Showing papers in "Journal of Algebraic Combinatorics in 2019"

Depth and regularity modulo a principal ideal

Abstract: We study the relationship between depth and regularity of a homogeneous ideal I and those of (I, f) and I : f, where f is a linear form or a monomial. Our results have several interesting consequences on depth and regularity of edge ideals of hypergraphs and of powers of ideals.

The power graph of a torsion-free group

Abstract: The power graphP(G) of a group G is the graph whose vertex set is G, with x and y joined if one is a power of the other; the directed power graph $$\overrightarrow{P}(G)$$ has the same vertex set, with an arc from x to y if y is a power of x. It is known that, for finite groups, the power graph determines the directed power graph up to isomorphism. However, it is not true that any isomorphism between power graphs induces an isomorphism between directed power graphs. Moreover, for infinite groups the power graph may fail to determine the directed power graph. In this paper, we consider power graphs of torsion-free groups. Our main results are that, for torsion-free nilpotent groups of class at most 2, and for groups in which every non-identity element lies in a unique maximal cyclic subgroup, the power graph determines the directed power graph up to isomorphism. For specific groups such as $$\mathbb {Z}$$ and $$\mathbb {Q}$$ , we obtain more precise results. Any isomorphism $$P(\mathbb {Z})\rightarrow P(G)$$ preserves orientation, so induces an isomorphism between directed power graphs; in the case of $$\mathbb {Q}$$ , the orientations are either all preserved or all reversed. We also obtain results about groups in which every element is contained in a unique maximal cyclic subgroup (this class includes the free and free abelian groups), and about subgroups of the additive group of $$\mathbb {Q}$$ and about $$\mathbb {Q}^n$$ .

Flagged Grothendieck polynomials

Abstract: We show that the flagged Grothendieck polynomials defined as generating functions of flagged set-valued tableaux of Knutson et al. (J Reine Angew Math 630:1–31, 2009) can be expressed by a Jacobi–Trudi-type determinant formula generalizing the work of Hudson–Matsumura (Eur J Comb 70:190–201 2018). We also introduce the flagged skew Grothendieck polynomials in these two expressions and show that they coincide.

Generalized minimum distance functions

Abstract: Using commutative algebra methods, we study the generalized minimum distance function (gmd function) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. The number of solutions that a system of homogeneous polynomials has in any given finite set of projective points is expressed as the degree of a graded ideal. If $${\mathbb {X}}$$ is a set of projective points over a finite field and I is its vanishing ideal, we show that the gmd function and the Vasconcelos function of I are equal to the rth generalized Hamming weight of the corresponding Reed–Muller-type code $$C_{\mathbb {X}}(d)$$ of degree d. We show that the generalized footprint function of I is a lower bound for the rth generalized Hamming weight of $$C_{\mathbb {X}}(d)$$ . Then, we present some applications to projective nested Cartesian codes. To give applications of our lower bound to algebraic coding theory, we show an interesting integer inequality. Then, we show an explicit formula and a combinatorial formula for the second generalized Hamming weight of an affine Cartesian code.

Oriented hypergraphic matrix-tree type theorems and bidirected minors via Boolean order ideals

Abstract: Restrictions of incidence preserving path maps produce oriented hypergraphic All Minors Matrix-tree Theorems for Laplacian and adjacency matrices. The images of these maps produce a locally signed graphic, incidence generalization, of cycle covers and basic figures that correspond to incidence-k-forests. When restricted to bidirected graphs, the natural partial ordering of maps results in disjoint signed Boolean lattices whose minor calculations correspond to principal order ideals. As an application, (1) the determinant formula of a signed graphic Laplacian is reclaimed and shown to be determined by the maximal positive-circle-free elements, and (2) spanning trees are equivalent to single-element order ideals.

Categorical relations between Langlands dual quantum affine algebras: doubly laced types

Abstract: We prove that the Grothendieck rings of category $$\mathcal {C}^{(t)}_Q$$ over quantum affine algebras $$U_q'(\mathfrak {g}^{(t)})$$ $$(t=1,2)$$ associated with each Dynkin quiver Q of finite type $$A_{2n-1}$$ (resp $$D_{n+1}$$ ) are isomorphic to one of the categories $$\mathcal {C}_{\mathscr {Q}}$$ over the Langlands dual $$U_q'({^L}\mathfrak {g}^{(2)})$$ of $$U_q'(\mathfrak {g}^{(2)})$$ associated with any twisted adapted class $$[\mathscr {Q}]$$ of $$A_{2n-1}$$ (resp $$D_{n+1}$$ ) This results provide simplicity-preserving correspondences on Langlands duality for finite-dimensional representation of quantum affine algebras, suggested by Frenkel–Hernandez

Construction and nonexistence of strong external difference families

Abstract: Strong external difference families (SEDFs) were introduced by Paterson and Stinson as a more restrictive version of external difference families. SEDFs can be used to produce optimal strong algebraic manipulation detection codes. We characterize the parameters $$(v, m, k, \lambda )$$ of a nontrivial SEDF that is near-complete (satisfying $$v=km+1$$ ). We construct the first known nontrivial example of a $$(v, m, k, \lambda )$$ SEDF having $$m > 2$$ . The parameters of this example are (243, 11, 22, 20), giving a near-complete SEDF, and its group is $$\mathbb {Z}_3^5$$ . We provide a comprehensive framework for the study of SEDFs using character theory and algebraic number theory, showing that the cases $$m=2$$ and $$m>2$$ are fundamentally different. We prove a range of nonexistence results, greatly narrowing the scope of possible parameters of SEDFs.

New Cameron-Liebler line classes with parameter $\frac{q^2+1}{2}$

Abstract: New families of Cameron–Liebler line classes of $$\mathrm{PG}(3,q)$$ , $$q\ge 7$$ odd, with parameter $$(q^2+1)/2$$ are constructed.

Heisenberg algebra, wedges and crystals

Abstract: We explain how the action of the Heisenberg algebra on the space of q-deformed wedges yields the Heisenberg crystal structure on charged multipartitions, by using the Boson–Fermion correspondence and looking at the action of the Schur functions at $$q=0$$ . In addition, we give the explicit formula for computing this crystal in full generality.

On computing Schur functions and series thereof

Abstract: In this paper, we present two new algorithms for computing all Schur functions $$s_\kappa (x_1,\ldots ,x_n)$$ for partitions $$\kappa $$ such that $$|\kappa |\le N$$ . For nonnegative arguments, $$x_1,\ldots ,x_n$$ , both algorithms are subtraction-free and thus each Schur function is computed to high relative accuracy in floating point arithmetic. The cost of each algorithm per Schur function is $$\mathscr {O}(n^2)$$ .

Puncturing maximum rank distance codes

Abstract: We investigate punctured maximum rank distance codes in cyclic models for bilinear forms of finite vector spaces. In each of these models, we consider an infinite family of linear maximum rank distance codes obtained by puncturing generalized twisted Gabidulin codes. We calculate the automorphism group of such codes, and we prove that this family contains many codes which are not equivalent to any generalized Gabidulin code. This solves a problem posed recently by Sheekey (Adv Math Commun 10:475–488, 2016).

The spectra of lifted digraphs

Abstract: We present a method to derive the complete spectrum of the lift $$\varGamma ^\alpha $$ of a base digraph $$\varGamma $$, with voltage assignment $$\alpha $$ on a (finite) group G. The method is based on assigning to $$\varGamma $$ a quotient-like matrix whose entries are elements of the group algebra $$\mathbb {C}[G]$$, which fully represents $$\varGamma ^{\alpha }$$. This allows us to derive the eigenvectors and eigenvalues of the lift in terms of those of the base digraph and the irreducible characters of G. Thus, our main theorem generalizes some previous results of Lovasz and Babai concerning the spectra of Cayley digraphs.

On the minimal positive standardizer of a parabolic subgroup of an Artin–Tits group

Abstract: The minimal standardizer of a curve system on a punctured disk is the minimal positive braid that transforms it into a system formed only by round curves. We give an algorithm to compute it in a geometrical way. Then, we generalize this problem algebraically to parabolic subgroups of Artin–Tits groups of spherical type and we show that, to compute the minimal standardizer of a parabolic subgroup, it suffices to compute the pn-normal form of a particular central element.

Spectrahedrality of hyperbolicity cones of multivariate matching polynomials

Abstract: The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application, we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Branden). Finally, we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.

Remarks on singular Cayley graphs and vanishing elements of simple groups

Abstract: Let $$\Gamma $$ be a finite graph and let $$A(\Gamma )$$ be its adjacency matrix. Then $$\Gamma $$ is singular if $$A(\Gamma )$$ is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. Here we investigate this problem for Cayley graphs $$\mathrm{Cay}(G,H)$$ when G is a finite group and when the connecting set H is a union of conjugacy classes of G. In this situation, the singularity problem reduces to finding an irreducible character $$\chi $$ of G for which $$\sum _{h\in H}\,\chi (h)=0.$$ At this stage, we focus on the case when H is a single conjugacy class $$h^G$$ of G; in this case, the above equality is equivalent to $$\chi (h)=0$$. Much is known in this situation, with essential information coming from the block theory of representations of finite groups. An element $$h\in G$$ is called vanishing if $$\chi (h)=0$$ for some irreducible character $$\chi $$ of G. We study vanishing elements mainly in finite simple groups and in alternating groups in particular. We suggest some approaches for constructing singular Cayley graphs.

On supersolvable and nearly supersolvable line arrangements

Abstract: We introduce a new class of line arrangements in the projective plane, called nearly supersolvable, and show that any arrangement in this class is either free or nearly free. More precisely, we show that the minimal degree of a Jacobian syzygy for the defining equation of the line arrangement, which is a subtle algebraic invariant, is determined in this case by the combinatorics. When such a line arrangement is nearly free, we discuss the splitting types and the jumping lines of the associated rank two vector bundle, as well as the corresponding jumping points, introduced recently by S. Marchesi and J. Valles. As a by-product of our results, we get a version of the Slope Problem, valid over the real and the complex numbers as well.

Reflexive polytopes arising from partially ordered sets and perfect graphs

Abstract: Reflexive polytopes which have the integer decomposition property are of interest. Recently, some large classes of reflexive polytopes with integer decomposition property coming from the order polytopes and the chain polytopes of finite partially ordered sets are known. In the present paper, we will generalize this result. In fact, by virtue of the algebraic technique on Grobner bases, new classes of reflexive polytopes with the integer decomposition property coming from the order polytopes of finite partially ordered sets and the stable set polytopes of perfect graphs will be introduced. Furthermore, the result will give a polyhedral characterization of perfect graphs. Finally, we will investigate the Ehrhart $$\delta $$ -polynomials of these reflexive polytopes.

The second largest eigenvalues of some Cayley graphs on alternating groups

Abstract: Let $$A_n$$ denote the alternating group of degree n with $$n\ge 3$$ . The alternating group graph $$AG_n$$ , extended alternating group graph $$EAG_n$$ and complete alternating group graph $$CAG_n$$ are the Cayley graphs $$\mathrm {Cay}(A_n,T_1)$$ , $$\mathrm {Cay}(A_n,T_2)$$ and $$\mathrm {Cay}(A_n,T_3)$$ , respectively, where $$T_1=\{(1,2,i),(1,i,2)\mid 3\le i\le n\}$$ , $$T_2=\{(1,i,j),(1,j,i)\mid 2\le i

Infinitesimal unitary Hopf algebras and planar rooted forests

Abstract: Infinitesimal bialgebras were introduced by Joni and Rota. An infinitesimal bialgebra is at the same time an algebra and coalgebra, in such a way that the comultiplication is a derivation. Twenty years after Joni and Rota, Aguiar introduced the concept of an infinitesimal (non-unitary) Hopf algebra. In this paper, we study infinitesimal unitary bialgebras and infinitesimal unitary Hopf algebras, in contrary to Aguiar’s approach. Using an infinitesimal version of the Hochschild 1-cocycle condition, we prove, respectively, that a class of decorated planar rooted forests is the free cocycle infinitesimal unitary bialgebra and free cocycle infinitesimal unitary Hopf algebra on a set. As an application, we obtain that the planar rooted forests are the free cocycle infinitesimal unitary Hopf algebra on the empty set.

On 2-distance-transitive circulants

Abstract: A complete classification is given of 2-distance-transitive circulants, which shows that a 2-distance-transitive circulant is a cycle, a Paley graph of prime order, a regular complete multipartite graph, or a regular complete bipartite graph of order twice an odd integer minus a 1-factor.

Edge-transitive uniface embeddings of bipartite multi-graphs

Abstract: It is shown that a bipartite multi-graph $${\varGamma }$$ has an orientably edge-transitive embedding with a single face if and only if $${\varGamma }=\mathbf{K}_{m,n}^{(\lambda )}$$ such that $$\gcd (m,n)=1$$ and mn is even whenever $$\lambda $$ is even. A consequence of this shows that each orientable surface carries a simple edge-transitive map with a single face.

Uniform column sign-coherence and the existence of maximal green sequences

Abstract: In this paper, we prove that each matrix in $$M_{m\times n}({\mathbb {Z}}_{\ge 0})$$ is uniformly column sign-coherent (Definition 2.2 (ii)) with respect to any $$n\times n$$ skew-symmetrizable integer matrix (Corollary 3.3 (ii)). Using such matrices, we introduce the definition of irreducible skew-symmetrizable matrix (Definition 4.1). Based on this, the existence of maximal green sequences for skew-symmetrizable matrices is reduced to the existence of maximal green sequences for irreducible skew-symmetrizable matrices.

New families of optimal high-energy ternary sequences having good correlation properties

Abstract: We employ algebraic methods to provide some new constructions of what we call optimal high-energy ternary sequences (a sequence with entries in $$\{0,1,-1\}$$ with a single zero, having optimal correlation properties). Our motivation for these constructions stems from their usefulness in several areas related to communication and radar systems.

Defective dual varieties for real spectra

Abstract: We introduce an invariant of a finite point configuration $$A \subset \mathbb {R}^{1+n}$$ which we denote the cuspidal form of A. We use this invariant to extend Esterov’s characterization of dual-defective point configurations to exponential sums; the dual variety associated with A has codimension at least 2 if and only if A does not contain any iterated circuit.

Stable Grothendieck rings of wreath product categories

Abstract: Let k be an algebraically closed field of characteristic zero, and let $${\mathcal {C}} = {\mathcal {R}} -\hbox {mod}$$ be the category of finite-dimensional modules over a fixed Hopf algebra over k. One may form the wreath product categories $$ {\mathcal {W}}_{n}({\mathcal {C}}) = ( {\mathcal {R}} \wr S_n)-\hbox {mod}$$ whose Grothendieck groups inherit the structure of a ring. Fixing distinguished generating sets (called basic hooks) of the Grothendieck rings, the classification of the simple objects in $$ {\mathcal {W}}_{n}({\mathcal {C}}) $$ allows one to demonstrate stability of structure constants in the Grothendieck rings (appropriately understood), and hence define a limiting Grothendieck ring. This ring is the Grothendieck ring of the wreath product Deligne category $$S_t({\mathcal {C}})$$ . We give a presentation of the ring and an expression for the distinguished basis arising from simple objects in the wreath product categories as polynomials in basic hooks. We discuss some applications when $$ {\mathcal {R}} $$ is the group algebra of a finite group, and some results about stable Kronecker coefficients. Finally, we explain how to generalise to the setting where $${\mathcal {C}}$$ is a tensor category.

Combinatorial wall-crossing and the Mullineux involution

Abstract: In this paper, we define the combinatorial wall-crossing transformation and the generalized column regularization on partitions and prove that a certain composition of these two transformations has the same effect on the one-row partition (n). As corollaries we explicitly describe the quotients of the partitions which arise in this process. We also prove that the one-row partition is the unique partition that stays regular at any step of the wall-crossing transformation.

Bases of the quantum matrix bialgebra and induced sign characters of the Hecke algebra

Abstract: We combinatorially describe entries of the transition matrices which relate monomial bases of the zero-weight space of the quantum matrix bialgebra. This description leads to a combinatorial rule for evaluating induced sign characters of the type A Hecke algebra $$H_n(q)$$ at all elements of the form $$(1 + T_{s_{i_1}}) \cdots (1 + T_{s_{i_m}})$$ , including the Kazhdan–Lusztig basis elements indexed by 321-hexagon-avoiding permutations. This result is the first subtraction-free rule for evaluating all elements of a basis of the $$H_n(q)$$ -trace space at all elements of a basis of $$H_n(q)$$ .

Seven combinatorial problems around isolated quasihomogeneous singularities

Abstract: This paper proposes seven combinatorial problems around formulas for the characteristic polynomial and the spectral numbers of an isolated quasihomogeneous hypersurface singularity. One of them is a new conjecture on the characteristic polynomial. It is an amendment to an old conjecture of Orlik on the integral monodromy of an isolated quasihomogeneous singularity. The search for a combinatorial proof of the new conjecture led us to the seven purely combinatorial problems.

Semimodularity and the Jordan–Hölder theorem in posets, with applications to partial partitions

Abstract: Lattice-theoretical generalizations of the Jordan–Holder theorem of group theory give isomorphisms between finite maximal chains with same endpoints. The best one has been given by Czedli and Schmidt (after Gratzer and Nation), and it applies to semimodular lattices and gives a chain isomorphism by iterating up and down the perspectivity relation between intervals $$[x\wedge y,x]$$ and $$[y,x\vee y]$$ where x covers $$x\wedge y$$ and $$x\vee y$$ covers y. In this paper, we extend to arbitrary (and possibly infinite) posets the definitions of standard semimodularity and of the slightly weaker “Birkhoff condition”, following the approach of Ore (Bull Amer Math Soc 49(8):567–568, 1943). Instead of perspectivity, we associate tags to the covering relation, a more flexible approach. We study the finiteness and length constancy of maximal chains under both conditions and obtain Jordan–Holder theorems. Our theory is easily applied to groups, to closure ranges of an arbitrary poset, and also to five new order relations on the set of partial partitions of a set (i.e. partitions of its subsets), which do not constitute lattices.

Brill-Noether theory of curves on P1xP1: tropical and classical approach

Abstract: The gonality sequence (dr)r>1 of a smooth algebraic curve comprises the minimal degrees dr of linear systems of rank r. We explain two approaches to compute the gonality sequence of smooth curves in P1 ×P1: a tropical and a classical approach. The tropical approach uses the recently developed Brill–Noether theory on tropical curves and Baker’s specialization of linear systems from curves to metric graphs [1]. The classical one extends the work [12] of Hartshorne on plane curves to curves on P1 × P1.