Pavle Blagojevic, Vladimir Grujic, Rade Zivaljevic
备注：
This is an updated version of the paper. In this version some results (Proposition 1.7., Theorem 1.8, Theorem 1.9, Theorem 1.11) are now reformulated in the greater generality (over integer coefficients). Moreover, we now interpret Theorems 1.8 and 1.11 as a generalization of classical Steenrod's theorem to the case symmetric products of (simple) diagrams of spaces
Using the topological technique of diagrams of spaces, we calculate the homology of the union and the complement of finite arrangements of subspaces of the form $D + SP^{n-d}(X)$ in symmetric products $SP^n(X)$ where $D\in SP^d(X)$. As an application we include a computation of the homology of the homotopy end space of the open manifold $SP^n(M_{g,k})$, where $M_{g,k}$ is a Riemann surface of genus $g$ punctured at $k$ points, a problem which was originally motivated by the study of commutative $(m+k,m)$-groups.
The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This article has two objectives: On one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we intro…
For every irreducible complex representation~$π_λ$ of the symmetric group~$§_n$, we construct, in a canonical way, a so-called intrinsic hyperplane arrangement~$\A_λ$ in the space of~$π_λ$. This arrangement is a direct generalization of the classical braid arrangement (which is the special case of our construction corresponding to the natural representation of~$§_n$), has a natural description in terms of invariant subspaces of Young subgroups, and enjoys a number of remarkable properties.
Originally in 1954 the Tutte polynomial was a bivariate polynomial associated to a graph in order to enumerate the colorings of this graph and of its dual graph at the same time. However the Tutte polynomial reveals more of the internal structure of a graph, and contains even other specializations from other sciences like the Jones polynomial in Knot theory, the partition function of the Pott model in statistical physics, and the reliability polynomial in network theory. In this article, we study the Tutte polynomial associated to more general objects which are the arrangements of hyperplanes.…
An m-free hyperplane arrangement is a generalization of a free arrangement. Holm asked the following two questions: (1)Does m-free imply (m+1)-free for any arrangement? (2)Are all arrangements m-free for m large enough? In this paper, we characterize m-freeness for product arrangements, while we prove that all localizations of an m-free arrangement are m-free. From these results, we give answers to Holm's questions.
We calculate the skew-symmetric cohomology of the complement of a discriminantal hyperplane arrangement with coefficients in local systems arising in the context of the representation theory of the Lie algebra sl_2. For a discriminantal arrangement in C^k, the skew-symmetric cohomology is nontrivial in dimension k-1 precisely when the "master function" which defines the local system on the complement has nonisolated critical points. In symmetric coordinates, the critical set is a union of lines. Generically, the dimension of this nontrivial skew-symmetric cohomology group is equal to the numbe…
We investigate toric varieties defined by arrangements of hyperplanes and call them strongly symmetric. The smoothness of such a toric variety translates to the fact that the arrangement is crystallographic. As a result, we obtain a complete classification of this class of toric varieties. Further, we show that these varieties are projective and describe associated toric arrangements in these varieties.
A common generalization for the chromatic polynomial and the flow polynomial of a graph $G$ is the Tutte polynomial $T(G;x,y)$. The combinatorial meaning for the coefficients of $T$ was discovered by Tutte at the beginning of its definition. However, for a long time the combinatorial meaning for the values of $T$ is missing, except for a few values such as $T(G;i,j)$, where $1\leq i,j\leq 2$, until recently for $T(G;1,0)$ and $T(G;0,1)$. In this third one of a series of papers, we introduce product valuations, cartesian product arrangements, and multivariable characteristic polynomials, and ap…
Stanley introduced the chromatic symmetric function of a simple graph, which is a generalization of a chromatic polynomial. This is expressed in terms of the integer points of the complements of the corresponding graphic arrangement. Stanley proved a combinatorial reciprocity theorem for chromatic functions. This is considered as an Ehrhart-type reciprocity theorem for the graphic arrangement. We introduce the chromatic signed-symmetric function of a signed graph, an analogue of the chromatic symmetric function, by the integer points of the complements of the corresponding signed-graphic arran…
We consider the vanishing ideal of an arrangement of linear subspaces in a vector space and investigate when this ideal can be generated by products of linear forms. We introduce a combinatorial construction (blocker duality) which yields such generators in cases with a lot of combinatorial structure, and we present the examples that motivated our work. We give a construction which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. We also consider generic arrangements of p…
A family of homothets of an o-symmetric convex body K in d-dimensional Euclidean space is called a Minkowski arrangement if no homothet contains the center of any other homothet in its interior. We show that any pairwise intersecting Minkowski arrangement of a d-dimensional convex body has at most $2\cdot 3^d$ members. This improves a result of Polyanskii (arXiv:1610.04400). Using similar ideas, we also give a proof the following result of Polyanskii: Let $K_1,\dots,K_n$ be a sequence of homothets of the o-symmetric convex body $K$, such that for any $i<j$, the center of $K_j$ lies on the boun…
To every subspace arrangement X we will associate symmetric functions P[X] and H[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant H[X] specializes to the Tutte polynomial T[X]. Billera, Jia and Reiner recently introduced a quasi-symmetric function F[X] (for matroids) which behaves valuatively with respect to matroid base polytope decompositions. We will define a quasi-symmetric function G[X] for polymatroids which has thi…
We give combinatorial models for the homotopy type of complements of elliptic arrangements (i.e., certain sets of abelian subvarieties in a product of elliptic curves). We give a presentation of the fundamental group of such spaces and, as an application, we treat the case of ordered configuration spaces of elliptic curves. Our models are finite polyhedral CW complexes, and our combinatorial tools of choice are acyclic categories (small categories without loops). As a stepping stone, we give a characterization of which acyclic categories arise as face categories of polyhedral CW complexes.
We investigate the line arrangement that results from intersecting d complete flags in C^n. We give a combinatorial description of the matroid T_{n,d} that keeps track of the linear dependence relations among these lines. We prove that the bases of the matroid T_{n,3} characterize the triangles with holes which can be tiled with unit rhombi. More generally, we provide evidence for a conjectural connection between the matroid T_{n,d}, the triangulations of the product of simplices Delta_{n-1} x Δ_{d-1}, and the arrangements of d tropical hyperplanes in tropical (n-1)-space. Our work provides a …
We extend the notion of face rings of simplicial complexes and simplicial posets to the case of finite-length (possibly infinite) simplicial posets with a group action. The action on the complex induces an action on the face ring, and we prove that the ring of invariants is isomorphic to the face ring of the quotient simplicial poset under a mild condition on the group action. We also identify a class of actions on simplicial complexes that preserve the homotopical Cohen-Macaulay property under quotients. When the acted-upon poset is the independence complex of a semimatroid, the $h$-polynomia…
We construct bundles $E_k(\A,\F) \to M$ over the complement $M$ of a complex hyperplane arrangement \A, depending on an integer $k \geq 1$ and a set $\F=\{f_1, \ldots, f_μ\}$ of continuous functions $f_i \colon M \to \C$ whose differences are nonzero on $M$, generalizing the configuration space bundles arising in the Lawrence-Krammer-Bigelow representation of the pure braid group. We display such families \F\ for rank two arrangements, reflection arrangements of types $A_\ell$, $B_\ell$, $D_\ell$, $F_4$, and for arrangements supporting multinet structures with three classes, with the resulting…
We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study logarithmic derivations and critical set varieties of arrangements in a way which is symmetric with respect to matroid duality. Our main result exhibits the variety of the ideal of pairs as a subspace arrangement whose components correspond to cyclic flats of the arrangement. As a corollary, we are able to give geometric explanations of some freeness and projec…
Answering a question of Füredi and Loeb (1994), we show that the maximum number of pairwise intersecting homothets of a $d$-dimensional centrally symmetric convex body $K$, none of which contains the center of another in its interior, is at most $O(3^d d\log d)$. If $K$ is not necessarily centrally symmetric and the role of its center is played by its centroid, then the above bound can be replaced by $O(3^d\binom{2d}{d}d\log d)$. We establish analogous results for the case where the center is defined as an arbitrary point in the interior of $K$. We also show that in the latter case, one can al…
The cohomology of the configuration space of n points in R^3 admits a symmetric group action and has been shown to be isomorphic to the regular representation. One way to prove this is by defining an S^1-action whose fixed point set is the complement of the braid arrangement and using equivariant cohomology to show that the cohomology of the configuration space is the associated graded algebra of the cohomology of the arrangement complement. In this paper, we will extend this result to the setting of affine subspace arrangements coming from real hyperplane arrangements using similar methods. W…
Given a collection of $t$ subspaces in an $n$-dimensional $\mathbb{K} $-vector space $W$ we can associate to them $t$ vanishing ideals in the symmetric algebra $\mathcal{S}(W^*) = \mathbb{K}[x_1,x_2,\dots,x_n]$. As a subspace is defined by a set of linear equations, its vanishing ideal is generated by linear forms so it is a linear ideal. Conca and Herzog showed that the Castelnuovo-Mumford regularity of the product of $t$ linear ideals is equal to $t$. Derksen and Sidman showed that the Castelnuovo-Mumford regularity of the intersection of $t$ linear ideals is at most $t$ and they also showed…
We study curves consisting of unions of projective lines whose intersections are given by graphs. Under suitable hypotheses on the graph, these so-called \emph{graph curves} can be embedded in projective space as line arrangements. We discuss property $N_p$ for these embeddings and are able to produce products of linear forms that generate the ideal in certain cases. We also briefly discuss questions regarding the higher-dimensional subspace arrangements obtained by taking the secant varieties of graph curves.
We study enumerative questions on the moduli space $\mathcal{M}(L)$ of hyperplane arrangements with a given intersection lattice $L$. Mnëv's universality theorem suggests that these moduli spaces can be arbitrarily complicated; indeed it is even difficult to compute the dimension $D =\dim \mathcal{M}(L)$. Embedding $\mathcal{M}(L)$ in a product of projective spaces, we study the degree $N=\mathrm{deg} \mathcal{M}(L)$, which can be interpreted as the number of arrangements in $\mathcal{M}(L)$ that pass through $D$ points in general position. For generic arrangements $N$ can be computed combinat…
A finite simple graph $Γ$ determines a quotient $P_Γ$ of the pure braid group, called a graphic arrangement group. We analyze homomorphisms of these groups defined by deletion of sets of vertices, using methods developed in prior joint work with R. Randell. We show that, for a $K_4$-free graph $Γ$, a product of deletion maps is injective, embedding $P_Γ$ in a product of free groups. Then $P_Γ$ is residually free, torsion-free, residually torsion-free nilpotent, and acts properly on a CAT(0) cube complex. We also show $P_Γ$ is of homological finiteness type $F_{m-1}$, but not $F_m$, where $m$ i…
We study a family of symmetric polynomials that we refer to as the Boolean product polynomials. The motivation for studying these polynomials stems from the computation of the characteristic polynomial of the real matroid spanned by the nonzero vectors in $\mathbb{R}^n$ all of whose coordinates are either $0$ or $1$. To this end, one approach is to compute the zeros of the Boolean product polynomials over finite fields. The zero loci of these polynomials cut out hyperplane arrangements known as resonance arrangements, which show up in the context of double Hurwitz polynomials. By relating the …
Using an algorithm for computing the symmetric function Kronecker product of D-finite symmetric functions we find some new Kronecker product identities. The identities give closed form formulas for trace-like values of the Kronecker product.
The Heisenberg product is an associative product defined on symmetric functions which interpolates between the usual product and the Kronecker product. In 1938, Murnaghan discovered that the Kronecker product of two Schur functions stabilizes. We prove an analogous result for the Heisenberg product of Schur functions.
We provide a classification of multiplicity-free inner tensor products of irreducible characters of symmetric groups, thus confirming a conjecture of Bessenrodt. Concurrently, we classify all multiplicity-free inner tensor products of skew characters of the symmetric groups. We also provide formulae for calculating the decomposition of these tensor products.
We provide an explicit description of the quantum product of multi-symmetric functions using the elementary multi-symmetric functions introduced by Vaccarino.
