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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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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.

Let A_{1}, A_{2}, A_{3} be Hermitian n × n matrices with given lists of eigenvalues

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(A_{1}), λ(A_{2}), λ(A_{3}) such that

A_{1} + A_{2} = A_{3}

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

Let α = λ(A_{1}), β = λ(A_{2}) and γ = λ(A_{3}). One obvious necessary condition is the following
“trace equality”:

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

(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 = {i_{1} < i_{2} . . . < i_{r}} ⊂ {1, . . . , *n*} corresponds to a partition

of length at most r consisting of nonnegative integers.

Let α, β and γ be three partitions and c_{ℜβ}^{γ} be the corresponding Littlewood-Richardson coefficient. Set

The solution of the Horn’s problem is given by

([2]). A triple ( α, β, γ) is a triple of lists of eigenvalues of three Hermitian matrices A_{1}, A_{2}, A_{3} such that

A_{1} + A_{2} = A_{3}

*if and only if inequalities (1.1) are satisfied for all r < n and (I, J,K) R ^{n}_{r} and the “trace equality”*

*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 * R ^{n}_{r}* for a given n. In fact the explicit recursive answer to Horn’s problem can be given using larger set of triples

When r = 1, set* T ^{n}_{1}* =

The result of Klyatchko, Totaro, Knutson and Tao is that ( α, β, γ) is a triple of lists of eigenvalues of three Hermitian matrices A_{1}, A_{2}, A_{3} such that A_{1} + A_{2} = A_{3} if and only if inequality (1.1) holds for every triple

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 numbersM_{1}, M_{2}, . . ., M_{n} and γ R necessary and sufficient for the existence of Hermitian operators A_{1}, A_{2}, , . . . , A_{n} such that

and

For example when λ(A_{j}) = {0, 1} we have the following problem. Describe γ R such that there orthoprojectors P_{1}, P_{2}, . . .,P_{n} with

Let

§there are orthoprojections P_{J} such that

Clearly The following beautiful description of Σ_{n} was obtained in [7]:

Here vis a fixed point of the dynamical system

In this work the sets M_{1}, M_{2}, . . .,M_{n} will be finite. Even for finite M_{k} it can be very complicated
to describe such n-tuples of operators up to unitary equivalence if the cardinality of M_{k} 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 M_{j} = α^{(j)}. Let us consider the associative algebra defined by the following generators and relations :

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

where P_{k} is a polynomial with simple roots from the set M_{k}.

We can associate a star-shaped graph *G* with *n* rays of lengths m_{1}, . . . ,mncoming from a single center. We will label the vertices of this graph by the points of the sets M_{1}, . . . ,M_{n} to associated a labeled graph with the algebra A_{M1},...,M_{n},γ which completely determines the algebra. We can write A_{M1},...,M_{n},γ = A_{G,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 M_{k}{0} to the k-th ray such that they increase to the center. To the central vertex Â assigns value γ. We will fix some enumeration of the vertices of graph *G* and thus *X* will be identified with a vector
with m_{1} + · · · + m_{n} + 1 coordinates.

The following theorem reveals a remarkable connection between complexity of the algebra A_{G,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 (m_{1}· · · m_{n}) 2 {(2, 2, 2), (3, 3, 2), (3, 4, 2), (5, 3, 2)} and the extended Dynkin graphs are those for which ((m_{1}· · · m_{n}) 2 {(2, 2, 2, 2), (3, 3, 3), (4, 4, 2), (6, 3, 2)}.

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

*1. If G is a Dynkin graph, then algebra A _{G,X }is finite-dimensional for all X.*

*2. If G is extended Dynkin graph, then algebra A _{G,X} has quadratic growth for all X.*

*3. If G is non-Dynkin, then A _{G,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 Σm_{1}·m_{2}· · m_{n}of the parameters α^{(j)} _{k }, λ for which there exist *-representations of A_{M1,...,Mn,γ }, .

Let us call a *-representation π of the algebra A_{M1},...,M_{n},γ non-degenerate if spectrum of π(A_{k}) coincides with M_{k} for all k.

Consider the set

there is a non-degenerate ¤-representation of A_{G,X}}

which depends only on (m_{1}· · · m_{n} ).Every irreducible representation of algebra A_{M1,...,Mn,γ } is an irreducible non-degenerate *-representation of an algebra for some subsets Hence if and only if there exists

Henceforth we will denote the set Σm_{1}·m_{2}· · m_{n} by Σ(*G*). Irreducible representations of the algebras A_{G,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.

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

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

Let M_{g} be the set of vertices connected with g by an edge. Define operators

A representation ¦ in Rep(G,H) will be called locally-scalar if all operators Ag are scalar, i.e.A_{g} = α_{g}I_{Hg} . 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 V_{G} = R^{Gv} . Elements x of V_{G} we will call G-vectors. A vector x = (x_{g} ) is called positive, x > 0, if x ≠ 0 and x_{g} ¸ 0 for all g G_{v}. 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 A_{g} = f(g)I_{Hg} then the G-vector f = (f(g)) is called the *character* and Π is called f-representation in this case. The *support* G^{Π}_{v} of ¦ is {g G_{v}|Π(g) ≠ 0}.

A character of the locally-scalar representation Π is uniquely defined on the support G^{Π} _{ v} and non-uniquely on its complement. In the general case, denote by {f_{Π}} the set of characters of _{Π}. For each vertex g G_{v}, denote by σ_{g} the linear operator on V_{G} given by the formulae:

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

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

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 {f_{Π}}}. All representations Π from Rep(*G, d, f*) have the same support X = X_{d} = Let is the full subcategory with objects (Π, f) where *f(g)* > 0 if Put

Let us denote

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

These functors are equivalences of the categories. Let us denote factors), (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 Π_{g} of the graph

Here we will describe the connection of representations of the algebra A_{G,X} and locally-scalar graph representations. Let Π be a *-representation of the algebra A_{G,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} = Let be natural isometries. Then, in particular, Let denote the operator acting from to where and coefficients for a fixed S are defined by the following recursion: with initial data It is easy to check that if then the above recursion determines uniquely. Then is a self-adjoint operator. Moreover, 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 and γ appropriately ordered.

This correspondence is in fact an equivalence functor between category of non-degenerate *- representations of algebra A_{G,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

A finite-dimensional *-representation π of the algebra A_{G,X} such that for for will be called non-degenerate. By we will denote the full subcategory of non-degenerate representations in the category RepA_{G,X} of *-representations of the *-algebra A_{G,X} .

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

A locally-scalar representation of the graph G with the character corresponds to a non-degenerate representation of A_{G,X} with the character

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

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

The vector is called the *generalized dimension* of the representation π of the algebra A_{G,X}. Let for a non-degenerate representation of the algebra A_{G,X}. be the dimension of Π. It is easy to see that

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

The *symmetric bilinear form* is The vector 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 is equal to 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, *E*_{j}) ≤ 0 for all *j*.

For an extended Dynkin graph G a vertex j is called extending if δ_{j} = 1. The graph obtained by deleting extending vertex is the corresponding Dynkin graph. The set of *roots* is 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 is finite. Moreover, if *e* is an extending vertex then the set is a complete set of representatives
of the cosets from If α is a root then α + δ is again a root. We call a coset the δ-series and a coset If α is a root then its images under the action of
the group generated by will 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 h* _{G}* which depends only on the graph

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

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

For a vector we shall write v ≥s 0 if v_{j} > 0 for all j ≠ s and v_{s} = 0.

The equivalence functor assigns to every representation of generalized dimension (l_{1}, . . . , l_{n}) a unique locally-scalar representation of graph G with a character (x_{1}, . . . , x_{n}, x_{0}) and dimension (v_{1}, . . . , v_{n}, v_{0}). Let M_{f} denote the transition matrix which transform the vector *X* to (x_{1}, . . . , x_{n}, x_{0}), (where *v ^{t}* denote the transposed vector

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

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

• There exist k and t such that

(4.1)

or

(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 A _{G,X} in dimension v.*

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

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