c -Birkhoff polytopes

Cutting a polytope is a very natural way to produce new classes of interesting polytopes. Moreover, it has been very enlightening to explore which algebraic and combinatorial properties of the original polytope are hereditary to its subpolytopes obtained by a cut. In this work, we devote our attention to all the separating hyperplanes for some given polytope (integral and convex) and study the existence and classification of such hyperplanes. We prove the existence of separating hyperplanes for the order and chain polytopes for any finite posets that are not a single chain, and prove there are no such hyperplanes for any Birkhoff polytopes. Moreover, we give a complete separating hyperplane classification for the unit cube and its subpolytopes obtained by one cut, together with some partial classification results for order and chain polytopes.

For ad-dimensional convex lattice polytopeP, a formula for the boundary volume vol(∂P) is derived in terms of the number of boundary lattice points on the first ⌊d/2⌋ dilations ofP. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulas for thef-vector of a smooth polytope in dimensions three, four, and five. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.

It is proved that for a certain class of integer polytopes P the polynomial h(t ) which appears as the numerator in the Ehrhart series of P, when written as a rational function of t, is equal to the h-polynomial of a simplicial polytope and hence that its co-efficients satisfy the conditions of the g-theorem. This class includes the order polytopes of graded posets, previously studied by Reiner and Welker, and the Birkhoff polytope of doubly stochastic n × n matrices. In the latter case the unimodality of the coefficients of h (t ), which follows, was conjectured by Stanley in 1983.

Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al [SODA 2017] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it.Consider a labeling of the edges of the complete bipartite graph K-{n,n} with labels coming from F-2^d, that satisfies the following condition: For any simple cycle, the sum of the labels over its edges is nonzero. The minimal d where this is possible controls the alphabet size needed for maximally recoverable codes in nn grid topologies.Prior to the current work, it was known that d is between log(n)^2 and n log n. We improve both bounds and show that d is linear in n. The upper bound is a recursive construction which beats the random construction. The lower bound follows by first relating the problem to the independence number of the Birkhoff polytope graph, and then providing tight bounds for it using the representation theory of the symmetric group.

Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al. [Maximally recoverable codes for grid-like topologies, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 2092--2108] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it. Consider a labeling of the edges of the complete bipartite graph $K_{n,n}$, with labels coming from $\mathbb{F}_2^d$, that satisfies the following condition: for any simple cycle, the sum of the labels over its edges is nonzero. The minimal $d$ where this is possible controls the alphabet size needed for maximally recoverable codes in $n \times n$ grid topologies. Prior to the current work, it was known that $d$ is between $(\log n)^2$ and $n \log n$. We improve both bounds and show that $d$ is linear in $n$. The upper bound is a recursive construction which beats the random construction. The lower bound follows by first relating the problem to the independence number of the Birkhoff polytope graph, and then providing tight bounds for it using the representation theory of the symmetric group.

Pattern-avoiding polytopes

This work regards the order polytopes arising from the class of generalized snake posets and their posets of meet-irreducible elements. Among generalized snake posets of the same rank, we characterize those whose order polytopes have minimal and maximal volume. We give a combinatorial characterization of the circuits in related order polytopes and then conclude that all of their triangulations are unimodular. For a generalized snake word, we count the number of flips for the canonical triangulation of these order polytopes. We determine that the flip graph of the order polytope of the poset whose lattice of upper order ideals comes from a ladder is the Cayley graph of a symmetric group. Lastly, we introduce an operation on triangulations called twists and prove that twists preserve regular triangulations.Mathematics Subject Classifications: 52B20, 52B05, 52B12, 06A07Keywords: Order polytopes, triangulations, flow polytopes, circuits

Lattice bijections for string modules, snake graphs and the weak Bruhat order

The absolute order on the hyperoctahedral group B n is investigated. Using a poset fiber theorem, it is proved that the order ideal of this poset generated by the Coxeter elements is homotopy Cohen---Macaulay. This method results in a new proof of Cohen---Macaulayness of the absolute order on the symmetric group. Moreover, it is shown that every closed interval in the absolute order on B n is shellable and an example of a non-Cohen---Macaulay interval in the absolute order on D 4 is given. Finally, the closed intervals in the absolute order on B n and D n which are lattices are characterized and some of their important enumerative invariants are computed.

We construct an explicit isomorphism between (truncations of) quiver Hecke algebras and Elias–Williamson's diagrammatic endomorphism algebras of Bott–Samelson bimodules. As a corollary, we deduce that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are tautologically equal to the associated p-Kazhdan–Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. We hence give an elementary and more explicit proof of the main theorem of Riche–Williamson's recent monograph and extend their categorical equivalence to cyclotomic quiver Hecke algebras, thus solving Libedinsky–Plaza's categorical blob conjecture.

Let \(W\) be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator \(\mathsf{Pop}:W\to W\) to be the map that fixes the identity element and sends each nonidentity element \(w\) to the meet of the elements covered by \(w\) in the right weak order. When \(W\) is the symmetric group \(S_n\), \(\mathsf{Pop}\) coincides with the pop-stack-sorting map. Generalizing a theorem about the pop-stack-sorting map due to Ungar, we prove that \[\sup\limits_{w\in W}\left|O_{\mathsf{Pop}}(w)\right|=h,\] where \(h\) is the Coxeter number of \(W\) (with \(h=\infty\) if \(W\) is infinite) and \(O_f(w)\) denotes the forward orbit of \(w\) under a map \(f\). When \(W\) is finite, this result is equivalent to the statement that the maximum number of terms appearing in the Brieskorn normal form of an element of \(W\) is \(h-1\). More generally, we define a map \(f:W\to W\) to be compulsive if for every \(w\in W\), \(f(w)\) is less than or equal to \(\mathsf{Pop}(w)\) in the right weak order. We prove that if \(f\) is compulsive, then \(\sup\limits_{w\in W}|O_f(w)|\leq h\). This result is new even for symmetric groups. We prove that \(2\)-pop-stack-sortable elements in type \(B\) are in bijection with \(2\)-pop-stack-sortable permutations in type \(A\), which were enumerated by Pudwell and Smith. Claesson and Gu{\dh}mundsson proved that for each fixed nonnegative integer \(t\), the generating function that counts \(t\)-pop-stack-sortable permutations in type \(A\) is rational; we establish analogous results in types \(B\) and \(\widetilde A\).Mathematics Subject Classifications: 05E16, 37E15, 05A05Keywords: Pop-stack-sorting, Coxeter group, weak order, Coxeter number, compulsive map, regular language

Given a finite irreducible Coxeter group W with a fixed Coxeter element c, we define the Coxeter pop-tsack torsing operator Pop T :W→W by Pop T (w)=w·π T (w) -1 , where π T (w) is the join in the noncrossing partition lattice NC(w,c) of the set of reflections lying weakly below w in the absolute order. This definition serves as a “Bessis dual” version of the first author’s notion of a Coxeter pop-stack sorting operator, which, in turn, generalizes the pop-stack sorting map on symmetric groups. We show that if W is coincidental or of type D, then the identity element of W is the unique periodic point of Pop T and the maximum size of a forward orbit of Pop T is the Coxeter number h of W. In each of these types, we obtain a natural lift from W to the dual braid monoid of W. We also prove that W is coincidental if and only if it has a unique forward orbit of size h. For arbitrary W, we show that the forward orbit of c -1 under Pop T has size h and is isolated in the sense that none of the non-identity elements of the orbit have preimages lying outside of the orbit.

Although both noncrossing partitions and nonnesting partitions are uniformly enumerated for Weyl groups, the exact relationship between these two sets of combinatorial objects remains frustratingly mysterious. In this paper, we give a precise combinatorial answer in the case of the symmetric group: for any standard Coxeter element, we construct an equivariant bijection between noncrossing partitions under the Kreweras complement and nonnesting partitions under a Coxeter‐theoretically natural cyclic action we call the Kroweras complement. Our equivariant bijection is the unique bijection that is both equivariant and support‐preserving, and is built using local rules depending on a new definition of charmed roots. Charmed roots are determined by the choice of Coxeter element — in the special case of the linear Coxeter element , we recover one of the standard bijections between noncrossing and nonnesting partitions.

The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide with the infinite fully commutative reduced words and with infinite powers of Coxeter elements. We answer several open problems raised there by classifying minimal elements in all affine types and relating these elements to the classes of fully commutative and Coxeter elements. Interestingly, the infinite fully commutative elements correspond to the minuscule and cominuscule nodes of the Dynkin diagram, while the infinite Coxeter elements correspond to a single node, which we call the heavy node, in all affine types other than type $A$.

Positivity results for Stanley's character polynomials