Miniversal deformations of marked matrices

Miniversal Deformations of Marked Matrices Albert Compta, Josep Ferrer and Ferran Puerta Departament de Matem`atica Aplicada I. E.T.S. Ingenieria Industrial de Barcelona. UPC Diagonal 647. 08028 Barcelona. Spain e-mails: [email protected]c.es, [email protected]c.es, [email protected]c.es Abstract Given the set of square matrices M⊂Mn+m(C) that keep the subspace W=Cnx{0}⊂ Cn+minvariant, we obtain the implicit form of a miniversal deformation of a matrix a∈M, and we compute it explicitely when this matrix is marked (this is, if there is a permutation matrix p∈Mn+m(C) such that p−1ap is a Jordan matrix). We derive some applications to tackle the classical Carlson problem. 1 Introduction In [4] one proves that all the solutions to the Carlson problem appear in any neighbourhood of the simplest matrices, the so-called marked matrices. Studying the perturbations of this type of matrices is the central goal of this paper. More precisely, we recall that the Carlson problem consists in obtaining the possible Jordan invariants of a matrix of the form a=AC 0B when those of Aand Bare prescribed. Notice that the 0 block means that the corresponding subspace is a-invariant, Aand Bbeing the matrices of the restriction to this invariant subspace and of the corresponding quotient map, respectively, in the associated basis. As is well known one can assume that A,Bare nilpotent Jordan matrices. Then, trivial solutions for the Segre characterisitic of aare obtained by taking Cin such a way that ais marked; that is to say, abecomes a Jordan matrix by conjugation with a permutation matrix. As we have said above, any other solution can be realized by perturbing marked matrices; therefore, each solution is represented in the versal deformations of some of them. Versal deformation has been introduced by Arnold in [1] to study the variations of the invariants of a square matrix when its entries are perturbed, and thanks to a natural generalization contained in [12], the same technique has been applied to pairs, pencils, etc. ([5], [8],... ). In [7] versal deformations of invariant subspaces with regard to a fixed endomorphism are described. Here the subspace is prescribed; thus we are interested in a local canonical form of the differentiable families of endomorphisms having this invariant subspace, or equivalently, of square matrices as aabove. 1 2A. Compta, J. Ferrer and F. Puerta In particular, we characterize a miniversal deformation of a matrix of the form awith regard to the changes of basis which preserve the 0 block, and we compute it explicitely when ais marked nilpotent. We then derive the referred applications to tackle the Carlson problem. The organization of this paper is as follows. In section (2), we obtain the implicit form of a miniversal deformation of the matrix aby applying Arnold’s technique (2.10). In section (3), we apply this theorem to obtain an explicit form of a first miniversal deformation of a marked matrix (3.8) and in (3.13), we obtain a second miniversal deformation without repeated parameters. Finally, we study the relation between the obtained deformations and the Carlson problem in the last section. Particularly, the deformations that preserve the restriction Aand the quotient B or, in other words, the pair of partitions (γ,β) of their Segre characteristics are the deformations with the only non zero parameters in the right upper block. We note that we obtain matricial realizations of all the compatible Littlewood-Richardson sequences with the pair (γ,β)ofSegre characteristics among the deformations of a matricial realization of the Carlson compatible triple (γ∪β,γ,β) (4.4). By the union partition we mean the reordered union of two sets of partitions. We denote by Mthe set of matrices that preserves the subspace Cnx{0}⊂Cn+m, M={a∈Mn+m(C): a=AC 0B,A∈Mn(C),B ∈Mm(C),C ∈Mn,m(C)}. a∗will be the conjugate-transposed matrix of aand Gwill be the group of invertible matrices of M. Apartition α=(α1,α 2,...,α (α),0,...) is a non increasing sequence of non negative integers α1≥α2≥···≥α(α)>0 where (α)isitslength and |α|=α1+α2+···+α(α)its weight. The conjugate partition α∗=(α∗ 1,α ∗ 2,...) of the partition αis defined by α∗ j=#{1≤i≤(α): αi≥j}. Notice that α∗ 1=(α), (α∗)=α1,|α∗|=|α|,(α∗)∗=α. Let αand βbe two partitions. Then, the union partition α∪βis the partition obtained by reordering the union of the two sets of partitions. 2 Miniversal Deformations In order to study the perturbations of the numerical invariants of a square matrix with regard to the usual conjugation relation associated to changes of basis, Arnold introduces the so-called versal deformations in [1]. The starting point is the fact that the corresponding equivalence classes are orbits by the action of the linear group and, hence, they are submanifolds. Versal Miniversal Deformations of Marked Matrices 3 deformations can then be obtained as submanifolds which are transverse to the orbit of the given matrix. Arnold’s techniques can be generalized to other situations where this basic fact holds. Let us see that this is our case. Definition 2.1 We consider the action of the group Gon the differentiable manifold Mdefined by the conjugation G×M −→ M (p, a)−→ p∗a=p−1ap The orbit of the matrix a∈M,Oa={p∗a:p∈G}, is the equivalence class of a∈Mwith regard to the relation given by the group action. Definition 2.2 Let Vbe a manifold (for example, Mor G). A deformation of a∈Vis a differentiable map ϕ:Λ−→ V where Λis a neighbourhood of the origin in Cland ϕ(0) = a. We also say that the image ϕ(Λ) is a family of deformations of the central element a∈V. The set Λis called the basis of the deformation and lits dimension. We say that λiis a parameter of the deformation if λ=(λ1,λ 2,...,λ l)∈Λ. For example, every local parametrization of a submanifold S⊂Vwhere a∈Sis a deformation of a. We will simply say that Sis a deformation of a. A deformation is called “versal” if any other deformation is induced from it in the following sense: Definition 2.3 A deformation of a∈M,ϕ:Λ−→ M (Λ⊂Cr)iscalledversal if, given another deformation of a∈M,ψ:Γ−→ M , there is a neighbourhood of the origin Γ⊂Γ,a differentiable map ρ:Γ −→ Λand a deformation of I∈G,δ:Γ −→ G such that ψ(τ)=δ(τ)∗ϕ(ρ(τ)) ,∀τ∈Γ. It is called miniversal if it has the minimal dimension of all the versal deformations. Remark 2.4 It is enough to compute a miniversal deformation of a point of the orbit; then, a miniversal deformation of any other point of the same orbit is induced from it by means of the group action. The “closed orbit lemma ” ([12], p. 37) ensures that the referred starting point of Arnold’s techniques holds in our case: Proposition 2.5 For all a∈M, the orbit Oaby the action of the algebraic group G,isa submanifold of Mlocally closed where the boundary is the union of orbits of strictly smaller dimension. 4A. Compta, J. Ferrer and F. Puerta Now, we recall the key relation between “versality ” and “transversality ”. Definition 2.6 Let S⊂Vbe a submanifold of the manifold Vand ϕ:Λ−→ V be a differentiable map. We say that ϕis transverse to Sin λ∈Λif ϕ(λ)∈Sand the tangent space to Vin the point ϕ(λ)verifies Tϕ(λ)V=Imdϕλ+Tϕ(λ)S. In particular, if Λis a submanifold of V(and ϕis the inclusion), we say that it is transverse to Sin λ∈Λif TλV=TλΛ+TλS. We say that it is minitransverse ifthesumisadirectsum. As we have said above, the key point is the following proposition, proved in [1] for square matrices, and which can be generalized (for example [12]) to the cases like the ones here, where the equivalence classes are submanifolds given as orbits by the action of a Lie group. Proposition 2.7 A deformation ϕ:Λ−→ M of a∈Mis versal/miniversal if and only if it is transverse/minitransverse to the orbit Oain the origin O∈Λ. Corollary 2.8 A miniversal deformation of a∈Mis determined by any supplementary subspace of TaOain TaM=M. Namely, if {e1,e 2,...,e r}is a basis of a supplementary subspace of TaOain M, a miniversal deformation of a∈Mis ϕ(λ1,λ 2,...,λ r)=a+λ1e1+λ2e2+···+λrer. Moreover, ris the codimension of Oa. Finally, we recall the following result giving an explicit description of TaOa: Proposition 2.9 Let the matrix a∈Mand Oabe its orbit by the action of the group G; then, the tangent space to this orbit in the point a∈Mis TaOa={[a, p]:p∈M}, where [a, p]=ap −pa. Now, we are able to state and prove the main result of this section: Theorem 2.10 Let a=AC 0B∈M. Then, a miniversal deformation of this matrix is determined by the linear submanifold a+N, where Nis the subspace formed by the matrices x=XZ 0Y∈M verifying the conditions Miniversal Deformations of Marked Matrices 5 (1) A∗Z−ZB∗=0 (2) [A∗,X]−ZC∗=0 (3) [Y,B∗]−C∗Z=0 Proof. We consider the hermitien product in Mdefined by <x,p>=tr(xp∗) where x=XZ 0Yand p=PR 0Q. Because of (2.8), a miniversal deformation of ais given by a+N, where Nis the orthogonal subspace of TaOa. So, a matrix x∈Mwill be in Nif and only if <x,[a, p]>=0,∀p∈M. Since [a, p]=AP −PA AR+CQ −PC −RB 0BQ −QB , this condition is equivalent to tr(XP∗A∗−XA∗P∗+ZR∗A∗+ZQ∗C∗−ZC∗P∗−ZB∗R∗)+tr(YQ ∗B∗−YB ∗Q∗)=0 ∀p∈M. Then, because of the invariance of the trace by the circular permutations, the last condition is equivalent to tr(A∗XP∗−XA∗P∗+A∗ZR∗−ZC∗P∗−ZB∗R∗)+tr(C∗ZQ∗+B∗YQ ∗−YB ∗Q∗)=0 ∀p∈M. Getting the common factors out, it becomes tr((A∗X−XA∗−ZC∗)P∗+(A∗Z−ZB∗)R∗)+tr((C∗Z+B∗Y−YB ∗)Q∗)=0 ∀P∈Mn(C),Q∈Mm(C),R∈Mn,m(C), which is equivalent to <A∗X−XA∗−ZC∗A∗Z−ZB∗ 0C∗Z+B∗Y−YB ∗,PR 0Q>=0 ∀p∈M. Hence, x∈Nif and only if the first matrix is zero. 6A. Compta, J. Ferrer and F. Puerta 3 Obtention of Miniversal Deformations of Marked Matrices We recall that if fis an endomorphism of a finite dimensional vector space X,anf-invariant subspace Fof Xis said to be marked if there is a Jordan basis of Fthat can be extended to a Jordan basis of Xwith regard to f. Definition 3.1 We say that a∈Mis a marked matrix if Cn×{0}is an a-invariant marked subspace of Cn+m. Notice that if Aand Bare nilpotent Jordan matrices; this means that there is a permutation matrix p∈Gsuch that p−1ap is a nilpotent Jordan matrix. As we have said in the introduction, we will solve the equations in (2.10) explicitely in those cases when ais a nilpotent marked matrix. Because of (2.4), it is sufficient to obtain the versal deformation of any matrix in this orbit. It is easily see that any nilpotent marked matrix is equivalent to a matrix of the form described in the following definition: Definition 3.2 We say that a marked nilpotent matrix a∈Mis in canonical form if (1) A=diag(A1,...,A r), where A1,...,A rare nilpotent matrices in Jordan form of sizes γ1,...,γ rrespectively, and γ1+···+γr=n. (2) B=diag(B1,...,B s), where B1,...,B sare nilpotent matrices in Jordan form of sizes β1,...,β srespectively, and β1+···+βs=m. (3) C=[Ci,j]1≤i≤r,1≤j≤sCi,j ∈Mγi,βj(C)such that Cii =     0...01 0 0 . . . 0     if 1 ≤i≤ρ≤min(r, s) and Cij =0for any other cases. (4) γ1≥γ2≥···≥γρ. (5) βi≥βi+1 if 0<i<ρ γi=γi+1 (6) γρ+1 ≥γρ+2 ≥···≥γr. (7) βρ+1 ≥βρ+2 ≥···≥βs. We say that the matrix a=AC 0Bis of type (˜γ, ˜ β,ρ), where ˜γ=(γ1,...,γ r)and ˜ β= (β1,...,β s) Observe that ρis the number of chains in a Jordan basis of Cn×{0}with regard to Athat can be extended to chains of a Jordan basis of Cn+mwith regard to a. Also notice that ˜γand ˜ βhave the same elements as γand β, but they are not in non increasing order. Miniversal Deformations of Marked Matrices 7 Example 3.3 The next matrix ais a marked nilpotent matrix in canonical form of type ((3,2,1,3,1),(2,4,1,3,2),3): 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Then, the Segre characteristic of ais (5,6,2,3,3,2,1). When we solve the set of equations in (2.10), the following special type of Toeplitz matrices will often appear: Definition 3.4 (1) We say that a matrix X=(xi,j)∈Mγ,β(C)is a T-matrix if it is a Toeplitz matrix; that is to say, if it is constant along the diagonals. (2) If Xis a T-matrix such that all the diagonals from the (λ+1)th are zero (beginning to count from the right upper corner), we say that Xis a λT-matrix. (3) If Xis a λT-matrix, where λ=min(γ,β), we simply say that Xis a UTT-matrix (upper triangular Toeplitz matrix). For example,   123 412 041  ,00234 00023 ,    12 31 03 00     are 4T, 3T and 3T matrices, respectively, and   123 012 001  ,00012 00001 ,    12 01 00 00     are UTT-matrices. 8A. Compta, J. Ferrer and F. Puerta Definition 3.5 We say that a block matrix X=Xi,j1≤i≤r,1≤j≤sXi,j ∈Mγi,βj(C) is a block T-matrix if each block Xi,j is a T-matrix. We define a block UTT-matrix analogously. We are now going to solve equations (1), (2) and (3) in (2.10) when the matrix a∈Mis a marked nilpotent matrix in canonical form. Lemma 3.6 Let M∈Mγ(C)andN∈Mβ(C) be Jordan nilpotent non derogatory matrices. Then, a matrix Z∈Mγ,β(C) verifies the equation (1) M∗Z−ZN∗=0 if and only if Zis a UTT-matrix. Proof. It is clear that Z=(zi,j) verifies        z2,1z2,2... z 2,β z3,1z3,2... z 3,β . . .. . .. . . zγ,1zγ,2... z γ,β 00... 0        =         0z1,1z1,2... z 1,β−1 0z2,1z2,2... z 2,β−1 . . .. . .. . .. . . . . .. . .. . .. . . 0zγ,1zγ,2... z γ,β−1         , which is equivalent to zh, =zh+1,+1.So,Zis a T-matrix, and being zh,1=zγ,β−=0if h,  > 1, we conclude that Zis a UTT-matrix. In order to solve equations (1), (2) and (3) in (2.10), we decompose X,Y,Zinto blocks: X=Xi,j1≤i,j≤rXi,j ∈Mγi,γj(C) Y=Yt,k1≤t,k≤sYt,k ∈Mβt,βk(C) Z=Zi,k1≤i≤r,1≤k≤sZi,k ∈Mγi,βk(C). The next lemma follows immediately from (3.2): Lemma 3.7 With the notation in (3.2) the equations (1), (2) and (3) in (2.10) are equivalent to the following ones: (1) A∗ iZik −ZikB∗ k=0,1≤i≤r, 1≤k≤s (2) A∗ iXij −XijA∗ j=ZijC∗ jj ,1≤i≤r, 1≤j≤ρ (3) A∗ iXij −XijA∗ j=0,1≤i≤r, ρ<j ≤r (4) YtkB∗ k−B∗ tYtk =C∗ ttZtk ,1≤t≤ρ, 1≤k≤s (5) YtkB∗ k−B∗ tYtk =0,ρ<t≤s, 1≤k≤s. Miniversal Deformations of Marked Matrices 9 Consequently, in order to obtain the solution of the above set of equations, we are led to consider the following four sets of equations: (I) If 1 ≤i, j ≤ρ 1.I A∗ iZij −ZijB∗ j=0. 2.I A∗ iXij −XijA∗ j=ZijC∗ jj. 3.I YijB∗ j−B∗ iYij =C∗ iiZij. (II) If i>ρ,j≤ρ 1.II A∗ iZij −ZijB∗ j=0. 2.II A∗ iXij −XijA∗ j=ZijC∗ jj. 3.II YijB∗ j−B∗ iYij =0. (III) If i≤ρ,j>ρ 1.III A∗ iZij −ZijB∗ j=0. 2.III A∗ iXij −XijA∗ j=0. 3.III YijB∗ j−B∗ iYij =C∗ iiZij. (IV) If i, j > ρ 1.IV A∗ iZij −ZijB∗ j=0. 2.IV A∗ iXij −XijA∗ j=0. 3.IV YijB∗ j−B∗ iYij =0. The following theorem describes the corresponding solutions: Theorem 3.8 (First Miniversal Deformation) Let a∈Mbe a marked nilpotent matrix in canonical form of type (˜γ, ˜ β,p). Then, a miniversal deformation of a∈Mis a+Nwhere Nis the subspace formed by the matrices x=XZ OY such that (I) 1≤i, j ≤ρ a) If γi≤γjor βi≥βj Zij =0and Xij,Yij are UTT-matrices. b) If γi>γ jand βi<β j Zij are µijT-matrices where µij =min(γi−γj,β j−βi). Xij are (γj+µij)T-matrices and the diagonals γj+1,...,γ j+µij are equal to the diagonals 1,...,µ ij of Zij. Yij are (βi+µij)T-matrices and the diagonals βi+1,...,β i+µij are equal to the diagonals 1,...,µ ij of Zij. (II) i>ρ,j≤ρ a) If γi≤γj 16 A. Compta, J. Ferrer and F. Puerta Example 3.14 The new miniversal deformation in example (3.11) is 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ∗∗∗ ∗∗∗∗∗∗∗ ∗∗ ∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗ ∗∗ ∗∗ ∗∗∗ ∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗ 4 Relation with the Carlson Problem We recall that the Carlson problem asks about the Jordan invariants of a∈Mwhen A, B, C vary in such a way that the Jordan invariants of the restriction block and the quotient block are preserved. It is well-known that the problem can be reduced to the nilpotent case (and even to the particular case when Aand Bare Jordan matrices); this means that only the Segre characteristics are involved. Hence, we define: Definition 4.1 Let α,γ,βbe three partitions with |α|=n,|γ|=d,|β|=n−d. We say that αis Carlson-compatible with (γ,β)(or that the triple (α, γ, β)is Carlson-compatible) if there is a nilpotent matrix a=AC 0Bsuch that the Segre characteristics of a, A, B are α, γ, β respectively. Then we say that ais a Carlson-realization of (γ,β)or, more precisely, of the triple (α, γ, β). For example, the marked matrices in (3.2) are Carlson-realizations of (γ,β). In general, the matrices in (3.13) do not preserve the invariants of the initial one. However, because of Arnold’s deformations of a square matrix (see [1]), γand βare preserved if and only if X= 0 and Y= 0, respectively. Hence, the miniversal deformation in (3.13) gives a representation of the Carlson-realizations of (γ,β) near the initial matrix. More precisely: Proposition 4.2 Let a∈Mbe a marked nilpotent matrix in canonical form of type (˜γ, ˜ β,p) and ϕits miniversal deformation in (3.13). Given any deformation of a∈M,ψ:Γ−→ M , Miniversal Deformations of Marked Matrices 17 such that ψ(τ)is a Carlson-realization of (γ,β)for any τ∈Γ, there is a neighbourhood of the origin Γ⊂Γ, a differentiable map ρ3:Γ −→ Cµand a deformation of I∈G,δ:Γ −→ G such that ψ(τ)=(δ(τ))−1ϕ(0,0,ρ 3(τ))δ(τ). Proof. In general, for any deformation of a∈M, there are ˆx=ρ1(τ),ˆy=ρ2(τ),ˆz=ρ3(τ)such that ψ(τ)=(δ(τ))−1ϕ(ρ1(τ),ρ 2(τ),ρ 3(τ))δ(τ). If ψ(τ) is a realization of (γ,β), the restriction block and the quotient block in ϕ(ρ1(τ),ρ 2(τ),ρ 3(τ)) must have Segre characteristic γand βrespectively. As we have commented above, this is only possible if ρ1(τ)=0andρ2(τ)=0. In particular, we conclude with the following result: Corollary 4.3 Let abe a marked nilpotent matrix in canonical form of type (˜γ, ˜ β,p).Ifa is stable by the deformations that preserve the Segre characteristics of the restriction and the quotient, then (i) ρ=min(r, s). (ii) γi≥γi+1. (iii) βi≥βi+1. Proof. The number of parameters in Zmust be zero, that is to say µ=0. More generally, let us see that any Carlson-compatible partition with (γ,β) appears in the miniversal deformation (3.13) of a matrix aof type (γ,β,0) by taking X= 0 and Y= 0 (Notice that the above matrix ais a trivial Carlson-realization of the triple (γ∪β,γ,β)). This representation improves the well known ”condensation lemma” which asserts that any Carlson-compatible partition with a given pair (γ,β) can be realized by means of a matrix N(γ)C 0N(β), where N(γ), N(β) are nilpotent Jordan matrices having Segre characteristic γ,β, respectively; the only non zero entries in Care the ones placed in the rows which correspond to null rows in N(γ). Here we prove that several of these entries in Ccan be assumed to be zero. Theorem 4.4 Given a pair of partitions (γ,β), realizations of all the Carlson-compatible partitions with them are obtained by considering the miniversal deformation (3.13) for p=0,and taking X=0and Y=0. In particular, they are of the form N(γ)Z 0N(β), where N(γ),N(β)are nilpotent Jordan matrices having Segre characteristic γ,β respectively; the only non zero entries in Zare some of the ones placed in the rows which correspond to null rows in N(γ). In addition, the parameters in Zcan be taken as small as desired. Proof. In [4] it is shown that realizations of all Carlson-compatible partitions with (γ,β) occur in any neighbourhood of the marked ones. Hence, all of them appear in the set of the miniversal 18 A. Compta, J. Ferrer and F. Puerta deformations (3.13) when all possible types (˜γ, ˜ β,p) are considered for fixed (γ,β). Finally, notice that all these nilpotent marked matrices in canonical form of type (˜γ, ˜ β,p) appear in the miniversal deformation of the one of type (γ,β,0); it is sufficient to take all the entries valued 0, except some z1 ij, in such a way that there is at least a non zero entry for every iand for every j. Remark 4.5 In particular it follows that there are realizations of all the Carlson-compatible partitions with (γ,β)in any neighbourhood of the trivial one a=diag(N(γ),N(β)).Notice that, according to the notation in (3.2), this matrix is a marked nilpotent matrix in canonical form of type (γ,β,0). Example 4.6 Let γ=(3,3,2,1,1) and β=(4,3,2,2,1) be the Segre characteristics of example (3.3). Then, we have that γ∪β=(4,3,3,3,2,2,2,1,1,1). The family of deformations that preserves the pair (γ,β) and that has matricial realizations of all the compatible Littlewood-Richardson sequences is 1 1 1 1 1 1 1 1 1 1 1 1 ∗∗ ∗∗ ∗∗∗ ∗∗∗∗ ∗∗ ∗∗∗ ∗∗∗∗ ∗∗∗∗ ∗∗ ∗∗∗∗∗∗∗ ∗∗∗ ∗∗ (We have kept the same order of the blocks as in example (3.3)) Remark 4.7 As we have said above, the last example shows that theorem (4.4) improves the known ”condensation lemma”. Example 4.8 The marked nilpotent matrices in M6(C)withγ=β=(2,1) are p ˜γ˜ βSegre char. 0 (2,1) (2,1) (2,2,1,1) 1 (2,1) (2,1) (4,1,1) 1 (2,1) (1,2) (3,2,1) 1 (1,2) (2,1) (3,2,1) 1 (1,2) (1,2) (2,2,2) 2 (2,1) (2,1) (4,2) 2 (2,1) (1,2) (3,3) Miniversal Deformations of Marked Matrices 19 If we deform the first case preserving the pair (γ,β), we obtain:         000xyz 100000 0000tu 000000 000100 000000         which gives us the following Segre characteristic depending on the parameter’s values: Segre char. yu −zt y x z t u (4,2) =0 =0 (3,3) =0 0 (4,1,1) 0=0 (3,2,1) 0 0 =0 (*) (3,2,1) 0 0 0 (**) (2,2,2) 0 0 0 0 0 =0 (3,1,1,1) 0 0 =0 0 0 0 (2,2,1,1) 0 0 0 0 0 0 (*) z=0ot=0ou=0. (**) z=0ot=0. Therefore, they are realizations of all the Carlson-compatible partitions with γ=β=(2,1) (see (4.5)). In particular, the above marked ones are included, and the seventh is not marked. Moreover, such deformation gives the corresponding Littlewood-Richardson sequences of the partitions (2,1) and (2,1) (notice that partition (2,1) is auto-conjugate): (2,1) (2,2,1) (2,2,1,1)=(4,2)∗ (2,1) (2,2,1) (2,2,2)=(3,3)∗ (2,1) (3,1,1) (3,1,1,1)=(4,1,1)∗ (2,1) (3,2) (3,2,1)=(3,2,1)∗ (2,1) (3,1,1) (3,2,1)=(3,2,1)∗ (2,1) (3,2) (3,3)=(2,2,2)∗ (2,1) (4,1) (4,1,1)=(3,1,1,1)∗ (2,1) (4,1) (4,2)=(2,2,1,1)∗ References [1] V.I. Arnold,On Matrices Depending on Parameters, Uspekhi Mat. Nauk., 26 (1971), p. 101–114. [2] R. Bru; L. Rodman; H. Schneider,Extensions of Jordan Bases for Invariant Subspaces of a Matrix. Linear Algebra Appl., 150 (1991), p. 209–225. [3] A. Compta; J. Ferrer,A Geometric Approach to the Carlson Problem, SIAM Journal on Matrix Analysis Appl. Vol. 22, n. 1(2000), p. 258-275. 20 A. Compta, J. Ferrer and F. Puerta [4] A. Compta; J. Ferrer,Matricial Realizations of the Solutions of the Carlson Problem,Preprint (2001) [5] J. Ferrer; M.I. Garc´ ıa; F. 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