Minimal Set Of Generators Research Articles

Lieb-Schultz-Mattis Theorem for 1D Quantum Magnets with Antiunitary Translation and Inversion Symmetries.

We study quantum many-body systems in the presence of an exotic antiunitary translation or inversion symmetry involving time reversal. Based on a symmetry-twisting method and spectrum robustness, we propose that a half-integer spin chain that respects any of these two antiunitary crystalline symmetries in addition to the discrete Z_{2}×Z_{2} global spin-rotation symmetry must either be gapless or possess degenerate ground states. This explains the gaplessness of a class of chiral spin models not indicated by the Lieb-Schultz-Mattis theorem and its known extensions. Moreover, we present symmetry classes with minimal sets of generators that give nontrivial Lieb-Schultz-Mattis-type constraints, argued by the bulk-boundary correspondence in 2D symmetry-protected topological phases as well as lattice homotopy. Our results for detecting the ingappability of 1D quantum magnets from the interplay between spin-rotation symmetries and magnetic space groups are applicable to systems with a broader class of spin interactions, including Dzyaloshinskii-Moriya and triple-product interactions.

The admissible monomial bases for the polynomial algebra of five variables in some types of generic degrees

Let Pk be the graded polynomial algebra F2[x1,x2,…,xk] over the prime field of two elements, F2, with the degree of each xi being 1. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for Pk as a module over the mod-2 Steenrod algebra, A. In this paper, we explicitly determine a minimal set of A-generators for P5 in the case of the generic degrees n=2d+1−1 and n=2d+1−2 for all d⩾6.

The hit problem of three variables for the cohomology of the classifying space BE_(s ) as a module over the mod 3 Steenrod Algebra at some low degrees

The hit problem, set up by F. Peterson, finds a minimal set of generators for the polynomial algebra , as a module over the mod-2 Steenrod algebra. We study the extended hit problem for the cohomology of the classifying space over field , with at degrees d

Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings

Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings

Gröbner geometry for regular nilpotent Hessenberg Schubert cells

A regular nilpotent Hessenberg Schubert cell is the intersection of a regular nilpotent Hessenberg variety with a Schubert cell. In this paper, we describe a set of minimal generators of the defining ideal of a regular nilpotent Hessenberg Schubert cell in the type A setting. We show that these minimal generators are a Gröbner basis for an appropriate lexicographic monomial order. As a consequence, we obtain a new computational-algebraic proof, in type A, of Tymoczko's result that regular nilpotent Hessenberg varieties are paved by affine spaces. In addition, we prove that these defining ideals are complete intersections, are geometrically vertex decomposable, and compute their Hilbert series. We also produce a Frobenius splitting of each Schubert cell that compatibly splits all of the regular nilpotent Hessenberg Schubert cells contained in it. This work builds on, and extends, work of the second and third author on defining ideals of intersections of regular nilpotent Hessenberg varieties with the (open) Schubert cell associated to the Bruhat-longest permutation.

Minimal set of generators of ideals defining nilpotent orbit closures

Over a field of characteristic 0 0 , we construct a minimal set of generators of the defining ideals of closures of nilpotent conjugacy class in the set of n × n n \times n matrices. This modifies a conjecture of Weyman and provides a complete answer to it.

On a minimal set of generators for the algebra $H^*(BE_6; \mathbb F_2)$ as a module over the Steenrod algebra and applications

Let $\mathcal P_n \cong H^{*}\big(BE_n; \mathbb F_2 \big)$ be the graded polynomial algebra over the prime field of two elements $\mathbb F_2$, where $E_n$ is an elementary abelian 2-group of rank $n,$ and $BE_n$ is the classifying space of $E_n.$ We study the {\it hit problem}, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra $\mathcal P_{n},$ viewed as a module over the mod-2 Steenrod algebra $\mathcal{A}$. This problem remains unsolvable for $n>4,$ even with the aid of computers in the case of $n=5.$ By considering $\mathbb F_2$ as a trivial $\mathcal A$-module, then the hit problem is equivalent to the problem of finding a basis of $\mathbb F_2$-graded vector space $\mathbb F_2 {\otimes}_{\mathcal{A}}\mathcal P_{n}.$ This paper aims to explicitly determine an admissible monomial basis of the $ \mathbb{F}_{2}$-vector space $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n}$ in the generic degree $n(2^{r}-1)+2\cdot 2^{r},$ where $r$ is an arbitrary non-negative integer, and in the case of $n=6.$ As an application of these results, we obtain the dimension results for the polynomial algebra $\mathcal P_n$ in degrees $(n-1)\cdot(2^{n+u-1}-1)+\ell\cdot2^{n+u},$ where $u$ is an arbitrary non-negative integer, $\ell =13,$ and $n=7.$ Moreover, for any integer $r>1,$ the behavior of the sixth Singer algebraic transfer in degree $6(2^{r}-1)+2\cdot2^r$ is also discussed at the end of this paper. Here, the Singer algebraic transfer is a homomorphism from the homology of the Steenrod algebra to the subspace of $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n}$ consisting of all the $GL_n(\mathbb F_2)$-invariant classes. It is a useful tool in describing the homology groups of the Steenrod algebra, $\text{Tor}^{\mathcal A}_{n, n+*}(\mathbb F_2,\mathbb F_2).$

Mukai Conjecture

In algebraic geometry, syzygies roughly refers to the relations of defining equations of a projective variety embedded in a projective space. They are defined in terms of a minimal graded free resolution for the homogeneous coordinate ring over a polynomial ring. Since the resolution is unique up to isomorphism, it makes sense to talk about the number of a minimal set of generators of given degree in any syzygy module.

The integral Chow ring of , for odd

For any odd integer $d$, we give a presentation for the integral Chow ring of the stack $\mathcal {M}_{0}(\mathbb {P}^r, d)$, as a quotient of the polynomial ring $\mathbb {Z}[c_1,c_2]$. We describe an efficient set of generators for the ideal of relations, and compute them in generating series form. The paper concludes with explicit computations of some examples for low values of $d$ and $r$, and a conjecture for a minimal set of generators.

On a minimal set of generators for the algebra H ∗ ( B E d ; F 2 ) and its applications

We investigate the Peterson hit problem for the polynomial algebra \(\mathcal P_{d}\), viewed as a graded left module over the mod-\(2\) Steenrod algebra, \(\mathcal{A}\). For \(d>4\), this problem is still unsolved, even in the case of \(d=5\) with the help of computers. In this article, we study the hit problem for the case \(d=6\) in the generic degree \(6(2^{r}-1)+6.2^r\), with \(r\) an arbitrary non-negative integer. Furthermore, the behavior of the sixth Singer algebraic transfer in degree \(6(2^{r}-1)+6.2^r\) is also discussed at the end of this paper.

Calogero-Moser Spaces and the Invariants of Two Matrices of Degree 3

We find a minimal set of generators for the coordinate ring of Calogero-Moser space ${\boldsymbol {\mathcal {C}}}_{\boldsymbol {3}}$ and the algebraic relations among them explicitly. We give a new presentation for the algebra of 3×3 invariant matrices involving the defining relations of ${\mathbb {C}}[\boldsymbol {\mathcal {C}}_{\boldsymbol {3}}]$ . Furthermore, we find an explicit description of the GL3-invariant commuting variety of 3×3 matrices and its orbits under the action of the affine plane Cremona group.

Projective covers and minimal sets of generators over local rings

We describe the structure of projective covers of modules over a local ring, when such covers exist, and modules with minimal sets of generators. The case of modules over valuation rings is studied in more detail.

On robustness and related properties on toric ideals

A toric ideal is called robust if its universal Gröbner basis is a minimal set of generators, and is called generalized robust if its universal Gröbner basis equals its universal Markov basis (the union of all its minimal sets of binomial generators). Robust and generalized robust toric ideals are both interesting from both a commutative algebra and an algebraic statistics perspective. However, only a few nontrivial examples of such ideals are known. In this work, we study these properties for toric ideals of both graphs and numerical semigroups. For toric ideals of graphs, we characterize combinatorially the graphs giving rise to robust and to generalized robust toric ideals generated by quadratic binomials. As a by-product, we obtain families of Koszul rings. For toric ideals of numerical semigroups, we determine that one of its initial ideals is a complete intersection if and only if the semigroup belongs to the so-called family of free numerical semigroups. Hence, we characterize all complete intersection numerical semigroups which are minimally generated by one of its Gröbner basis and, as a consequence, all the Betti numbers of the toric ideal and its corresponding initial ideal coincide. Moreover, also for numerical semigroups, we prove that the ideal is generalized robust if and only if the semigroup has a unique Betti element and that there are only trivial examples of robust ideals. We finish the paper with some open questions.

The ring of invariants of pairs of 3 × 3 matrices

The defining relation on the minimal set of generators of the invariants of pairs of matrices of degree three has been established by Nakamoto. Furthermore, it has been simplified with another minimal set of generators by Aslaksen, Drensky and Sadikova using methods coming from representation theory. This work is devoted to obtain this defining relation in a completely different and novel way. We use the Poisson algebra structure on the ring of invariants of pairs of matrices to do so.

Free Covers and Minimal Sets of Generators

We continue our study of the relation between existence of projective covers and existence of minimal sets of generators for right modules over an arbitrary ring.

Minimal generators of annihilators of even neat elements in the exterior algebra footnotesize

This paper deals with an exterior algebra of a vector space whose base field is of positive characteristic. In this work, a minimal set of generators forming the annihilator of even neat elements of such an exterior algebra is exhibited. The annihilator of some special type of even neat elements is determined to prove the conjecture established in [3]. Moreover, a vector space basis for the annihilators under consideration is calculated.

Poisson algebra structure on the invariants of pairs of matrices of degree three

We provide a table of multiplication of the Poisson algebra on the minimal set of generators of the invariants of pairs of matrices of degree three.

The infinite dimensional unital Nambu–Poisson algebra of order 3

From a commutative associative algebra with a basis the infinite dimensional unital Nambu–Poisson algebra of order 3 is constructed, which is also a canonical Nambu–Poisson algebra, and its structures and derivations are discussed. It is proved that: (1) there is a minimal set of generators of consisting of six vectors; (2) the quotient 3-Lie algebra is simple; (3) four infinite dimensional 3-Lie algebras: the 3-Virasoro–Witt algebra (), and the 3-algebra can be embedded in the unital Nambu–Poisson algebra of order 3.

On the dimension of [formula omitted] as a module over Steenrod ring

We write P for the polynomial algebra in one variable over the finite field Z2 and P⊗t=Z2[x1,…,xt] for its t-fold tensor product with itself. We grade P⊗t by assigning degree 1 to each generator. We are interested in determining a minimal set of generators for the ring of invariants (P⊗t)Gt as a module over Steenrod ring, A2. Here Gt is a subgroup of the general linear group GL(t,Z2). An equivalent problem is to find a monomial basis of the space of “unhit” elements, Z2⊗A2(P⊗t)Gt in each t and degree n≥0. The structure of this tensor product is proved surprisingly difficult and has been not yet known for t≥5, even for the trivial subgroup Gt={e}. In the present paper, we consider the subgroup Gt={e} for t∈{5,6}, and obtain some new results on A2-generators of (P⊗t)Gt in some degrees. At the same time, some of their applications have been proposed. We also provide an algorithm in MAGMA for verifying the results. This study can be understood as a continuation of our recent works in [24,26].

The generators, relations and type of the Backelin semigroup

We present an explicit minimal set of generators for the defining ideal of the family of Backelin semigroups and find its Betti numbers. In particular, we compute the type of the semigroup as well, correcting a claim in the literature. Additionally, we show that the Betti numbers of the corresponding numerical semigroup ring coincide to those of its tangent cone.