ(xki2 , zxi1 ∧ xi3 ∧ · · · ∧ xik))×× det (ζ(xki, xj))This ends the proof.Let set Fx1 = Fx2 = · · · = Fxn = F then for 1 ≤ k1, k2, . . . , kn ≤ n, we get11Corollary 4.8DetF(F (zxi1 ∧ xi2 ∧ · · · ∧ xik))=∑1≤k1<···<kn≤n+m∏if(xki)××DetF(ζ(xki2 , zxi1 ∧ xi3 ∧ · · · ∧ xik))det (ζ(xki, xj))Proof – By applying the equality,DetF(f(xki1 )ζ(xki2 , zxi1 ∧ xi3 ∧ · · · ∧ xik))=∑σ3,...,σkF(σ3, · · · , σk) det(f(xki)ζ(xkj , zxi ∧ xσ1(i) ∧ · · · ∧ xσk(i)))=∏if(xki)DetF(ζ(xki2 , zxi1 ∧ xi3 ∧ · · · ∧ xik)).to identity 14, one obtains the result.Remark 4.9 Assume that L is the integer lattice. Then, if zx = x foreach x ∈ X and F(σ3, · · · , σk) = sign(σ3) . . . sign(σk) in Corollary 4.8, onerecovers Theorem 2 in [12]. Moreover, Proposition 4.5 generalizes the resultof Li[10].References[1] T. Apostol, Introduction to Analytic Number Theory, Springer (1976)[2] E. Altinisik, B.E. Sagan, N. Tuglu, GCD matrices, posets and noninter-secting paths, Linear and Multilinear Algebra 53(2) (2005) 75-84.[3] S. Beslin and S. Ligh, Another generalization of Smith’s determinant,Bull. Austral. Math. Soc, 40: 413-415 (1989)[4] A Cayley, On the theory of determinants, Trans. Cambridge Phil. Soc.VIII (1843), 1–16.[5] A Cayley, Me´moire sur les hyperde´terminants, Journ. Reine und Angew.Math., 30 (1846), 1–37.[6] A Cayley, On the theory of permutants, Cambridge and Dublin Math.Journal VII (1851), 40–51.12[7] A. Lascoux, Addition of 1, Se´minaire Lotharingien, Mars 04. 8p.[8] A. Lascoux, Multiplicative functions,http://www.combinatorics.net/lascoux/courses/dvi ps/Moebiusps.rar[9] D. H. Lehmer, The p dimensional analogue of Smith’s determinant,Amer. Math. Monthly 37:294-296 (1930).[10] Z. Li, The determinants of gcd matrices, Linear Algebra Appl., 134:137-143 (1990)[11] B. Lindstro¨m, Determinants on semilattices, Proc. Amer. Math. Soc. 20(1969), 207-208.[12] P. Haukkanen, Higher-Dimensionnal GCD matrices, Linear AlgebraAppl., 170:53-63 (1992)[13] C. Krattenthaler, Advanced determinant calculus: a complement, LinearAlgebra Appl., 411 (2005), 68-166.[14] L.H. Rice, P-way determinants with an application to transvectants,Amer. J. Math. 40:242-262 (1918)[15] H. J. S. Smith, On the value of certain arithmetical determinant, Proc.London Math. Soc. (1) 7 (1876), 208-212 (p. 62)[16] N P Sokolov, Spatial matrices and their applications (in Russian), Go-sudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960.[17] N P Sokolov, Introduction to the theory of multidimensional matrices(in Russian), Nukova Dumka, Kiev, 1972.13

Hyperdeterminants on semilattices

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