Search

Let R = k [ x 1 , ... , x n ] be a polynomial ring over a filed k . The edge ideal of a graph G is the monomial ideal, denoted by I G , whose generators correspond to the edges in G. The quotient ring R / I G is called the edge ring of G. In this paper, we study the minimal N - graded free resolution of edge ring R / I C n k of a k-partite n-crown graph C n k . We give a combinatorial formula for the graded Betti numbers in the linear strand of R / I C n k . We also determine the Hilbert series of the edge ring R / I C n k both in terms of i-faces of its independence complex and the graded Betti numbers of its edge ring. We then compare these Hilbert Series and derive a combinatorial formula for other graded Betti numbers of R / I C n k . As a consequence, we also determine the extremal graded Betti number of the edge ring of a k - partite n - crown graph R / I C n k . Moreover, we determine the projective dimension and the Castelnuovo–Mumford regularity of the edge rings of these graphs. [ABSTRACT FROM AUTHOR]

We consider the central Schur ring associated with the standard supercharacters of the adjoint group G (A) of a finite radical ring A , and define supercharacters of the subgroup C G (A) (σ) consisting of elements fixed by an involution of G (A) naturally defined by an (anti-)involution of A . In particular, we extend known results for unipotent subgroups of the classical finite Chevalley groups. [ABSTRACT FROM AUTHOR]

In this paper, we study the degenerate unipoly-Dedekind sums rising from the type 2 degenerate unipoly Bernoulli functions and degenerate unipoly-Bernoulli functions of arbitrary indices, respectively.

We introduce a new class of symmetric algebras, which we call hybrid algebras. This class contains on one extreme Brauer graph algebras, and on the other extreme general weighted surface algebras. We show that hybrid algebras are precisely the blocks of idempotent algebras of weighted surface algebras, up to socle deformations. More generally, for tame symmetric algebras whose Gabriel quiver is 2-regular, we show that the tree class of an arbitrary Auslander–Reiten component is Dynkin or Euclidean or one of the infinite trees A ∞ , A ∞ ∞ or D ∞ . [ABSTRACT FROM AUTHOR]

Let (hyp o n , ♯) be the hypoplactic monoid of finite rank n with Schützenberger's involution ♯ . In this paper, we exhibit a faithful representation of (hyp o n , ♯) as an involution monoid of upper triangular matrices over any semiring from a large class including the tropical semiring under the skew transposition. We then give a transparent combinatorial characterization of the word identities satisfied by (hyp o n , ♯) . Further, we prove that (hyp o n , ♯) is non-finitely based if and only if n = 2, 3 and give a polynomial time algorithm to check whether a given word identity holds in (hyp o n , ♯) . Communicated by Scott Chapman [ABSTRACT FROM AUTHOR]

The q-binomial coefficients are q-analogues of the binomial coefficients, counting the number of k-dimensional subspaces in the n-dimensional vector space F q n over F q. In this paper, we define a Euclidean analogue of q-binomial coefficients as the number of k-dimensional subspaces which have an orthonormal basis in the quadratic space (F q n , x 1 2 + x 2 2 + ⋯ + x n 2). We prove its various combinatorial properties compared with those of q-binomial coefficients. In addition, we formulate the number of subspaces of other quadratic types and study some related properties. [ABSTRACT FROM AUTHOR]

In this paper, we study the class of those Boolean functions that are coset leaders of first order Reed-Muller codes. We study their properties and try to better understand their structure (which seems complex), by studying operations on Boolean functions that can provide coset leaders (we show that these operations all provide coset leaders when the operands are coset leaders, and that some can even produce coset leaders without the operands being coset leaders). We characterize those coset leaders that belong to the well known classes of direct sums of monomial Boolean functions and Maiorana-McFarland functions. Since all the functions of Hamming weight at most 2 n - 2 are automatically coset leaders, we are interested in constructing infinite classes of coset leaders having possibly Hamming weight larger than 2 n - 2 . [ABSTRACT FROM AUTHOR]

This paper presents new results on the identities satisfied by the sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the varieties generated by the sylvester and the Baxter monoids have finite axiomatic rank, by giving a finite basis for them. [ABSTRACT FROM AUTHOR]

Let $\Omega _n$ be the ring of polynomial-valued holomorphic differential forms on complex n-space, referred to in physics as the superspace ring of rank n. The symmetric group ${\mathfrak {S}}_n$ acts diagonally on $\Omega _n$ by permuting commuting and anticommuting generators simultaneously. We let $SI_n \subseteq \Omega _n$ be the ideal generated by ${\mathfrak {S}}_n$ -invariants with vanishing constant term and study the quotient $SR_n = \Omega _n / SI_n$ of superspace by this ideal. We calculate the doubly-graded Hilbert series of $SR_n$ and prove an ‘operator theorem’, which characterizes the harmonic space $SH_n \subseteq \Omega _n$ attached to $SR_n$ in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results that were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Colmenarejo, Li, Machacek, Sulzgruber, Swanson, Wallach and Zabrocki.

Let Γ = (X , R) denote a finite, simple, connected, and undirected non-bipartite graph with vertex set X and edge set R . Fix a vertex x ∈ X , and define R f = R \ { y z ∣ ∂ (x , y) = ∂ (x , z) } , where ∂ denotes the path-length distance in Γ . Observe that the graph Γ f = (X , R f) is bipartite. We say that Γ supports a uniform structure with respect to x whenever Γ f has a uniform structure with respect to x. Assume that Γ is a distance-regular graph with classical parameters (D , q , α , β) with q ≤ 1 . Recall that q is an integer, which is not equal to 0 or - 1 . The purpose of this paper is to study when Γ supports a uniform structure with respect to x. The main result of the paper is a complete classification of graphs with classical parameters with q ≤ 1 and D ≥ 4 that support a uniform structure with respect to x. [ABSTRACT FROM AUTHOR]

In this paper, we explicitly give combinatorial formulas for the regularity of powers of edge ideals, reg (I (D) k) , of weighted oriented unmixed forests D whose leaves are sinks ( V + (D) are sinks). This combinatorial formula is a piecewise linear function of k, for k ≥ 1 . [ABSTRACT FROM AUTHOR]

Ordered blueprints are algebraic objects that generalize monoids and ordered semirings, and F 1 ± -algebras are ordered blueprints that have an element ϵ that acts as - 1 . In this work we introduce an analogue of the exterior algebra for F 1 ± -algebras that provides a new cryptomorphism for matroids. We also show how to recover the usual exterior algebra if the F 1 ± -algebra comes from a ring, and the Giansiracusa Grassmann algebra if the F 1 ± -algebra comes from an idempotent semifield. [ABSTRACT FROM AUTHOR]

This paper investigates the interplay between the algebraic properties of the rings, the combinatorial properties of their corresponding zero-divisor graphs, and the associated Hermitian matrix of such graphs. For a finite ring R, its zero-divisor graph may contain both directed edges and undirected edges; such graphs are called mixed graphs. The Hermitian matrices of mixed graphs are natural generalizations of the adjacency matrices of undirected graphs. In this paper, we completely determine the structure and the Hermitian eigenvalues of the zero-divisor graph Γ (D × R) by using the structure and the Hermitian eigenvalues of the zero-divisor graph Γ (R) . As applications, we investigate Γ (D × R) for some special R and extend some known results on this topic. [ABSTRACT FROM AUTHOR]

We discover a new property of the stochastic colored six-vertex model calledflip-invariance. We use it to show that for a given collection of observablesof the model, any transformation that preserves the distribution of eachindividual observable also preserves their joint distribution. This generalizesrecent shift-invariance results of Borodin-Gorin-Wheeler. As limiting cases, weobtain similar statements for the Brownian last passage percolation, theKardar-Parisi-Zhang equation, the Airy sheet, and directed polymers. Our proofrelies on an equivalence between the stochastic colored six-vertex model andthe Yang-Baxter basis of the Hecke algebra. We conclude by discussing therelationship of the model with Kazhdan-Lusztig polynomials and positroidvarieties in the Grassmannian.

We prove that the non-backtracking random walk on Ramanujan graphs with large girth exhibits the fastest possible cutoff with a bounded window. [ABSTRACT FROM AUTHOR]

A new hierarchy of operads over the linear spans of δ -cliffs, which are some words of integers, is introduced. These operads are intended to be analogues of the operad of permutations, also known as the associative symmetric operad. We obtain operads whose partial compositions can be described in terms of intervals of the lattice of δ -cliffs. These operads are very peculiar in the world of the combinatorial operads since, despite the relative simplicity for their construction, they are infinitely generated and they have nonquadratic and nonhomogeneous nontrivial relations. We provide a general construction for some of their quotients. We use it to endow the spaces of permutations, m-increasing trees, c-rectangular paths, and m-Dyck paths with operad structures. The operads on c-rectangular paths admit, as Koszul duals, operads generalizing the duplicial and triplicial operads. [ABSTRACT FROM AUTHOR]

Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are strong Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of surfaces with orbifold points, establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of their q-Cartan matrices. [ABSTRACT FROM AUTHOR]

We give a description of the class of endomorphisms in the semiring U T M n (S) of upper triangular matrices over an additively idempotent semiring S. The endomorphisms α such that α (E i j) , where Eij is a matrix unit, is a (0,1)-matrix are discussed. Our focus of investigation concerns the idempotent endomorphisms of this type. In the last theorem we prove that the number of idempotents is the (n + 1)-th Catalan number C n + 1 . [ABSTRACT FROM AUTHOR]

We extend the facial weak order from finite Coxeter groups to central hyperplane arrangements. The facial weak order extends the poset of regions of a hyperplane arrangement to all its faces. We provide four non-trivially equivalent definitions of the facial weak order of a central arrangement: (1) by exploiting the fact that the faces are intervals in the poset of regions, (2) by describing its cover relations, (3) using covectors of the corresponding oriented matroid, and (4) using certain sets of normal vectors closely related to the geometry of the corresponding zonotope. Using these equivalent descriptions, we show that when the poset of regions is a lattice, the facial weak order is a lattice. In the case of simplicial arrangements, we further show that this lattice is semidistributive and give a description of its join-irreducible elements. Finally, we determine the homotopy type of all intervals in the facial weak order. [ABSTRACT FROM AUTHOR]

Given a homogeneous ideal I in a polynomial ring over a field, one may record, for each degree d and for each polynomial f ∈ I d , the set of monomials in f with nonzero coefficients. These data collectively form the tropicalization of I. Tropicalizing ideals induces a "matroid stratification" on any (multigraded) Hilbert scheme. Very little is known about the structure of these stratifications. In this paper, we explore many examples of matroid strata, including some with interesting combinatorial structure, and give a convenient way of visualizing them. We show that the matroid stratification in the Hilbert scheme of points (P 1) [ k ] is generated by all Schur polynomials in k variables. We end with an application to the T-graph problem of (A 2) [ n ] ; classifying this graph is a longstanding open problem, and we establish the existence of an infinite class of edges. [ABSTRACT FROM AUTHOR]

Zamolodchikov periodicity is a property of T- and Y-systems, arising in the thermodynamic Bethe ansatz. Zamolodchikov integrability was recently considered as its affine analog in our joint work with P. Pylyavskyy. Here we prove periodicity and integrability for similar discrete dynamical systems based on the cube recurrence, also known as the discrete BKP equation. The periodicity part was conjectured by Henriques in 2007. [ABSTRACT FROM AUTHOR]

Let k be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a path on n vertices. We show that both depth and Stanley depth have the same values and can be given in terms of k and n. If n≣0, k + 1, k + 2, . . . , 2k(mod(2k + 1)), then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a cycle on n vertices and tight bounds otherwise, in terms of n and k. We also compute lower bounds for the Stanley depth of the edge ideals associated to the kth power of a path and a cycle and prove a conjecture of Herzog for these ideals.

Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G. We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results. [ABSTRACT FROM AUTHOR]

We introduce a new type of discrete quantum walks, called vertex-face walks, based on orientable embeddings. We first establish a spectral correspondence between the transition matrix U and the vertex-face incidence structure. Using the incidence graph, we derive a formula for the principal logarithm of U 2 , and find conditions for its underlying digraph to be an oriented graph. In particular, we show this happens if the vertex-face incidence structure forms a partial geometric design. We also explore properties of vertex-face walks on the covers of a graph. Finally, we study a non-classical behavior of vertex-face walks. [ABSTRACT FROM AUTHOR]