The faces of the braid arrangement form a monoid. The associated monoid algebra -- the face algebra -- is well-studied, especially in relation to card shuffling and other Markov chains. In this paper, we explore the action of the symmetric group on the face algebra from the perspective of invariant theory. Bidigare proved the invariant subalgebra of the face algebra is (anti)isomorphic to Solomon's descent algebra. We answer the more general question: what is the structure of the face algebra as a simultaneous representation of the symmetric group and Solomon's descent algebra? Special cases of our main theorem recover the Cartan invariants of Solomon's descent algebra discovered by Garsia-Reutenauer and work of Uyemura-Reyes on certain shuffling representations. Our proof techniques involve the homology of intervals in the lattice of set partitions.
We establish counting formulas and bijections for deformations of the braid arrangement. Precisely, we consider real hyperplane arrangements such that all the hyperplanes are of the form $x\_i-x\_j=s$ for some integer $s$. Classical examples include the braid, Catalan, Shi, semiorder and Linial arrangements, as well as graphical arrangements. We express the number of regions of any such arrangement as a signed count of decorated plane trees. The characteristic and coboundary polynomials of these arrangements also have simple expressions in terms of these trees. We then focus on certain "well-b…
The class of Worpitzky-compatible subarrangements of a Weyl arrangement together with an associated Eulerian polynomial was recently introduced by Ashraf, Yoshinaga and the first author, which brings the characteristic and Ehrhart quasi-polynomials into one formula. The subarrangements of the braid arrangement, the Weyl arrangement of type $A$, are known as the graphic arrangements. We prove that the Worpitzky-compatible graphic arrangements are characterized by cocomparability graphs. Our main result yields new formulas for the chromatic and graphic Eulerian polynomials of cocomparability gra…
We establish strong factorization for pairs of smooth fans which are refined by the braid arrangement fan. Our method uses a correspondence between cones and preposets.
Bernardi has given a general formula for the number of regions of a deformation of the braid arrangement as a signed sum over boxed trees. We prove that each set of boxed trees which share an underlying (rooted labeled plane) tree contributes 0, +1, or -1 to this sum, and we give an algorithm for computing this value. For Ish-type arrangements, we further construct a sign-reversing involution which reduces Bernardi's signed sum to the enumeration of a set of (rooted labeled plane) trees. We conclude by explicitly enumerating the trees corresponding to the regions of Ish-type arrangements which…
We study a certain discrete differentiation of piecewise-constant functions on the adjoint of the braid hyperplane arrangement, defined by taking finite-differences across hyperplanes. In terms of Aguiar-Mahajan's Lie theory of hyperplane arrangements, we show that this structure is equivalent to the action of Lie elements on faces. We use layered binary trees to encode flags of adjoint arrangement faces, allowing for the representation of certain Lie elements by antisymmetrized layered binary forests. This is dual to the well-known use of (delayered) binary trees to represent Lie elements of …
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.
We show that the toric variety of the permutohedron (=permutohedral space) has the structure of a cocommutative bimonoid in species, with multiplication/comultiplication given by embedding/projecting-onto boundary divisors. In terms of Losev-Manin's description of permutohedral space as a moduli space, multiplication is concatenation of strings of Riemann spheres and comultiplication is forgetting marked points. In this way, the bimonoid structure is an analog of the cyclic operad structure on the moduli space of genus zero marked curves. Covariant/contravariant data on permutohedral space is …
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…
Depth-Bounds for Neural Networks via the Braid Arrangement
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学研究所）；University of Technology Nuremberg（纽伦堡技术大学）；Freie Universität Berlin（柏林自由大学）；University of Twente（埃因霍温理工大学）
We analyze the faces of the dual Koch arrangement, which is the arrangement of $2^s + 1$ lines obtained by projective duality from the Koch chain $K_s$. In particular, we show that this line arrangement does not contain any $k$-gons for $k > 5$, and that the number of pentagons is $3 \cdot 2^{s-1} - 3$.
In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in $\mathbb{R}^n$. There are at least two different bijective explanations of this formula, one by Pak and Stanley, another by Athanasiadis and Linusson. In 1996, Athanasiadis used the finite field method to derive a formula for the number of $k$-dimensional faces of the Shi arrangement for any $k$. Until now, the formula of Athanasiadis did not have a bijective explanation. In this paper, we extend a bijection for regions defined by Bernardi to obtain a bijection be…
It is an open question to give a combinatorial interpretation of the Falk invariant of a hyperplane arrangement, i.e. the third rank of successive quotients in the lower central series of the fundamental group of the arrangement. In this article, we give a combinatorial formula for this invariant in the case of hyperplane arrangements that are complete lift representation of certain gain graphs. As a corollary, we compute the Falk invariant for the cone of the braid, Shi, Linial and semiorder arrangements.
Institut für Mathematik, Technische Universität Berlin（柏林工业大学数学研究所）；HUN-REN Alfréd Rényi Institute of Mathematics and ELTE Eötvös Loránd University（匈牙利科学院阿尔弗雷德·雷尼数学研究所和布达佩斯罗兰大学）
Characteristic elements of the Tits algebra of a real hyperplane arrangement carry information about the characteristic polynomial. We present this notion and its basic properties, and apply it to derive various results about the characteristic polynomial of an arrangement, from Zaslavsky's formulas to more recent results of Kung and of Klivans and Swartz. We construct several examples of characteristic elements, including one in terms of intrinsic volumes of faces of the arrangement.
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…
