On the property of subalgebras of evolution algebras

ON THE PROPERTY OF SUBALGEBRAS OF EVOLUTION ALGEBRAS L.M. CAMACHO, A.KH. KHUDOYBERDIYEV, B.A. OMIROV Abstract. In this paper we study subalgebras of complex finite dimensional evolution algebras. We obtain the classification of nilpotent evolution algebras whose any subalgebra is an evolution subalgebra with a basis which can be extended to a natural basis of algebra. Moreover, we formulate three conjectures related to description of such non-nilpotent algebras. Mathematics Subject Classification 2010: 17D92, 17D99. Key Words and Phrases: evolution algebra, evolution subalgebra, nitpotency. 1. Introduction Nowadays, the algebraic approach is effectively used in the study of the genetics, dynamical systems in population biology. In 20s and 30s of the last century the new object was introduced to mathematics, which was the product of interactions between Mendelian genetics and mathematics. One of the first scientist who gave an algebraic interpretation of the “ ×” sign, which indicated sexual reproduction was Serebrowsky [13]. Etherington introduced the formal language of abstract algebra to the study of the genetics [6]-[7]. An algebraic approach in genetics consists of the study of various types of genetic algebras (like algebras of free, ”self-reproductive” and bisexual populations, Bernstein algebras). Until 1980s, the most comprehensive reference in this area was W¨orz-Busekros’s book [15]. A good survey on algebraic structure of genetic inheritance is the Reed’s article [11]. More recent results, such as genetic evolution in genetic algebras, can be found in the Lyubich’s book [10]. Recently in the book of J.P. Tian [14] a new type of evolution algebra was introduced. This algebra describes some evolution laws of the genetics. The study of evolution algebras constitutes a new subject both in algebra and the theory of dynamical systems. In the Tian’s book a foundation of the framework of the theory of evolution algebras is established and some applications of evolution algebras in the theory of stochastic processes and genetics are discussed. Evolution algebras are in general non-associative and do not belong to any of the well-known classes of non-associative algebras. In fact, nilpotency, right nilpotency and solvability might be interpreted in a biological way as a various types of vanishing (“deaths”) populations. Although an evolution algebra is an abstract system, it gives an insight for the study of non-Mendelian genetics. For instance, an evolution algebra can be applied to the inheritance of organelle genes, one can predict, in particular, all possible mechanisms to establish the homoplasmy of cell populations. Recently, Rozikov and Tian [12] studied algebraic structures of evolution algebras associated with Gibbs measures defined on some graphs. In the papers [2], [5], [9] derivations, some properties of chain of evolution algebras and dibaricity of evolution algebras were studied. Certain algebraic properties of evolution algebras (like right nilpotency, nilpotency and solvability etc.) in terms of matrix of structural constants have been investigated in [1], [3], [4]. It is remarkable that a subalgebra and an ideal of a genetic algebra of population, biologically can be interpreted correspondingly as a subpopulation and a dominant subpopulation with respect to mating. 1 2 L.M. CAMACHO, A.KH. KHUDOYBERDIYEV, B.A. OMIROV This paper is devoted to study of subalgebras of finite dimensional evolution algebras. In order to achieve our goal we organize the paper as follows. In Section 2, we give some necessary notions and preliminary results about evolution algebras. We consider several types of subalgebras of evolution algebras and present examples of difference of such subalgebras as well. Section 3 is devoted to description of evolution algebras of permutations satisfying that any subalgebra is evolution subalgebra with a natural basis which can be extended to a natural basis of the algebra (condition P). For the list of two-dimensional evolution algebras we identify their subalgebras. In Section 4, we classify the nilpotent complex evolution algebras satisfying the condition P. In Section 5, we formulate three conjectures related to the description of such non-nilpotent algebras. Through the paper all algebras are assumed complex and finite dimensional. 2. Preliminaries. In this section we give necessary definitions and preliminaries results for understanding main results of the paper. Let us define the main object of this work - evolution algebra. Definition 2.1. [14] Let Ebe an algebra over a field F. If it admits a basis {e1, e2,...}such that ei·ej= 0 for i 6=j, ei·ei=X k ai,kekfor any i, then algebra Eis called evolution algebra. The basis {e1, e2,...}is said to be natural basis of evolution algebra E. It is remarkable that this type of algebra depends on natural basis {e1, e2,...}. We denote by A= (aij ) the matrix of the structural constants of the evolution algebra E. Definition 2.2. [14] Let Ebe an evolution algebra and E1be a subspace of E. If E1has a natural basis {ei|i∈Λ1}which can be extended to a natural basis {ej|j∈Λ}of E, then E1is called an evolution subalgebra, where Λ1and Λ are index sets and Λ1is a subset of Λ. In fact, for the linear subspace E1of evolution algebra Ewe can consider three conceptions of subalgebras. (1) E1is subalgebra in ordinary sense; (2) E1is subalgebra and there exists a natural basis of E1; (3) E1is subalgebra and there exists a natural basis of E1which can be extended to a natural basis of E. Note that Definition 2.2 agrees with the third conception of subalgebra. Below we present examples which show that conceptions 1 - 3 are different in general. Example 2.3. Let Ebe a three dimensional evolution algebra with a natural basis {e1, e2, e3}and the table of multiplication e1·e1=e1+e2, e2·e2=−e1−e2, e3·e3=e2+e3. It is not difficult to see that E1=< e1+e2, e2+e3>is a subalgebra, but E1is not an evolution subalgebra (that is, there does not exist a natural basis of E1). Indeed, if we assume the contrary, i.e., in the subspace E1there exists a natural basis {f1, f2}, then f1=α1(e1+e2) + α2(e2+e3), f2=β1(e1+e2) + β2(e2+e3). SUBALGEBRAS OF EVOLUTION ALGEBRAS 3 with α1β2−α2β16= 0. From the condition f1·f2= 0 we derive α1=α2= 0 or α2=β2= 0 or β1=β2= 0. Consequently, we get a contradiction with the assumption that {f1, f2}is a natural basis of E1. Example 2.4. Let Ebe a three dimensional evolution algebra with a natural basis {e1, e2, e3}and the following table of multiplication e1·e1=e1+e2+e3, e2·e2=−e1−e2+e3, e3·e3= 0. It is not difficult to see that E1=< e1+e2, e3>is an evolution algebra with a natural basis {e1+e2, e3}, but this basis can not be extended to a natural basis of evolution algebra E. If we assume that there exists a natural basis {f1, f2}of E1such that {f1, f2, f3}is a natural basis of E, then f1=α1(e1+e2) + α2e3, f2=β1(e1+e2) + β2e3, f3=γ1e1+γ2e2+γ3e3. From conditions f1·f3=f2·f3= 0 we deduce α1=β1= 0 or γ1=γ2= 0.Therefore, we get a contradiction with the assumption that {f1, f2, f3}is a basis. For the sake of convenience, we introduce the following definition. Definition 2.5. An evolution algebra Eis said to satisfy a condition Pif any subalgebra of Eis an evolution subalgebra with a natural basis which can be extended to a natural basis of E. In [14] the conditions for basis transformations that preserve naturalness of the basis are given. The relation between the matrices of structure constants in a new and old natural basis is established in terms of new defined operation on matrices, as well. Since the relation is not practical for our further purposes, we give the following brief version of isomorphism. Let us consider non-singular linear transformation Tof a given natural basis {e1,...,en}with a matrix (tij)1≤i,j≤nin this basis and fi= n X j=1 tijej,1≤i≤n. This transformation is isomorphism if and only if fi·fj= 0 for all i6=j. In the following theorem we present a list (up to isomorphism) of 2-dimensional evolution algebras. Theorem 2.6. [4] Any 2-dimensional non-abelian evolution algebra Eis isomorphic to one of the following, pairwise non-isomorphic, algebras: (1) dim E2= 1 •E1:e1e1=e1, •E2:e1e1=e1, e2e2=e1, •E3:e1e1=e1+e2, e2e2=−e1−e2, •E4:e1e1=e2. (2) dim E2= 2 •E5:e1e1=e1+a2e2, e2e2=a3e1+e2,1−a2a36= 0, where E5(a2, a3)∼ =E′ 5(a3, a2), •E6:e1e1=e2, e2e2=e1+a4e2, where for a46= 0,E6(a4)∼ =E6(a′ 4)⇔a′ 4 a4= cos 2πk 3+isin 2πk 3for some k= 0,1,2. 4 L.M. CAMACHO, A.KH. KHUDOYBERDIYEV, B.A. OMIROV Consider the following k-dimensional evolution algebras ESk:(ei·ei=ei+1,1≤i≤k−1, ek·ek=e1,ENk:(ei·ei=ei+1,1≤i≤k−1, ek·ek= 0. In [8], the authors describe a complex evolution algebra En,π(a1, a2,...,an) with a basis {e1, e2,...,en}and the table of multiplications as follows: (ei·ei=aieπ(i),1≤i≤n, ei·ej= 0, i 6=j, where πis an element of the group of permutations Sn. Namely, the following assertion is true. Theorem 2.7. An arbitrary evolution algebra En,π(a1, a2,...,an)is isomorphic to a direct sum of evolution algebras ESp1, ESp2, . . . , ESps, ENk1, ENk2, . . . , ENkr,i.e., En,π(a1, a2,...,an)∼ =ESp1⊕ESp2⊕···⊕ESps⊕ENk1⊕ENk2⊕···⊕ENkr, where s X i=1 pi+ r X j=1 kj=n We introduce the following sequence: Ek= k−1 X i=1 EiEk−i, k ≥1 Definition 2.8. An evolution algebra Eis called nilpotent if there exists n∈Nsuch that En= 0 and the minimal such number is called index of nilpotency. Theorem 2.9. [1] Let Ebe an n−dimensional evolution algebra. Then Eis nilpotent if and only if the matrix of structure constants Acan be transformed by permutation of the natural basis to the following form: A=         0a12 a13 . . . a1n 0 0 a23 . . . a2n 0 0 0 . . . a3n . . .. . .. . ..... . . 0 0 0 ... 0         . The next theorem gives the classification of evolution algebras with maximal possible index of nilpotency. Theorem 2.10. [1] Any n-dimensional complex evolution algebra with maximal index of nilpotency is isomorphic to one of pairwise non-isomorphic algebras with the following matrix of structural constants              0 1 a13 . . . a1,n−10 0 0 1 . . . a2,n−10 0 0 0 . . . a3,n−10 . . .. . .. . .··· . . .. . . 0 0 0 ··· 0 1 0 0 0 ··· 0 0              , where one of non-zero aij can be chosen equal to 1. SUBALGEBRAS OF EVOLUTION ALGEBRAS 5 The set of all evolution algebras whose matrices of structural constants have the form of Theorem 2.10 will be denoted by ZNn. 3. Main result First we investigate which evolution algebras of the list of Theorem 2.6 satisfy (or not) the condition P. Proposition 3.1. Evolution algebras E1and E4satisfy the condition P. Proof. Since Eis a two dimensional, then any non-trivial subalgebra of Eis one-dimensional. Let E′ 1be an one-dimensional subalgebra of E1and E′ 1=< x > with x=A1e1+A2e2. Consider x·x= (A1e1+A2e2)·(A1e1+A2e2) = A2 1e1 On the other hand, x·x=αx =α(A1e1+A2e2). Then A2 1=αA1and αA2= 0. •If α= 0,then A1= 0 and {e2}is the basis of E′ 1.Obviously, this basis is extendable to the natural basis {e1, e2}of E1. •If α6= 0,then A1=α, A2= 0 and {e1}is the basis of E′ 1,which is also extendable to the natural basis of E1. The assertion of proposition regarding the algebra E4is carried out in a similar way.  Proposition 3.2. Evolution algebras E2, E3, E5and E6do not satisfy the condition P. Proof. 1. Let E′ 2be a one-dimensional subalgebra of E2and E′ 2=< x > with x=A1e1+A2e2.The equality x·x=αx implies A2 1+A2 2=αA1and αA2= 0.We are seeking a subalgebra with a natural basis that can not be extended to a basis of the algebra. Thus, α= 0. We set A2=iA1. Then x=A1(e1+ie2).Let us assume that the basis {x}can be extended to the natural basis of E2,that is, there exists y∈E2such that {x, y}is a natural basis of E2.Let y=B1e1+B2e2,then 0 = x·y=A1(e1+ie2)·(B1e1+B2e2) = A1(B1+iB2)e1. Hence B2=iB1and we obtain a contradiction with the linear independence of elements xand y. Thus, the evolution algebra E2does not satisfy the condition P. 2. Let E′ 3=< x > be a one-dimensional subalgebra of E3with x=A1e1+A2e2. Putting A1=A2= 1 we conclude that E′ 3=< x > is a subalgebra. Let us assume that x=e1+e2can be extended to a natural basis of E3,then there exists y=B1e1+B2e2,such that {x, y}is a natural basis of E3. From the following equality 0 = x·y= (e1+e2)·(B1e1+B2e2) = (B1−B2)e1+ (B1−B2)e2, we derive B2=B1,which is a contradiction with condition of {x, y}being a basis. Therefore, evolution algebra E3does not satisfy the condition P. 6 L.M. CAMACHO, A.KH. KHUDOYBERDIYEV, B.A. OMIROV 3. The element x=A1e1+A2e2forms a basis of a one-dimensional subalgebra of E5. Therefore, αx =x·xfor some α∈C. Note, that the condition dimE5= 2 implies x·x6= 0 (consequently α6= 0). Without loss of generality we can assume that α= 1. Then x=x·xdeduce    A2 1+A2 2a3=A1, A2 1a2+A2 2=A2.(3.1) It is not difficult to check that the system of equation (3.1) has a solution A1, A2such that A1A26= 0. Indeed, if a2=a3= 0,then A1=A2= 1 is a solution of the equation (3.1). Let us assume that (a2, a3)6= (0,0) then, without loss of generality, we can suppose a36= 0.Then from the equation (3.1) we have A2=A1 a3((a2a3−1)A1+ 1),(3.2) A3 1+2 a2a3−1A2 1+a3+1 a2a3−1A1−1 (a2a3−1)2= 0.(3.3) Note that the equation (3.3) with respect to A1has three solutions and one of them does not equal to −1 a2a3−1.Recall that all solutions equal to −1 a2a3−1has the following cubic equation A3 1+3 a2a3−1A2 1+3 (a2a3−1)2A1+1 (a2a3−1)3= 0. Therefore, the equation (3.1) has a solution A1, A2with A1A26= 0.Consequently, there exists a subalgebra E′ 5=< x > with x=A1e1+A2e2,where A1A26= 0. The basis of this subalgebra can not be extended to a natural basis of E5.Indeed, if y=B1e1+B2e2 with condition that {x, y}is a natural basis of E, then 0 = x·y= (A1e1+A2e2)·(B1e1+B2e2) = (A1B1+A2B2a3)e1+ (A1B1a2+A2B2)e2, which implies A1B1+A2B2a3= 0, A1B1a2+A2B2= 0. Since A1A2(1 −a2a3)6= 0,we get B1=B2= 0.It is a contradiction with condition of {x, y}being a basis. Therefore, the two dimensional evolution algebra E5does not satisfy the condition P. 4. The assertion that the algebra E6does not satisfy the condition Pis carried out by applying similar arguments as for the algebra E5. Next, we present a result on preservation of the property Pfor a direct sum of evolution algebra which satisfy the condition Pand abelian algebra. Proposition 3.3. Let Ebe an n−dimensional evolution algebra which satisfies the condition P. Then the evolution algebra E⊕Ckalso satisfies the condition P. Proof. Let {e1, e2,...,en, h1, h2,...,hk}be a basis of E⊕Ckand Mbe an s-dimensional subalgebra of E⊕Ck.We set {x1, x2,...,xs}as a basis of Mand xi= n P j=1 βi,jej+ k P j=1 γi,jhj. Consider xi·xj= n X t=1 βi,tβj,t n X k=1 at,kek,1≤i, j ≤s. SUBALGEBRAS OF EVOLUTION ALGEBRAS 7 Since xi·xjbelong to M, then the elements n P t=1 βi,tβj,t n P k=1 at,kekare expressed by linear combinations of elements yi= n P j=1 βi,jej,1≤i≤s. Consider N=< y1, y2,...,ys>. It is easy to see that Nis a subalgebra of Eof dimension s′≤s. For the sake of convenience, by renumeration of indexes, we can assume that basis of Nis {y1, y2,...,y′ s}. If s′=s, then using conditions of proposition we can find a natural basis {y1, y2,...,ys, z1, z2,..., zn−s}of E. Thus the following basis {x1, x2,...,xs, z1, z2,...,zn−s, h1, h2...,hk}is a natural basis of E⊕Ck. If s′< s, then by elementary transformation of matrices we conclude       β1,1. . . β1,n γ1,1. . . γ1,k β2,1. . . β2,n γ2,1. . . γ2,k . . .. . .. . .. . .. . .. . . βs,1. . . βs,n γs,1. . . γs,k      ∼               β1,1. . . β1,n γ1,1. . . γ1,k β2,1. . . β2,n γ2,1. . . γ2,k . . .. . .. . .. . .. . .. . . βs′,1. . . βs′,n γs′,1. . . γs′,k 0... 0γ′ s′+1,1. . . γ′ s′+1,k . . .. . .. . .. . .. . .. . . 0... 0γ′ s,1. . . γ′ s,k               . Hence, the following elements x′ i=         n P j=1 βi,jej+ k P j=1 γi,jhj1≤i≤s′ k P j=1 γ′ i,jhjs′+ 1 ≤i≤s form a natural basis of M. Now, we show that this basis is extendable to a natural basis of E⊕Ck.Due to Nbeing a subalgebra of E, we derive the existence of a natural basis {y1, y2,...,ys′, z1, z2,...,zn−s′}of E. It is not difficult to check that the following basis {x′ 1, x′ 2,...,x′ s′, z1, z2,...,zn−s′, x′ s′+1, x′ s′+2,...,x′ s, h′ 1, h′ 2...,hk+s′−s} is a natural basis of E⊕Ck,where {h′ 1, h′ 2...,hk+s′−s}are the complementary basis elements to {x′ s′+1, x′ s′+2,...,x′ s}in Ck. Let Ebe an n-dimensional evolution algebra such that E=E1⊕E2, where E1and E2are the evolution subalgebras of E. Proposition 3.4. Let Ebe a algebra satisfying the condition P. Then the subalgebras E1and E2also satisfy the condition P. Proof. Let E′ 1be a subalgebra of E1,then E′ 1is a subalgebra of E. Therefore there exist a natural basis {e′ 1, e′ 2,...,e′ m}of E′ 1which can be extended to a natural basis {e′ 1, e′ 2,...,e′ m, xm+1, xm+2,...,xn}of E. Since E=E1⊕E2,then xj=yj+zjwith yj∈E1, zj∈E2, m + 1 ≤j≤n. From e′ i·xk= 0 and xk·xt= 0 we deduce e′ i·yk= 0 and yk·yt=zk·zt= 0.Since {e′ 1, e′ 2,...,e′ m, xm+1, xm+2,...,xn} is a basis of E, then any element of E1belongs to < e′ 1, e′ 2,...,e′ m, ym+1, ym+2,...,yn> . From the elements ym+1, ym+2,...,ynwe choose some such that {e′ 1, e′ 2,...,e′ m, yj1, yj1,...,yjk}is a basis of E1. Thus, E1satisfies the condition P. 8 L.M. CAMACHO, A.KH. KHUDOYBERDIYEV, B.A. OMIROV The next example shows that the converse assertion of Proposition 3.4 is not true in general. Example 3.5. Let Ebe a 4−dimensional evolution algebra defined by a direct sum of two-dimensional evolution algebras E1and E2, where E1:e1·e1=e2;E2:e3·e3=e4. Clearly, E1and E2are algebras satisfying the condition P, but Enot. Indeed, the subalgebra L=< e1+e3, e2+e4>is not an evolution subalgebra. In the following proposition we identify evolution algebras with the condition Pamong the algebras of the type En,π(a1, a2,...,an). Proposition 3.6. Let Ebe an n-dimensional evolution algebra of the type En,π(a1, a2,...,an)which satisfies the condition P. Then Eis isomorphic to one of the following non-isomorphic algebras: ES1⊕Cn−1, ENs⊕Cn−s, ES1⊕ENs⊕Cn−s−1. Proof. Let Ebe an algebra of the type En,π(a1, a2,...,an), then by Theorem 2.7 we have E∼ =ESp1⊕ESp2⊕···⊕ESps⊕ENk1⊕ENk2⊕···⊕ENkr. Proposition 3.4 we obtain that the algebras ESpiand ENkisatisfy the condition P. If there exists pj≥2 with 1 ≤j≤sthen, we have ESpj:   ei·ei=ei+1,1≤i≤pj−1, epj·epj=e1, This algebra does not satisfy the condition P, because the one-dimensional subalgebra < x > with x=e1+e2+···+epjis not an evolution subalgebra. Thus, pj= 1 for any j∈ {1,...,s}. If there exist iand jsuch that pi=pj= 1,then from Example 3.5 we conclude that Edoes not satisfy the condition P. Therefore, we can assume p1= 1 and pj= 0 for 2 ≤j≤s. Let us suppose that there exist iand jsuch that ki≥2, kj≥2.Without loss of generality we can suppose i= 1, j = 2 and k1≥k2.We denote {e1, e2,...,ek1}and {f1, f2,...,fk2}the basis of ENk1 and ENk2, respectively. Then M=< x1, x2,...,xk2>with xi=ek1−k2+i+fi,1≤i≤k2form a subalgebra of Ewith the following products xi·xi=xi+1,1≤i≤k2−1, xk2·xk2= 0. It is not difficult to check that Mis not an evolution subalgebra. Thus, we get a contradiction with the assumption that there exist iand jsuch that ki≥2, kj≥2.Therefore, we can assume kj= 1 for 2≤j≤r. Since EN1is a one-dimensional algebra with trivial multiplication, then by Proposition 3.3 it is enough to consider the case s=r= 1,that is, we reduce the study to ESp⊕ENkwith p∈ {0,1}. •In the case of p= 1 and k= 1 we obtain the algebra ES1⊕Cn−1; •In the case of p= 1 and k≥2 we obtain the algebra ES1⊕ENk⊕Cn−k−1; •In the case of p= 0,we obtain the algebra ENk⊕Cn−k. It is not difficult to check that all obtained algebras ES1⊕Cn−1, ENs⊕Cn−s, ES1⊕ENs⊕Cn−s−1 satisfy the condition P. SUBALGEBRAS OF EVOLUTION ALGEBRAS 9 4. Nilpotent case. Let Ebe an n-dimensional non-abelian evolution algebra with a natural basis {e1, e2,...,en}.By transformation of the basic elements we get the following table of multiplication e2 i6= 0,1≤i≤k, e2 i= 0, k + 1 ≤i≤n, k ≤n. (4.1) We consider the notation given in Theorem 2.9. Proposition 4.1. Let rank(A)< k. Then Edoes not satisfy the condition P. Proof. We shall prove the statement of proposition by the contrary. Let us assume that rank(A) = s < k, then there exist the indexes i1, i2,...,issuch that the elements e2 i1, e2 i2,...,e2 isare linearly independent. For the sake of convenience we shall assume that e2 1, e2 2,...,e2 sare linearly independent. Consider the non-trivial linear combination α1e2 1+α2e2 2+···+αse2 s+αs+1e2 s+1 = 0. Since αs+1 6= 0 (otherwise we obtain trivial linear combination) we get e2 s+1 =−α1 αs+1 e2 1−α2 αs+1 e2 2−···− αs αs+1 e2 s. Due to existence αi6= 0 for some 1 ≤i≤s, without loss of generality, we can assume α16= 0. For the element x=√α1e1+√α2e2+···+√αses+√αs+1es+1 we have x·x= 0.Hence, < x > is an one-dimensional subalgebra. Consequently, there exist a natural basis {x, y2, y3,...,yn}of E. Let us introduce the following denotations yi= n X j=1 βi,jej,2≤i≤n. Consider 0 = x·yi= ( s+1 X j=1 √αjej)·( n X j=1 βi,jej) = s+1 X j=1 √αjβi,je2 j= s X j=1 √αjβi,je2 j−√αs+1βi,s+1 s X j=1 αj αs+1 e2 j= s X j=1 √αjβi,j −√αs+1βi,s+1 αj αs+1 e2 j. Thus, √αjβi,j −√αs+1βi,s+1 αj αs+1 = 0,2≤i≤n, 1≤j≤s. (4.2) For j= 1 in the restrictions (4.2) we obtain βi,s+1 =rαs+1 α1 βi,1,2≤i≤n. We have that {x, y2, y3,...,yn}and {e1, e2,...,en}are two bases of E. Then the matrix of change of basis has the following form: B=             √α1... √αs√αs+1 0... 0 β2,1. . . β2,s qαs+1 α1β2,1β2,s+2 . . . β2,n β3,1. . . β3,s qαs+1 α1β3,1β3,s+2 . . . β3,n . . .. . .. . .. . .. . .. . .. . . βn,1. . . βn,s qαs+1 α1βn,1βn,s+2 . . . βn,n             .