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.     

Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions

We define an equivalence relation on integer compositions and show that two ribbon Schur functions are identical if and only if their defining compositions are equivalent in this sense. This equivalence is completely determined by means of a factorization for compositions: equivalent compositions have factorizations that differ only by reversing some of the terms. As an application, we can derive identities on certain Littlewood-Richardson coefficients.Finally, we consider the cone of symmetric functions having a nonnnegative representation in terms of the fundamental quasisymmetric basis. We show the Schur functions are among the extremes of this cone and conjecture its facets are in bijection with the equivalence classes of compositions.     

Effective computation of singularities of parametric affine curves

Let k be a field of characteristic zero and f(t),g(t) be polynomials in k[t]. For a plane curve parameterized by x=f(t),y=g(t), Abhyankar developed the notion of Taylor resultant (Mathematical Surveys and Monographs, Vol. 35, American Mathematical Society, Providence, RI, 1990) which enables one to find its singularities without knowing its defining polynomial. This concept was generalized as D-resultant by Yu and Van den Essen (Proc. Amer. Math. Soc. 125(3) (1997) 689), which works over an arbitrary field. In this paper, we extend this to a curve in affine n-space parameterized by x₁=f₁(t),…,x_n=f_n(t) over an arbitrary ground field k, where f₁,…,f_n∈k[t]. This approach compares to the usual approach of computing the ideal of the curve first. It provides an efficient algorithm of computing the singularities of such parametric curves using Gröbner bases. Computational examples worked out by symbolic computation packages are included.     

Graded tensor products

We study the polynomial identities of regular algebras, introduced in [A. Regev, T. Seeman, Z₂-graded tensor products of P.I. algebras, J. Algebra 291 (2005) 274-296]. For example, a finite-dimensional algebra is regular if it has a basis whose multiplication table satisfies some commutation relations. The matrix algebra M_n(F) over the field F is regular, which is closely related to M_n(F) being Z_n-graded. We study the polynomial identities of various types of tensor products of such algebras. In particular, using the theory of Hopf algebras, we prove a far reaching extension of the A⊗B theorem for Z₂-graded PI algebras.     

Differential operator specializations of noncommutative symmetric functions

Let K be any unital commutative Q-algebra and z=(z₁,…,zn) commutative or noncommutative free variables. Let t be a formal parameter which commutes with z and elements of K. We denote uniformly by K《z》 and K?t?《z》 the formal power series algebras of z over K and K?t?, respectively. For any α?1, let D[α]《z》 be the unital algebra generated by the differential operators of K《z》 which increase the degree in z by at least α−1 and the group of automorphisms Ft(z)=z−Ht(z) of K?t?《z》 with o(Ht(z))?α and Ht=0(z)=0. First, for any fixed α?1 and , we introduce five sequences of differential operators of K《z》 and show that their generating functions form an NCS (noncommutative symmetric) system [W. Zhao, Noncommutative symmetric systems over associative algebras, J. Pure Appl. Algebra 210 (2) (2007) 363-382] over the differential algebra D[α]《z》. Consequently, by the universal property of the NCS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218-348, MR1327096; see also hep-th/9407124], we obtain a family of Hopf algebra homomorphisms , which are also grading-preserving when Ft satisfies certain conditions. Note that the homomorphisms SFt above can also be viewed as specializations of NCSFs by the differential operators of K《z》. Secondly, we show that, in both commutative and noncommutative cases, this family SFt (with all n?1 and ) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions [I. Gessel, Multipartite P-partitions and inner products of skew Schur functions, in: Contemp. Math., vol. 34, 1984, pp. 289-301, MR0777705; C. Malvenuto, C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (3) (1995) 967-982, MR1358493; Richard P. Stanley, Enumerative Combinatorics II, Cambridge University Press, 1999] are also discussed.     

Noncommutative symmetric functions and an amazing matrix

We present a simple way to derive the results of Diaconis and Fulman [P. Diaconis, J. Fulman, Foulkes characters, Eulerian idempotents, and an amazing matrix, arXiv:1102.5159] in terms of noncommutative symmetric functions.     

Additive functions on arithmetic progressions with large moduli

In this paper we study additive functions on arithmetic progressions with large moduli. We are able to improve some former results given by Elliott.     

A k-tableau characterization of k-Schur functions

We study k-Schur functions characterized by k-tableaux, proving combinatorial properties such as a k-Pieri rule and a k-conjugation. This new approach relies on developing the theory of k-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of k-Schur Littlewood-Richardson coefficients contains the 3-point Gromov-Witten invariants; structure constants for the quantum cohomology ring.     

The Hilbert functions which force the Weak Lefschetz Property

The purpose of this note is to characterize the finite Hilbert functions which force all of their artinian algebras to enjoy the Weak Lefschetz Property (WLP). Curiously, they turn out to be exactly those (characterized by Wiebe in [A. Wiebe, The Lefschetz property for componentwise linear ideals and Gotzmann ideals, Comm. Algebra 32 (12) (2004) 4601-4611]) whose Gotzmann ideals have the WLP.This implies that, if a Gotzmann ideal has the WLP, then all algebras with the same Hilbert function (and hence lower Betti numbers) have the WLP as well. However, we will answer in the negative, even in the case of level algebras, the most natural question that one might ask after reading the previous sentence: If A is an artinian algebra enjoying the WLP, do all artinian algebras with the same Hilbert function as A and Betti numbers lower than those of A have the WLP as well?Also, as a consequence of our result, we have another (simpler) proof of the fact that all codimension 2 algebras enjoy the WLP (this fact was first proven in [T. Harima, J. Migliore, U. Nagel, J. Watanabe, The weak and strong Lefschetz properties for Artinian K-algebras, J. Algebra 262 (2003) 99-126], where it was shown that even the Strong Lefschetz Property holds).     

Attached primes and Matlis duals of local cohomology modules

Let J be an ideal of a noetherian local ring R. We show new results on the set of attached primes of a local cohomology module . To prove our results we establish and use new relations between the set of attached primes of a local cohomology module and the set of associated primes of the Matlis dual of the same local cohomology module. Received: 17 March 2006     

Generalized symmetric functions and invariants of matrices

We generalize the classical isomorphism between symmetric functions and invariants of a matrix. In particular, we show that the invariants over several matrices are given by the abelianization of the symmetric tensors over the free associative algebra. The main result is proved by finding a characteristic free presentation of the algebra of symmetric tensors over a free algebra. The author is supported by research grant Politecnico di Torino n.119, 2004.     

On the critical ideals of graphs

We introduce some determinantal ideals of the generalized Laplacian matrix associated to a digraph G, that we call critical ideals of G. Critical ideals generalize the critical group and the characteristic polynomials of the adjacency and Laplacian matrices of a digraph. The main results of this article are the determination of some minimal generator sets and the reduced Gröbner basis for the critical ideals of the complete graphs, the cycles and the paths. Also, we establish a bound between the number of trivial critical ideals and the stability and clique numbers of a graph.     

Forms for meromorphic functions

An infinite circular arithmetic is defined and used to define power sum and centered forms for meromorphic functions. The properties of these forms are investigated and it is shown that both the power sum and the centered forms form including chains with respect to their orders and that they possess a kind of quadratic convergence with respect to the domain circle. An example of the use of the forms is given.     

A basic class of symmetric orthogonal functions with six free parameters

In a previous paper, we introduced a basic class of symmetric orthogonal functions (BCSOF) by an extended theorem for Sturm-Liouville problems with symmetric solutions. We showed that the foresaid class satisfies the differential equation

     

Direct products of simple modules over Dedekind domains

Let R be a (commutative) Dedekind domain and let the R-module M be a direct product of simple R-modules. Then any homomorphism from a closed submodule K of M to M can be lifted to M. Received: 9 December 2002     

Multigraded Betti numbers of simplicial forests

We prove that multigraded Betti numbers of a simplicial forest are always either 0 or 1. Moreover a nonzero multidegree appears exactly in one homological degree in the resolution. Our work generalizes work of Bouchat [2] on edge ideals of graph trees.     

Sections of zero dimensional ideals over a Noetherian ring

Let A be a commutative Noetherian ring and be an ideal containing a monic polynomial such that A[T]/I is zero dimensional. Suppose the conormal module I/I ² is generated by r elements over A[T]/I. Then a set of r generators of can be lifted to a r generating set of I. A part of this work is done at the Abdus Salam, International Centre for Theoretical Physics, Trieste, Italy. Received: 12 March 2006 Revised: 29 January 2007     

The symmetric group given by a Gröbner basis

We present a Gröbner basis associated with the symmetric group of degree n, which is determined by a strong generating set of the symmetric group and is defined by means of a term ordering with the elimination property.