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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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A Horn type spectral problem 1

Stanislav POPOVYCH

Department of Mathematics, Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden E-mail: [email protected]

*Corresponding Author:
Stanislav POPOVYCH
Department of Mathematics
Chalmers University of Technology
SE-412 96 G¨oteborg, Sweden
E-mail:[email protected]

Received date: December 20, 2007; Revised date: March 19, 2008

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Abstract

The famous Horn’s problem is about the possible eigenvalue list of a sum of two Hermitian matrices with prescribed eigenvalue lists. The Spectral Problem is to describe possible spectra for an irreducible finite family of Hermitian operators with the sum being a scalar operator. In case when spectra consist of finite number of points the complexity of the problem depends on properties of some rooted tree. We will consider the cases for which the explicit answer on the Spectral Problem can be obtained.

Introduction

Let A1, A2, A3 be Hermitian n × n matrices with given lists of eigenvalues

image

The well-known classical problem about the spectrum of a sum of two Hermitian n×n matrices (Alfred Horn’s problem) is to describe possible values of λ(A1), λ(A2), λ(A3) such that

A1 + A2 = A3

This problem was solved recently by Klyachko, Totaro, Knutson and Tao [2]. We shall briefly recall the solution below.

Let α = λ(A1), β = λ(A2) and γ = λ(A3). One obvious necessary condition is the following “trace equality”:

image

It turns out that necessary and sufficient conditions can be given in terms of linear inequalities of the form:

image(1.1)

where I, J,K are certain subsets of {1 , . . . , i} of the same cardinality r < n.

To describe such triples (I, J,K) we recall the standard correspondence between subsets and partitions. A subset I = {i1 < i2 . . . < ir} ⊂ {1, . . . , n} corresponds to a partition

image

of length at most r consisting of nonnegative integers.

Let α, β and γ be three partitions and cℜβγ be the corresponding Littlewood-Richardson coefficient. Set

image

The solution of the Horn’s problem is given by

Theorem 1.1

([2]). A triple ( α, β, γ) is a triple of lists of eigenvalues of three Hermitian matrices A1, A2, A3 such that

A1 + A2 = A3

if and only if inequalities (1.1) are satisfied for all r < n and (I, J,K) Rnr and the “trace equality”

image

holds. In fact the resulting system of inequalities together with “trace equality” is a complete and independent set of conditions.

The Littlewood-Richardson rule provides an algorithm to compute the sets Rnr for a given n. In fact the explicit recursive answer to Horn’s problem can be given using larger set of triples Tnr instead of Rnr .That this is possible was conjectured by Horn in 1962 and proved by Klyachko, Totaro, Knutson and Tao (see overview [2]). For a triple (I, J,K) of subsets of {1, . . . , n}, Horn defined sets Tnr of triples (I, J,K) of subsets of {1, . . . , n} of the same cardinality r, by the following recursive procedure. Set

image

When r = 1, set Tn1 = Un1 . Otherwise, let

image

The result of Klyatchko, Totaro, Knutson and Tao is that ( α, β, γ) is a triple of lists of eigenvalues of three Hermitian matrices A1, A2, A3 such that A1 + A2 = A3 if and only if inequality (1.1) holds for every triple

image

Spectral Problem

A modification of the Horn’s problem is the Spectral Problem posed in [5]. The Spectral Problem is to describe a connection between subsets of real numbersM1, M2, . . ., Mn and γ R necessary and sufficient for the existence of Hermitian operators A1, A2, , . . . , An such that

image

and

 

image

For example when λ(Aj) = {0, 1} we have the following problem. Describe γ R such that there orthoprojectors P1, P2, . . .,Pn with

image

Let

§imagethere are orthoprojections PJ such that image

Clearlyimage The following beautiful description of Σn was obtained in [7]:

image

Here image vis a fixed point of the dynamical system

image

In this work the sets M1, M2, . . .,Mn will be finite. Even for finite Mk it can be very complicated to describe such n-tuples of operators up to unitary equivalence if the cardinality of Mk is large enough. More precisely the corresponding x-algebras defined below may be x-wild (see [10]).

Let us stress here that an essential difference with Horn’s classical problem is that we do not fix the dimension of H and the spectral multiplicities.

The Spectral Problem can be stated in terms of *-representations of *-algebras introduced in [9]. Namely, let α(j) = (α(j) 1 , α(j) 2 , . . . , α(j) decreasing coefficients. Put Mj = α(j). Let us consider the associative algebra defined by the following generators and relations :

image

Here e is the identity of the algebra. This is a *-algebra if we declare all generators to be self-adjoint. Equivalently this algebra can be given by the following generators and relations

image

where Pk is a polynomial with simple roots from the set Mk.

We can associate a star-shaped graph G with n rays of lengths m1, . . . ,mncoming from a single center. We will label the vertices of this graph by the points of the sets M1, . . . ,Mn to associated a labeled graph with the algebra AM1,...,Mn,γ which completely determines the algebra. We can write AM1,...,Mn,γ = AG,X where X is the labeling of the graph G, i.e. X assigns real values to each vertex of the graph in such a way that X assigns all values from Mk{0} to the k-th ray such that they increase to the center. To the central vertex  assigns value γ. We will fix some enumeration of the vertices of graph G and thus X will be identified with a vector with m1 + · · · + mn + 1 coordinates.

The following theorem reveals a remarkable connection between complexity of the algebra AG,X and the properties of the graph G. Namely the complexity depends on whether G is a Dynkin or a non-Dynkin graph. Recall that the Dynkin graphs are those for which (m1· · · mn) 2 {(2, 2, 2), (3, 3, 2), (3, 4, 2), (5, 3, 2)} and the extended Dynkin graphs are those for which ((m1· · · mn) 2 {(2, 2, 2, 2), (3, 3, 3), (4, 4, 2), (6, 3, 2)}.

Theorem 2.1

([11]). For a given graph G the following holds:

1. If G is a Dynkin graph, then algebra AG,X is finite-dimensional for all X.

2. If G is extended Dynkin graph, then algebra AG,X has quadratic growth for all X.

3. If G is non-Dynkin, then AG,X contains the free algebra with 2 generators (hence it has the exponential growth).

Clearly the Spectral Problem is equivalent to a problem of a description of the set Σm1·m2· · mnof the parameters α(j) k , λ for which there exist *-representations of AM1,...,Mn,γ , .

Let us call a *-representation π of the algebra AM1,...,Mn,γ non-degenerate if spectrum of π(Ak) coincides with Mk for all k.

Consider the set

image there is a non-degenerate ¤-representation of AG,X}

which depends only on (m1· · · mn ).Every irreducible representation of algebra AM1,...,Mn,γ is an irreducible non-degenerate *-representation of an algebra image for some subsets imageHence image if and only if there exists imageimage

Henceforth we will denote the set Σm1·m2· · mn by Σ(G). Irreducible representations of the algebras AG,X associated with the Dynkin graph G exist only in certain dimensions that are bounded from above (see [8]). In [3, 4] we have given a complete description of Σ(G) for all Dynkin graphs G and an algorithm for finding all irreducible representations.

Coxeter functors

The main tools for our classification are the Coxeter functors for locally-scalar graph representations. First we will recall a connection between category of *-representation of algebra AG,X associated with the graph G and locally-scalar representations of the graph G. For more details see [3].

Henceforth we will use definitions, notations and results about representations of graphs in the category of Hilbert spaces found in [8].

A graph G consists of a set of vertices Gv, a set of edges Ge and a map " from Ge into the set of one- and two-element subsets of Gv (the edge is mapped into the set of incident vertices). We will consider only connected finite graphs without cycles. Fix a decomposition of Gv of the form image such that for each α Ge one of the vertices from image belongs toimage and the other to image. Vertices in image will be called even, and those in the setimage odd.

Representation of G associates with each vertex g 2 Gv a Hilbert space image and with each edge image such that image a pair of mutually adjoint operators imageimage where image We now construct a category Rep(G,H). Its objects are the representations of the graph G in H. A morphism image is a family image of operators image such that image

Let Mg be the set of vertices connected with g by an edge. Define operators

image

A representation ¦ in Rep(G,H) will be called locally-scalar if all operators Ag are scalar, i.e.Ag = αgIHg . The full subcategory Rep(G,H), the objects of which are locally-scalar representations, will be denoted by RepG and called the category of locally-scalar representations of the graph G.

Denote VG = RGv . Elements x of VG we will call G-vectors. A vector x = (xg ) is called positive, x > 0, if x ≠ 0 and xg ¸ 0 for all g Gv. Denote V+ G = {x VG|x > 0}. If Π is a finite dimensional representation of the graph G then the G-vector (d(g)), where d(g) = dim Π(g) is called the dimension of Π. If Ag = f(g)IHg then the G-vector f = (f(g)) is called the character and Π is called f-representation in this case. The support GΠv of ¦ is {g Gv|Π(g) ≠ 0}.

A character of the locally-scalar representation Π is uniquely defined on the support GΠ v and non-uniquely on its complement. In the general case, denote by {fΠ} the set of characters of Π. For each vertex g Gv, denote by σg the linear operator on VG given by the formulae:

image

The mapping σg is called the reflection at the vertex g. The composition of all reflections at odd vertices is denoted by image (it does not depend on the order of the factors), and at all even vertices by image A Coxeter transformation is image The transformation image is called an odd (even) Coxeter map. Let us adopt the following notations for compositions of the Coxeter maps: image

If d(g) is the dimension of a locally-scalar graph representation Π, then

image

For d Z+G and f V+G , consider the full subcategory Rep(G, d, f) in RepG (here Z+G is the set of positive integer G-vectors), with the set of objects Ob Rep(G, d, f) = {Π dim Π(g) = d(g), f {fΠ}}. All representations Π from Rep(G, d, f) have the same support X = Xd = imageLet imageimageimage is the full subcategory with objects (Π, f) where f(g) > 0 if image Put image

Let us denote

image

The even and odd Coxeter reflection functors are defined in [8],

image

These functors are equivalences of the categories. Let us denote image factors), image (k factors), if the compositions exist. Using these functors, an analog of Gabriel’s theorem for graphs and their locally-scalar representations has been proven in [8]. In particular, it has been proved that any locally-scalar graph representation decomposes into a direct sum (finite or infinite) of finite dimensional indecomposable representations, and all indecomposable representations can be obtained by odd and even Coxeter reflection functors starting from the simplest representations Πg of the graph imageimage

Here we will describe the connection of representations of the algebra AG,X and locally-scalar graph representations. Let Π be a *-representation of the algebra AG,X in space H of dimension n. Let P(s)j denote projection π(p(s)j ). Let us define projections R(s)j = P(s)1 + . . . + P(s) j , and subspaces H(s) j = image Let image be natural isometries. Then, in particular, imageimage Let image denote the operator image acting from imagetoimage where image and coefficients image for a fixed S are defined by the following recursion: image with initial data image It is easy to check that if image then the above recursion determines image uniquely. Then image is a self-adjoint operator. Moreover,image Operators V(s)j together with their conjugate give raise to a locally-scalar representation of the graph G with a character with coefficients of imageand γ appropriately ordered.

This correspondence is in fact an equivalence functor between category of non-degenerate *- representations of algebra AG,X and the category of non-degenerate locally-scalar representations of the graph G.

Since we will be concerned with extended Dynkin star-shaped graphs we will simplify simplify notations and consider only graphs with three rays. This will exclude graph for which the formulae are analogues and are left to be recovered by the reader.

So we will use notations α, β, γ instead of instead of α(1), α(2), α(3). By X we will denote the vector image

Definition 3.1.

A finite-dimensional *-representation π of the algebra AG,X such that image for image for imageimagewill be called non-degenerate. By image we will denote the full subcategory of non-degenerate representations in the category RepAG,X of *-representations of the *-algebra AG,X .

The above functor transform the representation of the algebra with character image to locally-scalar representation of the graph G with the following character image and on the first ray imageimage It is clear by analogy how to define f on other two rays.

A locally-scalar representation of the graph G with the characterimage corresponds to a non-degenerate representation of AG,X with the character image

Here xj = 0 if j ≤ 0. Analogously one can find βj and image We will denote image The mentioned above functor image acts between categories image and RepG, see [4].

The representation Π is unitary equivalent to an irreducible representation from the image of the functor image if and only if image

The vector imageis called the generalized dimension of the representation π of the algebra AG,X. Let image for a non-degenerate representation of the algebra AG,X. image be the dimension of Π. It is easy to see that

image

Spectral problem for algebras associated with extended Dynkin graphs

Let us recall a few facts about root systems associated with extended Dynkin diagrams. Let G be a simple connected graph. Then its Tits form is

image

The symmetric bilinear form is image The vector image is called sincere if each component is non-zero.

It is well known that for Dynkin graphs (and only for them) bilinear form (·, ·) is positive definite. The form is positive semi-definite for extended Dynkin graphs. And in the letter case Rad image is equal to image where δ is a minimal imaginary root. For other graphs (which are neither Dynkin nor extended Dynkin) there are vectors α ≥ 0 such that q(α) < 0 and (α, Ej) ≤ 0 for all j.

For an extended Dynkin graph G a vertex j is called extending if δj = 1. The graph obtained by deleting extending vertex is the corresponding Dynkin graph. The set of roots is image A root α is real if q(α) = 1 and imaginary if q(α) = 0. Every root is either positive or negative, i.e. all coordinates are simultaneously non-negative or non-positive.

It is known that for an extended Dynkin graph the set image is finite. Moreover, if e is an extending vertex then the set image is a complete set of representatives of the cosets from image If α is a root then α + δ is again a root. We call a coset image the δ-series and a coset image If α is a root then its images under the action of the group generated by imagewill be called a Coxeter series or C-series for short. It turns out that each C-series decomposes into a finite number of δ-series or 2δ -series of roots.

Note that to find formulae of the locally-scalar representations of a given extended Dynkin graph we need to consider two principally different cases: the case when the vector of generalized dimension is a real root and the case when it is an imaginary root. In the letter case the vector of parameters X must be orthogonal to a imaginary root. Hence X must belong to a ceratin hyperplane hG which depends only on the graph G.

It is know (see [11]) that in case X hG the dimension of any irreducible representation is bounded (by 2 for image, by 3 for image, by 4 for image and by 6 for image). Thus in case X hG we can describe the set of admissible parameters X using Horn’s inequalities. In caseimage the dimension of any irreducible locally-scalar representation is a real root. In what follows we will relay on the following result [6].

Theorem 4.1.

Let π be an irreducible non-degenerate *-representation of the algebra AG,X,¸λ associated with an extended Dynkin graph G and image be the corresponding locally-scalar representation of the graph G. Then either generalized dimension d of image is a singular root or vectorparameter image

For a vector image we shall write v ≥s 0 if vj > 0 for all j ≠ s and vs = 0.

The equivalence functor image assigns to every representation image of generalized dimension (l1, . . . , ln) a unique locally-scalar representation of graph G with a character (x1, . . . , xn, x0) and dimension (v1, . . . , vn, v0). Let Mf denote the transition matrix which transform the vector X to (x1, . . . , xn, x0), image (where vt denote the transposed vector v). Let Md be the transition matrix which transforms generalized dimension (v1, . . . , vn, v0) of a graph representation to generalized dimension (1, . . . , ln) of the corresponding algebra representation,i.e. image Further we will omit t superscript and image instead of Mf vt.

Theorem 4.2.

Let G be an extended Dynkin graph and π be a non-degenerate irreducible *- representation of a generalized dimension v of AG,X for some character X. Then one of the two possibilities holds:

image where δ is the minimal imaginary root of the root system associated with G.

• There exist k and t such that

image(4.1)

or

image(4.2)

(depending on the parity of k + t). Moreover, systems of inequalities (4.1), (4.2) are necessary and sufficient conditions for existence of representation of AG,X in dimension v.

The explicit answers to the Spectral Problem for all extended Dynkin graphs will appear in [6].

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