Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over ${\mathbb{Q}}$ .     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

A bound on the generalized competition index of a primitive matrix using Boolean rank

For a positive integer m where 1?m?n, the m-competition index (generalized competition index) of a primitive digraph is the smallest positive integer k such that for every pair of vertices x and y, there exist m distinct vertices v₁,v₂,…,v_m such that there are directed walks of length k from x to v_i and from y to v_i for 1?i?m. The m-competition index is a generalization of the scrambling index and the exponent of a primitive digraph. In this study, we determine an upper bound on the m-competition index of a primitive digraph using Boolean rank and give examples of primitive Boolean matrices that attain the bound.     

The Ramsey numbers of large cycles versus wheels

In this paper we show that the Ramsey number R(C_n,W_m)=2n-1 for even m and n?5m/2-1.     

Path-kipas Ramsey numbers

For two given graphs F and H, the Ramsey number R(F,H) is the smallest positive integer p such that for every graph G on p vertices the following holds: either G contains F as a subgraph or the complement of G contains H as a subgraph. In this paper, we study the Ramsey numbers , where P_n is a path on n vertices and is the graph obtained from the join of K₁ and P_m. We determine the exact values of for the following values of n and m: 1?n?5 and m?3; n?6 and (m is odd, 3?m?2n-1) or (m is even, 4?m?n+1); 6?n≤7 and m=2n-2 or m?2n; n?8 and m=2n-2 or m=2n or (q·n-2q+1?m?q·n-q+2 with 3?q?n-5) or m?(n-3)²; odd n?9 and (q·n-3q+1?m?q·n-2q with 3?q?(n-3)/2) or (q·n-q-n+4?m?q·n-2q with (n-1)/2?q?n-4). Moreover, we give lower bounds and upper bounds for for the other values of m and n.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Number theory of matrix semigroups

Unlike factorization theory of commutative semigroups which are well-studied, very little literature exists describing factorization properties in noncommutative semigroups. Perhaps the most ubiquitous noncommutative semigroups are semigroups of square matrices and this article investigates the factorization properties within certain subsemigroups of M_n(Z), the semigroup of n×n matrices with integer entries. Certain important invariants are calculated to give a sense of how unique or non-unique factorization is in each of these semigroups.     

Ideals of varieties parameterized by certain symmetric tensors

The ideal of a Segre variety P^n₁×?×P^n_t?P^{(n₁+1)?(n_t+1)−1} is generated by the 2-minors of a generic hypermatrix of indeterminates (see [H.T. Hà, Box-shaped matrices and the defining ideal of certain blowup surface, J. Pure Appl. Algebra 167 (2-3) (2002) 203-224. MR1874542 (2002h:13020)] and [R. Grone, Decomposable tensors as a quadratic variety, Proc. Amer. Math. 43 (2) (1977) 227-230. MR0472853 (57 #12542)]). We extend this result to the case of Segre-Veronese varieties. The main tool is the concept of “weak generic hypermatrix” which allows us to treat also the case of projection of Veronese surfaces from a set of general points and of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2.     

Construction of determinantal representation of trigonometric polynomials

For a pair of n×n Hermitian matrices H and K, a real ternary homogeneous polynomial defined by F(t,x,y)=det(tI_n+xH+yK) is hyperbolic with respect to (1,0,0). The Fiedler conjecture (or Lax conjecture) is recently affirmed, namely, for any real ternary hyperbolic polynomial F(t,x,y), there exist real symmetric matrices S₁ and S₂ such that F(t,x,y)=det(tI_n+xS₁+yS₂). In this paper, we give a constructive proof of the existence of symmetric matrices for the ternary forms associated with trigonometric polynomials.     

Riesz potentials and integral geometry in the space of rectangular matrices

Riesz potentials on the space of rectangular n×m matrices arise in diverse “higher rank” problems of harmonic analysis, representation theory, and integral geometry. In the rank-one case m=1 they coincide with the classical operators of Marcel Riesz. We develop new tools and obtain a number of new results for Riesz potentials of functions of matrix argument. The main topics are the Fourier transform technique, representation of Riesz potentials by convolutions with a positive measure supported by submanifolds of matrices of rank<m, the behavior on smooth and L^p functions. The results are applied to investigation of Radon transforms on the space of real rectangular matrices.     

Primitive digraphs with smallest large exponent

A primitive digraph D on n vertices has large exponent if its exponent, γ(D), satisfies α_n?γ(D)?w_n, where α_n=⌊w_n/2⌋+2 and w_n=(n-1)²+1. It is shown that the minimum number of arcs in a primitive digraph D on n?5 vertices with exponent equal to α_n is either n+1 or n+2. Explicit constructions are given for fixed n even and odd, for a primitive digraph on n vertices with exponent α_n and n+2 arcs. These constructions extend to digraphs with some exponents between α_n and w_n. A necessary and sufficient condition is presented for the existence of a primitive digraph on n vertices with exponent α_n and n+1 arcs. Together with some number theoretic results, this gives an algorithm that determines for fixed n whether the minimum number of arcs is n+1 or n+2.     

Legendrian tori and the semi-classical limit

Let X be CPn or a compact smooth quotient of the n-dimensional complex hyperbolic space, n>1. Let L be a hermitian holomorphic line bundle (with hermitian connection) on X chosen as follows: if X=CPn then L is the hyperplane bundle, and in the second case L is chosen so that L⊗(n+1)=KX⊗E, where KX is the canonical line bundle and E is a flat line bundle. The unit circle bundle P in L^∗ is a contact manifold. Let k^′ be a fixed positive integer. We construct certain Legendrian tori in P (the construction depends, in particular, on the choice of k^′) and sequences {uk}, k=k^′m, , of holomorphic sections of L⊗k associated to these tori. We study asymptotics of the norms ‖ukk_‖ as m→+∞ and, in particular, apply this result to construct explicitly certain non-trivial holomorphic automorphic forms on the n-dimensional complex hyperbolic space. We obtain an n>1 analogue of the classical period formula (this is a well-known statement for automorphic forms on the upper half plane, n=1).     

On a Heilbronn-type problem

Let F be a convex figure with area |F| and let G(n,F) denote the smallest number such that from any n points of F we can get G(n,F) triangles with areas less than or equal to |F|/4. In this article, to generalize some results of Soifer, we will prove that for any triangle T, G(5,T)=3; for any parallelogram P, G(5,P)=2; for any convex figure F, if S(F)=6, then G(6,F)=4.     

Bernoulli matrix and its algebraic properties

In this paper, we define the generalized Bernoulli polynomial matrix B^(α)(x) and the Bernoulli matrix B. Using some properties of Bernoulli polynomials and numbers, a product formula of B^(α)(x) and the inverse of B were given. It is shown that not only B(x)=P[x]B, where P[x] is the generalized Pascal matrix, but also B(x)=FM(x)=N(x)F, where F is the Fibonacci matrix, M(x) and N(x) are the (n+1)×(n+1) lower triangular matrices whose (i,j)-entries are and , respectively. From these formulas, several interesting identities involving the Fibonacci numbers and the Bernoulli polynomials and numbers are obtained. The relationships are established about Bernoulli, Fibonacci and Vandermonde matrices.     

On 0-1 matrices and small excluded submatrices

We say that a 0-1 matrix A avoids another 0-1 matrix (pattern) P if no matrix P^′ obtained from P by increasing some of the entries is a submatrix of A. Following the lead of (SIAM J. Discrete Math. 4 (1991) 17-27; J. Combin. Theory Ser. A 55 (1990) 316-320; Discrete Math. 103 (1992) 233-251) and other papers we investigate n by n 0-1 matrices avoiding a pattern P and the maximal number ex(n,P) of 1 entries they can have. Finishing the work of [8] we find the order of magnitude of ex(n,P) for all patterns P with four 1 entries. We also investigate certain collections of excluded patterns. These sets often yield interesting extremal functions different from the functions obtained from any one of the patterns considered.     

Sets of nonnegative matrices with positive inhomogeneous products

Let X be a set of k×k matrices in which each element is nonnegative. For a positive integer n, let P(n) be an arbitrary product of n matrices from X, with any ordering and with repetitions permitted. Define X to be a primitive set if there is a positive integer n such that every P(n) is positive [i.e., every element of every P(n) is positive]. For any primitive set X of matrices, define the index g(X) to be the least positive n such that every P(n) is positive. We show that if X is a primitive set, then g(X)?2^k?2. Moreover, there exists a primitive set Y such that g(Y) = 2^k?2.     

Dimension and enumeration of primitive ideals in quantum algebras

In this paper, we study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a Theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of various quantum algebras is always even. Next we give a combinatorial criterion for a prime ideal that is invariant under the torus action to be primitive. We use this criterion to obtain a formula for the number of primitive ideals in the algebra of 2×n quantum matrices that are invariant under the action of the torus. Roughly speaking, this can be thought of as giving an enumeration of the points that are invariant under the induced action of the torus in the “variety of 2×n quantum matrices”. The first author thanks NSERC for its generous support. This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme held at the University of Edinburgh, by a Marie Curie European Reintegration Grant within the 7th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.