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Finite dimensional simple modules of quantum affine algebras of type A correspond to semistandard Young tableaux of rectangular shapes. In this paper, we classify all prime modules corresponding to 2-column semistandard Young tableaux. Moreover, we give a conjectural sufficient condition for a module corresponding to a tableau with more than two columns to be prime., Comment: This is a part of the original file arXiv:2303.05618v3. The original file arXiv:2303.05618v3 is splited into two parts. The other part of the original file arXiv:2303.05618v3 has the title: Tropical Geometry, Quantum Affine Algebras, and Scattering Amplitudes

Minimal kinematics identifies likelihood degenerations where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov-Huh. We characterize all choices of minimal kinematics on the moduli space $\mathcal{M}_{0,n}$. These choices are motivated by the CHY model in physics and they are represented combinatorially by 2-trees. We compute 2-tree amplitudes, and we explore extensions to non-planar on-shell diagrams, here identified with the hypertrees of Castravet-Tevelev., Comment: 18 pages

A detailed understanding of the moduli spaces $X(k,n)$ of $n$ points in projective $k-1$ space is essential to the investigation of generalized biadjoint scalar amplitudes, as discovered by Cachazo, Early, Guevara and Mizera (CEGM) in 2019. But in math, conventional wisdom says that it is completely hopeless due to the arbitrarily high complexity of realization spaces of oriented matroids. In this paper, we nonetheless find a path forward. We present a Lie-theoretic realization of color factors for color-dressed generalized biadjoint scalar amplitudes, formulated in terms of certain tilings of the real moduli space $X(k,n)$ and collections of logarithmic differential forms, resolving an important open question from recent work by Cachazo, Early and Zhang. The main idea is to replace the realization space decomposition of $X(k,n)$ with a large class of overlapping tilings whose topologies are individually relatively simple. So we obtain a collection of color-dressed amplitudes, each of which satisfies $U(1)$ decoupling separately. The essential complexity appears when they are all superposed., Comment: 24 pages, 3 figures. Fixed typos

Real, complex, and tropical algebraic geometry join forces in a new branch of mathematical physics called positive geometry. We develop the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties. Their connected components are derived from polyhedral spaces with Weyl group symmetries. We study their canonical forms and scattering amplitudes, and we solve the likelihood equations., Comment: 34 pages, 4 figures

In a recent paper, we defined generalized color orderings (GCO) and Feynman diagrams (GFD) to compute color-dressed generalized biadjoint amplitudes. In this work, we study the Cachazo-Early-Guevara-Mizera (CEGM) representation of generalized partial amplitudes and ``decoupling" identities. This representation is a generalization of the Cachazo-He-Yuan (CHY) formulation as an integral over the configuration space $X(k,n)$ of $n$ points on $\mathbb{CP}^{k-1}$ in generic position. Unlike the $k=2$ case, Parke-Taylor-like integrands are not enough to compute all partial amplitudes for $k>2$. Here we give a set of constraints that integrands associated with GCOs must satisfy and use them to construct all $(3,n<9)$ integrands, all $(3,9)$ integrands up to four undetermined constants, and $95 \%$ of $(4,8)$ integrands up to 24 undetermined constants. $k=2$ partial amplitudes are known to satisfy identities. Among them, the so-called $U(1)$ decoupling identities are the simplest ones. These are characterized by a label $i$ and a color ordering in $X(2,|[n]\setminus \{i\}|)$. Here we introduce decoupling identities for $k>2$ determined combinatorially using GCOs. Moreover, we identify the natural analog of $U(1)$ identities as those characterized by a pair of labels $i\neq j$, and a pair of GCOs, one in $X(k,|[n]\setminus \{i\}|)$ and the other in $X(k-1,|[n]\setminus \{j\}|)$. We call them {\it double extension} identities. We also provide explicit connections among different ways of representing GCOs, such as configurations of lines, configurations of points, and reorientation classes of uniform oriented matroids (chirotopes)., Comment: 40+17 pages, 17 figures. Ancillary data for GCOs, decoupling identities and integrands attached

The goal of this paper is to make a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. The connection allows us to study important and difficult questions in these areas: (1) We give a systematic construction of prime modules (including prime non-real modules) of quantum affine algebras using tropical geometry. We also introduce new objects which generalize positive tropical Grassmannians. (2) We propose a generalization of Grassmannian string integrals in physics, in which the integrand is no longer a finite, but rather an infinite product indexed by prime modules of a quantum affine algebra. We give a general formula of $u$-variables using prime tableaux (corresponding to prime modules of quantum affine algebras of type $A$) and Auslander-Reiten quivers of Grassmannian cluster categories. (3) We study limit $g$-vectors of cluster algebras. This is another way to obtain prime non-real modules of quantum affine algebras systematically. Using limit $g$-vectors, we construct new examples of non-real modules of quantum affine algebras., Comment: This is a part of the original file arXiv:2303.05618v3. The original file arXiv:2303.05618v3 is splited into two parts. The other part has the title: Classification of prime modules of quantum affine algebras corresponding to 2-column tableaux

The biadjoint scalar theory has cubic interactions and fields transforming in the biadjoint representation of ${\rm SU}(N)\times {\rm SU}\big({\tilde N}\big)$. Amplitudes are "color" decomposed in terms of partial amplitudes computed using Feynman diagrams which are simultaneously planar with respect to two orderings. In 2019, a generalization of biadjoint scalar amplitudes based on generalized Feynman diagrams (GFDs) was introduced. GFDs are collections of Feynman diagrams derived by incorporating an additional constraint of "local planarity" into the construction of the arrangements of metric trees in combinatorics. In this work, we propose a natural generalization of color orderings which leads to color-dressed amplitudes. A generalized color ordering (GCO) is defined as a collection of standard color orderings that is induced, in a precise sense, from an arrangement of projective lines on $\mathbb{RP}^2$. We present results for $n\leq 9$ generalized color orderings and GFDs, uncovering new phenomena in each case. We discover generalized decoupling identities and propose a definition of the "colorless" generalized scalar amplitude. We also propose a notion of GCOs for arbitrary $\mathbb{RP}^{k-1}$, discuss some of their properties, and comment on their GFDs. In a companion paper, we explore the definition of partial amplitudes using CEGM integral formulas.

We study the problem of factorization for residues of generalized biadjoint scalar scattering amplitudes $m^{(k)}_n$, introduced by Cachazo, Early, Guevara and Mizera (CEGM), involving multi-dimensional residues which factorize generically into $k$-ary products of lower-point generalized biadjoint amplitudes of the same type $m^{(k)}_{n_1}\cdots m^{(k)}_{n_k}$, where $n_1+\cdots +n_k = n+k(k-1)$, noting that smaller numbers of factors arise as special cases. Such behavior is governed geometrically by regular matroid subdivisions of hypersimplices and cones in the positive tropical Grassmannian, and combinatorially by collections of compatible decorated ordered set partitions, considered modulo cyclic rotation. We make a proposal for conditions under which this happens and we develop $k=3,4$ in detail. We conclude briefly to propose a novel formula to construct coarsest regular matroid subdivisions of all hypersimplices $\Delta_{k,n}$ and rays of the positive tropical Grassmannian, which should be of independent interest., Comment: 30 pages, many figures. v2: typos corrected

In the Grassmannian formulation of the S-matrix for planar $\mathcal{N}=4$ Super Yang-Mills, $N^{k-2}MHV$ scattering amplitudes for $k$ negative and $n-k$ positive helicity gluons can be expressed, by an application of the global residue theorem, as a signed sum over a collection of $(k-2)(n-k-2)$-dimensional residues. These residues are supported on certain positroid subvarieties of the Grassmannian $G(k,n)$. In this paper, we replace the Grassmannian $G(3,n)$ with its torus quotient, the moduli space of $n$ points in the projective plane in general position, and planar $\mathcal{N}=4$ SYM with generalized biadjoint scalar amplitudes $m^{(3)}_n$ as introduced by Cachazo-Early-Guevara-Mizera (CEGM). Whereas in the Grassmannian formulation residues of the Parke-Taylor form correspond to individual BCFW, or on-shell diagrams, we show that each such $(n-5)$-dimensional residue of $m^{(3)}_n$ is an entire biadjoint scalar partial amplitude $m^{(2)}_n$, that is a sum over all tree-level Feynman diagrams for a fixed planar order. We propose a generalization which would give rise to identifications of $m^{(2)}_n$ inside $m^{(k)}_n$ for $k\ge 4$, via $(k-2)(n-k-2)$-dimensional residues. Our proof for $k=3$ uses the CEGM formula for $m^{(3)}_n$; it predicts a new Minkowski sum realization of the associahedron in terms of certain positroid polytopes in the second hypersimplex $\Delta_{2,n}$., Comment: 30 pages, 1 figure. Ancillary notebook with data and calculations

In this paper, we study a novel behavior developed by certain tree-level scalar scattering amplitudes, including the biadjoint, NLSM, and special Galileon, when a subset of kinematic invariants vanishes without producing a singularity. This behavior exhibits properties which we call $\textit{smooth splitting}$ and $\textit{semi-locality}$. The former means that an amplitude becomes the product of exactly three amputated Berends-Giele currents, while the latter means that any two currents share one external particle. We call these smooth splittings 3-splits. In fact, there are exactly $\binom{n}{3}-n$ such 3-splits, one for each generic, interior triangle in a polygon; as they cannot be obtained from standard factorization, they are a new phenomenon in Quantum Field Theory. In fact, the resulting splitting is analogous to the one first seen in Cachazo-Early-Guevara-Mizera (CEGM) amplitudes which generalize standard cubic scalar amplitudes from their ${\rm Tr}\, G(2,n)$ formulation to ${\rm Tr}\, G(k,n)$, where ${\rm Tr}\, G(k,n)$ is the tropical Grassmannian. Along the way, we show how smooth splittings naturally lead to the discovery of mixed amplitudes in the NLSM and special Galileon theories and to novel BCFW-like recursion relations for NLSM amplitudes., Comment: 49 pages, 13 figures. V2. Corrected typos; added Section 6.7 on smooth 2-splits in k=3 CEGM generalized amplitudes

In this paper we study the role of planarity in generalized scattering amplitudes, through several closely interacting structures in combinatorics, algebraic and tropical geometry. The generalized biadjoint scalar amplitude, introduced recently by Cachazo-Early-Guevara-Mizera (CEGM), is a rational function of homogeneous degree $-(k-1)(n-k-1)$ in $\binom{n}{k}-n$ independent variables; its poles can be constructed directly from the rays of the positive tropical Grassmannian. We introduce for each pair of integers $(k,n)$ with $2\le k\le n-2$ a system of generalized positive roots which arises as a specialization of the planar basis of kinematic invariants. We prove that the higher root polytope $\mathcal{R}^{(k)}_{n-k}$ has volume the k-dimensional Catalan number $C^{(k)}_{n-k}$, via a flag unimodular triangulation into simplices, in bijection with noncrossing collections of $k$-element subsets. We also give a bijection between certain positroidal subdivisions, called tripods, of the hypersimplex $\Delta_{3,n}$ and noncrossing pairs of 3-element subsets that are not weakly separated. We show that the facets of the Planar Kinematics (PK) polytope, introduced recently by Cachazo and the author, are exactly the $\binom{n}{k}-n$ generalized positive roots. We show that the PK specialization of the generalized biadjoint amplitude evaluates to $C^{(k)}_{n-k}$. Looking forward, we give defining equations and conjecture explicit solutions using $(\mathbb{CP}^{n-k-1})^{\times (k-1)}$ via a notion of compatibility degree for noncrossing collections, for a two parameter family of generalized worldsheet associahedra $\mathcal{W}^+_{k,n}$. These specialize when $k=2$ to a certain dihedrally invariant partial compactification of the configuration space $M_{0,n}$ of $n$ distinct points in $\mathbb{CP}^{1}$. Many detailed examples are given throughout to motivate future work., Comment: New introduction, fixed typos. 75 pages, 19 figures

In this paper we prove that points in the space $X(k,n)$ of configurations of $n$ points in $\mathbb{CP}^{k-1}$ which are fixed under a certain cyclic action are the solutions to the generalized scattering equations on planar kinematics (PK). In the first part, we give a constructive upper bound: we show that these solutions inject into certain aperiodic k-element subsets of $\{1,\ldots, n\}$, and consequently that their number is bounded above by the number of Lyndon words with k one's and n-k zeros. The proof uses a somewhat surprising connection between the superpotential of the mirror of $G(n-k,n)$ and the generalized CHY potential on $X(k,n)$. We also check the recent conjecture that generalized biadjoint amplitudes evaluate to $k$-dimensional Catalan numbers on PK for several examples including $k=3$ and $n\leq 40$ and $(k,n)=(6,13)$. We then reformulate the CEGM generalized biadjoint scalar amplitude directly as a Laplace transform-type integral over ${\rm Trop}^+ G(k,n)$ and we use it to evaluate the amplitude on PK with the purpose of exhibiting how GFD's glue together. We initiate the study of two minimal lattice polytopal neighborhoods of the planar kinematics point. One of these, the rank-graded root polytope $\mathcal{R}_{k,n}$, in the case $k=2$, is a projection of the standard type A root polytope. The other, denoted $\Pi_{k,n}$, in the case $k=2$, is a degeneration of the associahedron. We check up to and including $\mathcal{R}_{3,9}$ and $\mathcal{R}_{4,9}$ that the relative volume of $\mathcal{R}_{k,n}$ is the multi-dimensional Catalan number $C^{(k)}_{n-k}$, hinting towards the possibility of deeper geometric and combinatorial interpretations of $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ near the PK point., Comment: V2. 60 pages, 5 figures. Added Appendix A on Global Schwinger Parametrization

In this paper, we continue our study of blade arrangements and the positroidal subdivisions which are induced by them on $\Delta_{k,n}$. A blade is a tropical hypersurface which is generated by a system of $n$ affine simple roots of type $SL_n$ that enjoys a cyclic symmetry. When placed at the center of a simplex, a blade induces a decomposition into $n$ maximal cells which are known as Pitman-Stanley polytopes. We introduce a complex $(B_{k,n},\partial)$ of weighted blade arrangements and we prove that the positive tropical Grassmannian surjects onto the top component of the complex, such that the induced weights on blades in the faces $\Delta_{2,n-(k-2)}$ of $\Delta_{k,n}$ are (1) nonnegative and (2) their support is weakly separated. We finally introduce a hierarchy of elementary weighted blade arrangements for all hypersimplices which is minimally closed under the boundary maps $\partial$, and apply our result to classify up to isomorphism type all rays of the positive tropical Grassmannian $\text{Trop}_+ G(3,n)$ for $n\le 9$., Comment: 30 pages, 1 figure

In this note we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all $k$ and $n$ on what we call the minimal kinematics. We prove that on the minimal kinematics, the scattering equations on the configuration space of $n$ points on $\mathbb{CP}^{k-1}$ has a unique solution, and that this solution is in the image of a Veronese embedding. The minimal kinematics is an all $k$ generalization of the one recently introduced by Early for $k=2$ and uses a choice of cyclic ordering. We conjecture an explicit formula for $m_n^{(k)}(\mathbb{I},\mathbb{I})$ which we have checked analytically through $n=10$ for all $k$. The answer is a simple rational function which has only simple poles; the poles have the combinatorial structure of the circulant graph ${\rm C}_n^{(1,2,\dots, k-2)}$. Generalized biadjoint amplitudes can also be evaluated using the positive tropical Grassmannian ${\rm Tr}^+{\rm G}(k,n)$ in terms of generalized planar Feynman diagrams. We find perfect agreement between both definitions for all cases where the latter is known in the literature. In particular, this gives the first strong consistency check on the $90\,608$ planar arrays for ${\rm Tr}^+{\rm G}(4,8)$ recently computed by Cachazo, Guevara, Umbert and Zhang. We also introduce another class of special kinematics called planar-basis kinematics which generalizes the one introduced by Cachazo, He and Yuan for $k=2$ and uses the planar basis recently introduced by Early for all $k$. Based on numerical computations through $n=8$ for all $k$, we conjecture that on the planar-basis kinematics $m_n^{(k)}(\mathbb{I},\mathbb{I})$ evaluates to the multidimensional Catalan numbers, suggesting the possibility of novel combinatorial interpretations. For $k=2$ these are the standard Catalan numbers.

In recent work of Cachazo, Guevara, Mizera and the author, a generalization of the biadjoint scattering amplitude $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ was introduced as an integral over the moduli space of $n$ points in $\mathbb{CP}^{k-1}$, with value a sum of certain rational functions on the kinematic space $\mathcal{K}_{k,n}$. It was shown there for $m^{(3)}(\mathbb{I}_6,\mathbb{I}_6)$ and later by Cachazo and Rojas that collections of poles appearing in $m^{(3)}(\mathbb{I}_7,\mathbb{I}_7)$ are compatible exactly when they are dual to collections of rays which generate the maximal faces of a polyhedral complex known as the (nonnegative) tropical Grassmannian. In this note, we derive a remarkable planar basis for the space of generalized kinematic invariants which coincides in the case $k=2$ with usual standard planar multi-particle basis for the kinematic space. We implement in Mathematica the action on formal linear combinations of planar matroid subdivisions of a boundary operator which, together with the planar basis, determines compatibility for any given poles appearing in the expansion of $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$, by computing a certain combinatorial non-crossing condition on the second hypersimplicial faces $\Delta_{2,n-(k-2)}$ of $\Delta_{k,n}$. The algorithms are implemented in an accompanying Mathematica notebook and are evaluated on existing tables of rays, in the form of tropical Plucker vectors, to tabulate the finest planar subdivisions of $\Delta_{3,8},\Delta_{3,9}$ and $ \Delta_{4,8}$, or equivalently the set of maximal cones for the corresponding nonnegative tropical Grassmannians., Comment: 26 pages, 6 figures; v2: attached Mathematica notebook

We study arrangements of slightly skewed tropical hyperplanes, called blades by A. Ocneanu, on the vertices of a hypersimplex $\Delta_{k,n}$, and we investigate the resulting induced polytopal subdivisions. We show that placing a blade on a vertex $e_J$ induces an $\ell$-split matroid subdivision of $\Delta_{k,n}$, where $\ell$ is the number of cyclic intervals in the $k$-element subset $J$. We prove that a given collection of $k$-element subsets is weakly separated, in the sense of the work of Leclerc and Zelevinsky on quasicommuting families of quantum minors, if and only if the arrangement of the blade $((1,2,\ldots, n))$ on the corresponding vertices of $\Delta_{k,n}$ induces a matroid (in fact, a positroid) subdivision. In this way we obtain a compatibility criterion for (planar) multi-splits of a hypersimplex, generalizing the rule known for 2-splits. We study in an extended example the case $(k,n) = (3,7)$ the set of arrangements of $(k-1)(n-k-1)$ weakly separated vertices of $\Delta_{k,n}$., Comment: 30 pages, 10 figures. v4: expanded introduction. To appear in Combinatorial Theory

We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ${\mathbb{CP}^1}$, to higher-dimensional projective spaces $\mathbb{CP}^{k-1}$. The standard, $k=2$ Mandelstam invariants, $s_{ab}$, are generalized to completely symmetric tensors $\textsf{s}_{a_1a_2\ldots a_k}$ subject to a `massless' condition $\textsf{s}_{a_1a_2\cdots a_{k-2}\,b\,b}=0$ and to `momentum conservation'. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the $k=3$ case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all `biadjoint amplitudes' for $(k,n)=(3,6)$ and find a direct connection to the tropical Grassmannian. This leads to the notion of $k=3$ Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for $k=2$, and provides analytic solutions analogous to the MHV ones., Comment: 27+7 pages. v2: typos corrected. Connection to trop G(3,7) added at end of section 4. Appendix with numerical seeds for all solutions to X(3,6) equations provided

In this paper we study an algebra that naturally combines two familiar operations in scattering amplitudes: computations of volumes of polytopes using triangulations and constructions of canonical forms from products of smaller ones. We mainly concentrate on the case of $G(2,n)$ as it controls both general MHV leading singularities and CHY integrands for a variety of theories. This commutative algebra has also appeared in the study of configuration spaces and we called it the $\Delta$-algebra. As a natural application, we generalize the well-known square move. This allows us to generate infinite families of new moves between non-planar on-shell diagrams. We call them sphere moves. Using the $\Delta$-algebra we derive familiar results, such as the KK and BCJ relations, and prove novel formulas for higher-order relations. Finally, we comment on generalizations to $G(k,n)$., Comment: 36+13 pages

We initiate the study of certain cyclically skewed tropical hyperplanes, called permutohedral blades by A. Ocneanu, and we make connections matroid theory, tropical geometry, moduli spaces and scattering amplitudes. We study two families of piecewise constant functions on $\mathbb{R}^{n-1}$ taking values in $\{0,1\}$ and $\lbrack n\rbrack =\{1,2,\ldots,n\}$, called respectively characteristic functions and graduated functions. We show that their codimension-one level sets exactly permutohedral blades. Using ring-theoretic arguments we show that a blade decomposes as a Minkowski sum of tripods and one-dimensional subspaces. For each triangulation of a cyclically oriented polygon there exists such a factorization. In the language of tropical geometry, this constructs a tropical hypersurface as a Minkowski sum of tropical lines. We use the principle of inclusion/exclusion to construct a collection of piecewise constant functions of blades which we enumerate by the dimension of the support of the function. On the induced $\mathbb{Q}$-vector space one has an induced grading; this grading is compatible with various quotient spaces appearing in algebra, topology and scattering amplitudes. This vector space maps homomorphically onto the so-called $\Delta$-algebra, which appears in the study of non-planar MHV leading singularities, leading to non-planar analogs of the square move for plabic graphs for $G(2,n)$, called sphere moves. We give a closed formula for the graded dimension of the basis. It is shown in an Appendix by Donghyun Kim that the coefficients appearing in the numerator of the generating function for the graded dimension are symmetric, and that they sum to $\frac{(2j)!}{j!}$., Comment: v3: rewritten, with major improvements to exposition. Added Proof of Conjecture 8.4 by Donghyun Kim

S. Whitehouse's lifts of the Eulerian representations of $S_n$ to $S_{n+1}$ are reinterpreted, topologically and ring-theoretically, building on the first author's work on A. Ocneanu's theory of permutohedral blades., Comment: Exposition improved; proof of Proposition 6 added. To appear in The Journal of Pure and Applied Algebra

In this note, we study the permutohedral geometry of the poles of a certain differential form introduced in recent work of Arkani-Hamed, Bai, He and Yan. There it was observed that the poles of the form determine a family of polyhedra which have the same face lattice as that of the permutohedron. We realize that family explicitly, proving that it in fact fills out the configuration space of a particularly well-behaved family of generalized permutohedra, the zonotopal generalized permutohedra, that are obtained as the Minkowski sums of line segments parallel to the root directions $e_i-e_j$. Finally we interpret Mizera's formula for the biadjoint scalar amplitude $m(\mathbb{I}_n,\mathbb{I}_n)$, restricted to a certain dimension $n-2$ subspace of the kinematic space, as a sum over the boundary components of the standard root cone, which is the conical hull of the roots $e_1-e_2,\ldots, e_{n-2}-e_{n-1}$., Comment: 23 pages, 7 figures

We study three finite-dimensional quotient vector spaces constructed from the linear span of the set of characteristic functions of permutohedral cones by imposing two kinds of constraints: (1) neglect characteristic functions of higher codimension permutohedral cones, and (2) neglect characteristic functions of non-pointed permutohedral cones. We construct an ordered basis which is canonical, in the sense that it has subsets which map onto ordered bases for the quotients. We present straightening relations to the canonical basis, and using Laplace transforms we obtain functional representations for each quotient space., Comment: 32 pages, 5 Figures; exposition improved

We formulate conjectures giving combinatorial interpretations of the Ehrhart $h^*$-vector, for hypersimplices, for dilated simplices and for generic cross-sections of cubes, in terms of certain decorated ordered set partitions. All were formulated and checked computationally during our graduate study at Penn State.

In this note, we apply combinatorial techniques from our Ph.D. thesis to study how generalized permutohedra may be represented functionally on Parke-Tayor factors and related rational functions. In any functional representation of polyhedral cones, in general certain homological information may be lost. The combinatorial relations of the Parke-Taylor factors lift homologically to generalized permutohedra. The 6-particle case contains several related layers of interesting geometric data: the Newton polytope for the polynomial numerator lifts the permutohedron in three variables, which is a hexagon, and the fraction itself provides a functional representation of certain neighborhoods of a vertex of a 5-dimensional weight permutohedron. The lift from fraction to generalized permutohedron was derived by comparing functional representations. We observe additionally that the numerator and its permutations satisfy a degree 3 polynomial relation which defines a classical projective variety known as the Segre cubic. We include in an extended Appendix selected Mathematica code which can be used to verify our computations independently.

We prove that every orthogonal Gelfand-Zeitlin algebra $U$ acts on its Gelfand-Zeitlin subalgebra $\Gamma$. Considering the dual module, we show that every Gelfand-Zeitlin character of $\Gamma$ is realizable in a $U$-module. We observe that the Gelfand-Zeitlin formulae can be rewritten using divided difference operators. It turns out that the action of the latter operators on $\Gamma$ gives rise to an explicit basis in a certain Gelfand-Zeitlin submodule of the dual module mentioned above. This gives, generically, both in the case of regular and singular Gelfand-Zeitlin characters, an explicit construction of simple modules which realize given Gelfand-Zeitlin characters., Comment: v3: some corrections following the referee reports

In this paper, we announce results from our thesis, which studies for the first time the categorification of the theory of generalized permutohedra. The vector spaces in the categorification are tightly constrained by certain continuity relations which appeared in physics in the mid 20th century. We describe here the action of the symmetric group on the vector spaces in this categorification. Generalized permutohedra are replaced by vector spaces of characteristic functions of polyhedral cones about faces of permutohedra, called plates, due to A. Ocneanu. The symmetric group acts on plates by coordinate permutation. In combinatorics, the Eulerian numbers count the number of permutations with a given number of ascent and descents. The classical Worpitzky identity expands a power $r^p$ as a sum of Eulerian numbers, with binomial coefficients. In our thesis, for the main result we generalize the classical Worpitzky identity to an isomorphism of symmetric group modules, corresponding geometrically to the tiling of a scaled simplex by unit hypersimplices. In the categorification, the volume of a hypersimplex is replaced by the complex-linear dimension of a vector space associated to it. The main technical aspect of the proof of the character formula for the simplex involves a partition of unity of a commutative algebra of translations on a discrete torus, and a certain modular Diophantine equation. A detailed paper is in preparation., Comment: 12 pages, 6 figures