Research Article | Open Access
A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots
Following our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posets . One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by Drozd [Funct. Anal. Appl. 8(1974), 219–225]. We are mainly interested in a Coxeter spectral classification of posets such that the symmetric Gram matrix is positive semidefinite, where is the incidence matrix of . Following the idea of Drozd mentioned earlier, we associate to its Coxeter matrix , its Coxeter spectrum , a Coxeter polynomial , and a Coxeter number. In case is positive semi-definite, we also associate to a reduced Coxeter number , and the defect homomorphism . In this case, the Coxeter spectrum is a subset of the unit circle and consists of roots of unity. In case is positive semi-definite of corank one, we relate the Coxeter spectral properties of the posets with the Coxeter spectral properties of a simply laced Euclidean diagram associated with . Our aim of the Coxeter spectral analysis of such posets is to answer the question when the Coxeter type of determines its incidence matrix (and, hence, the poset ) uniquely, up to a -congruency. In connection with this question, we also discuss the problem studied by Horn and Sergeichuk [Linear Algebra Appl. 389(2004), 347–353], if for any -invertible matrix , there is such that and is the identity matrix.
In the present paper, we continue our Coxeter spectral study of finite posets, started in , in a close connection with the Coxeter spectral technique introduced in [2–4] for acyclic edge-bipartite graphs or signed graphs in the sense of . We also follow some of the techniques of representation theory, graph combinatorics, and the spectral graph theory; see [6–31].
Here, we use the terminology and notation introduced in [1, 4, 26–28]. We denote by the set of nonnegative integers, the ring of integers, and the rational number field. Given , we view as a free abelian group and denote by the standard -basis of . Given an index set , we denote by the abelian group of all vectors , with integer coordinates , by the -algebra of all square by integral matrices, and by the identity matrix. In particular, , with , is the -algebra of all square by matrices. The group is called the (integral) general linear group. We say that two square rational matrices are -equivalent, or -congruent, (and denote ) if there is a matrix such that . By a poset we mean a finite partially ordered set with respect to a partial order relation . Following , a poset is called a one-peak poset if has a unique maximal element . A finite poset is uniquely determined by its incidence matrix , that is, the square matrix, as follows: Following an idea of Drozd  (developed in ), we have introduced in [1, 28] the Tits matrix of to be the integral matrix where is the set of all maximal elements of . Usually, we equip the elements of with a numbering; that is, is viewed as , . Throughout, we fix such a numbering and make the identifications and . The incidence matrix and the Tits matrix depend on the numbering of . Namely, if is obtained from by a permutation and is the permutation matrix of , then Note that any poset admits an upper-triangular numbering ; that is, implies . In this case, is an upper-triangular matrix with on the main diagonal, and, hence, , and , for any numbering .
Fix a numbering of elements of . Following [1, 28], by the Euler matrix of the poset we mean the inverse of . Following [3, 4], we call the symmetric adjacency matrix and the characteristic polynomial of the poset . The set of all real roots of is defined to be the (real) spectrum of the poset .
We denote by the incidence quadratic form, the Tits quadratic form, and the Euler quadratic form of defined by the formulae respectively, where , is the set of all maximal elements in , and is the Tits matrix of ; see (27) and [1, 28] for a definition. The matrices with rational coefficients, are called the symmetric incidence Gram matrix, the symmetric Tits-Gram matrix, and the symmetric Euler-Gram matrix of . The matrices with integer coefficients, are called the Tits adjacency matrix, and the Euler adjacency matrix of . The polynomials are called the characteristic polynomial of and the Euler-characteristic polynomial of , respectively.
Example 1. (a) If is the poset
then ; that is, the characteristic polynomial of coincides with the Euler-characteristic polynomial of .
(b) If is the poset (12) of the Dynkin type , then the characteristic polynomial of does not coincide with the Euler-characteristic polynomial of , because
Following [17, 33], we introduce the following definition.
Definition 2. (a) We define a poset to be positive (resp., nonnegative) if the incidence form of is positive (resp., nonnegative); that is, , for any nonzero (resp., , for any ).
(b) We define a poset to be principal if its incidence form is principal in the sense of [34, Definition 2.1]; that is, is nonnegative, not positive, and the kernel is an infinite cyclic subgroup of .
Following the main idea of the Coxeter spectral analysis of acyclic edge-bipartite graphs (signed graphs) presented in [3, 4], we study finite posets (with a fixed numbering ) by means of the Coxeter spectrum of , that is, the set of all eigenvalues of the Coxeter matrix of , or equivalently, the set of all roots of the Coxeter polynomial see (31) and . It follows from (4) that the Coxeter spectrum of and the spectrum of do not depend on the numbering of the elements of the poset .
It follows from [27, Section 2] (see also [16, 26, 32]) that the problem of finding canonical forms of matrices in , with respect to the elementary transformations from the set , is controlled by the Tits quadratic form in the following sense. For any , there is only a finite number -canonical forms of matrices in if and only if the form is weakly positive; that is, is positive, for all nonzero vectors . Moreover, there is one-to-one correspondence between the irreducible -canonical forms in and the vectors satisfying the equation . A solution of the problem is given in  and [1, Theorem 1.6]. A useful homological interpretation (in terms of the Euler characteristic) of the -bilinear Tits form (26) and -bilinear Euler form is given in [1, ]. The reader is referred to [6–8, 25] for a detailed study and a solution of other important matrix problems of high computational complexity that have many useful applications in representation theory; see [16, 26].
The nonsymmetric Gram matrix of any graph of Figure 1, with the set of vertices and the set of edges , is defined to be the matrix where , if there is an edge and . We set , if is empty or .
The Coxeter polynomial of any diagram does not depend on the numbering of the vertices in and is presented in (48). If and , the Coxeter polynomial of depends on the numbering of the vertices in and is one of the polynomials presented in , where for , and . In particular, if is even and , then and
Following [4, 21], we associate (in Section 2) to any principal poset a simply laced Euclidean diagram such that the incidence symmetric Gram matrix is -congruent to the symmetric Gram matrix of ; that is, there is a -invertible matrix such that .
One of the aims of the Coxeter spectral analysis of nonnegative finite posets is to study the question when the Coxeter type of a poset determines the matrix (and, hence, the poset ) uniquely, up to a -congruency. Here, we set , if is positive. In other words, we claim that, for any pair , of nonnegative one-peak posets, if and only if the incidence matrices and are -congruent. We also study the problem related with the results proved by Horn and Sergeichuk , if for any -invertible matrix , there exists such that and is the identity matrix; see [17, 18].
The main results of the present paper on nonnegative posets can be summarised as follows:
canonical equivalences between the incidences, Tits, and Euler quadratic form (and corresponding Coxeter transformations and Coxeter spectra) of any poset , established in Proposition 5;
a characterization of principal posets given in Section 3. We show that a connected poset is principal if and only if there exists a simply laced Euclidean diagram such that the symmetric Gram matrix of is -congruent to the symmetric Gram matrix of . Moreover, we show in Section 3 that, given a connected principal poset , the Coxeter spectrum is a subset of a unit circle , , and any is a root of unity;
a Coxeter spectral classification result (Corollary 11) asserting that, given a pair , of one-peak principal posets with at most elements, the following conditions are equivalent:(3a),(3b),(3c) and ,(3d) the incidence matrix is -congruent to the incidence matrix ; that is, there is a -invertible matrix such that .
In Section 3, we study principal posets by means of the defect and the reduced Coxeter number, and in Section 4, we present a framework for the study of nonnegative posets of corank by means of their defect and the reduced Coxeter number. Examples are given in Sections 3–5.
2. A Framework for the Coxeter Spectral Analysis of Finite Posets
The quadratic wanderings on finite posets studied in  are playing a key role in the representation theory of posets, algebras, and coalgebras, as well as in the algebraic combinatorics of posets; see [6, 9–14, 16, 24–26, 28, 31, 32, 36–39]. Except for the incidence wandering and the Euler wanderings defined by the incidence matrix (2), with and a fixed numbering , as well as the Euler matrix , we study in [1, 26–28] the Tits wandering defined by the Tits matrix of (see [28, ]), that is, the Gram matrix of the Tits -bilinear form given by where is the set of all maximal elements in the poset and . We call the Tits quadratic form of .
Note that, given a one-peak poset of the form , with a unique maximal element , we have where is the incidence matrix of the poset ; see . Note that .
Now, we show that, in the Coxeter spectral study of finite posets , we can use the Coxeter spectral technique introduced in [2, 4], for the edge-bipartite graphs (signed graphs ), and developed in [2, 34, 40] for the matrix morsifications of unit quadratic forms.
Following [3, 4, 24], by an edge-bipartite graph (bigraph, in short), we mean a pair , where is a finite nonempty set of vertices and is a finite set of edges equipped with a bipartition such that the set of edges connecting the vertices and does not contain edges lying in , for each pair of vertices , and either or . Note that the edge-bipartite graphs can be viewed as signed multigraphs satisfying a separation property; see [4, 5].
We visualize as a graph in a Euclidean space , , with the vertices numbered by the integers ; usually, we simply write . An edge in is visualised as a continuous one , and an edge in is visualised as a dotted one . A bigraph is said to be loop-free if it has no loops.
We view any finite graph as an edge-bipartite one by setting and , for each pair of vertices .
To any loop-free edge-bipartite graph , with a fixed numbering of its vertices, we associate the upper-triangular nonsymmetric Gram matrix of the form (20), with , where , if there is an edge and , , if there is an edge and . We set , if is empty or . Since is loop-free, we have and the main diagonal of consists of unities.
Following , we call positive (resp., nonnegative), if the symmetric Gram matrix of is positive definite (resp., positive semidefinite).
Following , we associate to any loop-free edge-bipartite graph , with , the Coxeter spectrum defined to be the spectrum of the Coxeter (-Gram) matrix the Coxeter polynomial the Coxeter transformation , given by , the Coxeter number (the order of in the automorphism group of , i.e., the minimal integer such that ), the -bilinear Gram form of given by , and the integral unit quadratic form Conversely, following Ovsienko , to any integral unit form we associate the loop-free bigraph of as follows (see also [34, 41]):(a)the vertices of are the integers ,(b)two vertices are joined by continuous edges of the form if is negative, and by dotted edges of the form , if is positive,(c)there is no edge between and , if , or .
To any poset , with a fixed numbering of its points, we associate the following three edge-bipartite graphs: where , , and are the bigraphs of the quadratic forms , , and , respectively; see (7). More precisely, the bigraphs (33) are defined as follows.(i)The set of vertices of each of the bigraphs , , and is the enumerated set .(ii)There is an edge in , if or holds in .(iii)There is an edge in , if and are not maximal in and or holds in . There is an edge in , if holds and is maximal in .(iv)Let be the Euler matrix of . There is an edge (resp., ) in , if or (resp., or ).
We call , , and the incidence bigraph of , the Tits bigraph of , and the Euler bigraph of , respectively, (with respect to the numbering ).
The following simple lemma is of importance.
Lemma 3. Assume that is a finite poset with a fixed numbering , and let , , be the loop-free edge-bipartite graphs associated with in (33).(a)The symmetric Gram matrices , , are -congruent to the symmetric Gram matrices , , , respectively. The rank of each of the symmetric Gram matrices , , does not depend of the numbering and coincides with the common rank .(b).(c)The poset is positive (resp., nonnegative) if and only if the bigraph (and , ) is positive (resp., nonnegative).(d)The poset is principal if and only if the bigraph and , is principal.
Proof. For the proof of (a), we recall that the Gram matrices , , , , , are invariant, up to -congruency, under permutations of the elements . Since admits an upper-triangular numbering and , then (a) follows. The proof of remaining statements is left to the reader.
Definition 4. Let be a finite poset, with a fixed numbering .(a) We associate with the following three Coxeter matrices:(a1) the (incidence) Coxeter matrix ;(a2) the Coxeter-Tits matrix ;(a3) the Coxeter-Euler matrix . Moreover, we define the following three Coxeter transformations:(a4) the (incidence) Coxeter transformation of ;(a5) the Coxeter-Tits transformation of ;(a6) the Coxeter-Euler transformation of , by the following formulae:
(b) The integral polynomial
is called the Coxeter polynomial of the poset .(c) The Coxeter spectrum of is the set of all eigenvalues of the matrix , or, equivalently, the set of all roots of the Coxeter polynomial .(d) The order of the Coxeter transformation is called the Coxeter number of the poset . In other words, is the minimal integer such that . We set , if , for any .(e) Assume that is nonnegative. The Coxeter type of is defined to be the pair if is positive, and the triple if is not positive, where is the reduced Coxeter number of in the sense of Theorems 10 and 18.
The following proposition shows that equality (35) holds.
Proposition 5. Let be a finite poset, with a fixed numbering , let be the incidence, Tits, and Euler quadratic form of , and let be the corresponding Coxeter transformations.(a) The following equalities hold and , and the following diagrams are commutative (36) where , , , and are the group isomorphisms defined by the formulae and , for .(b), , and .(c) The Coxeter number of the poset coincides with the Coxeter number of . Moreover, and .(d) Assume that is connected and nonnegative.(d1) If the numbering is upper-triangular and is the bigraph (33) associated to , then and .(d2) The Coxeter type of does not depend on the numbering .(d3) The Coxeter spectrum is a subset of a unit circle , and any is a root of unity.(d4) The poset is positive if and only if .
Proof. The first equality is obvious, and the second one follows by a direct calculation. Hence, (b) follows and, consequently, the diagrams (36) are commutative; see [1, Proposition 3.13]. Hence, the statement (c) follows from the commutativity of the diagrams (36).(d1) We recall from Section 1 that, given two numberings and of elements in , we have , where is the permutation matrix of a permutation . Hence, (d1) easily follows.(d2) It is sufficient to note that the incidence matrix is upper triangular. Hence, and .
To prove (d3) and (d4), we recall from  and [3, Proposition 2.6] that the Coxeter spectrum of any matrix morsification of a nonnegative bigraph is a subset of the unit circle and any is a root of unity (see also [41, 42]). Moreover, is positive iff . Assume that is connected and nonnegative. Then, the bigraph (33) associated to is nonnegative, is a morsification of , and , because the incidence matrix is quasitriangular and [4, Proposition 2.2] applies. This completes the proof.
Corollary 6. For any poset , equality (35) holds.
Proof. Apply Proposition 5(b).
The following example shows that the correspondence defined in (33) does not preserve the Coxeter types of and . In particular, it shows that the equality does not hold in general and the Coxeter polynomial depends on the numbering of , whereas the Coxeter polynomial does not depend on the numbering of .
Example 7. Consider the poset such that its Hasse quiver has the form (37) By a permutation of the elements in , we get (38) Note that the first numbering is upper-triangular, whereas the second one is not upper-triangular.
3. Principal Posets
We recall that a poset is principal if its incidence unit form is principal in the sense of [34, Definition 2.1]; that is, is nonnegative and not positive, and the kernel is an infinite cyclic subgroup of .
We start with the following useful observation.
Lemma 8. Assume that is a connected principal poset.(a) The Coxeter number of is infinite.(b) The Coxeter spectrum is a subset of a unit circle , , and any is a root of unity.(c) If , then and , where(i), , ,(ii), , and are as in Proposition 5.
Proof. (a) By Proposition 5(d2), is independent of the numbering of . Then, without loss of generality, we may suppose that the numbering of is upper-triangular. Then, by Lemma 3(d) and Proposition 5(d1), the Coxeter number coincides with the Coxeter number of the principal edge-bipartite graph associated with in (33). Then, (a) is a consequence of [3, Proposition 3.12].
The statements (b) and (c) follow by applying Proposition 5 and the commutative diagram (36).
Proposition 9. Let be a connected poset, , and let be the symmetric incidence Gram matrix of , the symmetric Tits-Gram matrix of , and the symmetric Euler-Gram matrix of , respectively. The following five conditions are equivalent.(a)The poset is principal.(b)The Gram matrix is positive indefinite of rank .(c)The Tits quadratic form of is nonnegative and , for some nonzero vector . (d)The Euler quadratic form of is nonnegative and , for some nonzero vector .(e)If is any of the symmetric Gram matrices of , then there exists a simply laced Euclidean diagram (uniquely determined by ) such that the matrix is -congruent to the symmetric Gram matrix of the Euclidean diagram ; that is, there is a -invertible matrix such that.
Proof. (a)(b) If and