Search ResultsShowing 1-20 of 61
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arXiv:2410.19454 (Published 2024-10-25)
On combinatorial descriptions of faces of the cone of supermodular functions
Comments: 47 pagesFive different ways of combinatorial description of non-empty faces of the cone of supermodular functions on the power set of a finite basic set $N$ are introduced. Their identification with faces of the cone of supermodular games allows one to associate to them certain polytopes in $\mathbb{R}^{N}$, known as cores (of these games) in context of cooperative game theory, or generalized permutohedra in context of polyhedral geometry. Non-empty faces of the supermodular cone then correspond to normal fans of those polytopes. This (basically) geometric way of description of faces of the cone then leads to the combinatorial ways of their description. The first combinatorial way is to identify the faces with certain partitions of the set of enumerations of $N$, known as rank tests in context of algebraic statistics. The second combinatorial way is to identify faces with certain collections of posets on $N$, known as (complete) fans of posets in context of polyhedral geometry. The third combinatorial way is to identify the faces with certain coverings of the power set of $N$, introduced relatively recently in context of cooperative game theory under name core structures. The fourth combinatorial way is to identify the faces with certain formal conditional independence structures, introduced formerly in context of multivariate statistics under name structural semi-graphoids. The fifth way is to identify the faces with certain subgraphs of the permutohedral graph, whose nodes are enumerations of $N$. We prove the equivalence of those six ways of description of non-empty faces of the supermodular cone. This result also allows one to describe the faces of the polyhedral cone of (rank functions of) polymatroids over $N$ and the faces of the submodular cone over $N$.
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arXiv:2409.05219 (Published 2024-09-08)
Boolean, Free, and Classical Cumulants as Tree Enumerations
Comments: 18 pages, 7 figuresCategories: math.CODefant found that the relationship between a sequence of (univariate) classical cumulants and the corresponding sequence of (univariate) free cumulants can be described combinatorially in terms of families of binary plane trees called troupes. Using a generalization of troupes that we call weighted troupes, we generalize this result to allow for multivariate cumulants. Our result also gives a combinatorial description of the corresponding Boolean cumulants. This allows us to answer a question of Defant regarding his troupe transform. We also provide explicit distributions whose cumulants correspond to some specific weighted troupes.
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arXiv:2409.02341 (Published 2024-09-04)
Combinatorial description of Lusztig $q$-weight multiplicity
Comments: 7 pagesSubjects: 05E10We conjecture a precise relationship between Lusztig $q$-weight multiplicities for type $C$ and Kirillov-Reshetikhin crystals. We also define $\mathfrak{gl}_n$-version of $q$-weight multiplicity for type $C$ and conjecture the positivity.
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arXiv:2408.05015 (Published 2024-08-09)
An algebraic approach to Erdős-Ko-Rado sets of flags in spherical buildings II
Categories: math.COWe continue our investigation of Erd\H{o}s-Ko-Rado (EKR) sets of flags in spherical buildings. In previous work, we used the theory of buildings and Iwahori-Hecke algebras to obtain upper bounds on their size. As the next step towards the classification of the maximal EKR-sets, we describe the eigenspaces for the smallest eigenvalue of the opposition graphs. We determine their multiplicity and provide a combinatorial description of spanning sets of these subspaces, from which a complete description of the maximal Erd\H{o}s-Ko-Rado sets of flags may potentially be found. This was recently shown to be possible for type $A_n$, $n$ odd, by Heering, Lansdown, and the last author by making use of the current work.
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arXiv:2406.15803 (Published 2024-06-22)
Root polytopes, flow polytopes, and order polytopes
Comments: 32 pages, 12 figures, comments welcome!In this paper we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\{e_i-e_j \ \vert \ i \neq j\} \cup \{\pm e_i\}$ in $\mathbb{R}^n$, where $e_1,\dots,e_n$ is the standard basis of $\mathbb{R}^n$. Such a polytope can be encoded by a quiver $Q$ with vertices $V \subseteq \{v_1,\dots,v_n\} \cup \{\star\}$, where each edge $v_j\to v_i$ or $\star \to v_i$ or $v_i\to \star$ gives rise to the point $e_i-e_j$ or $e_i$ or $-e_i$, respectively; we denote the corresponding polytope as $\operatorname{Root}(Q)$. These polytopes have been studied extensively under names such as edge polytope and root polytope. We show that if the quiver $Q$ is star-connected (or strongly-connected if there is no $\star$ vertex) then the root polytope $\operatorname{Root}(Q)$ is reflexive and terminal; we moreover give a combinatorial description of the facets of $\operatorname{Root}(Q)$. We also show that if $Q$ is planar, then $\operatorname{Root}(Q)$ is (integrally equivalent to the) polar dual of the flow polytope of the dual quiver $Q^{\vee}$. Finally we consider the case that $Q$ comes from the Hasse diagram of a ranked poset $P$, and show that $\operatorname{Root}(Q)$ is polar dual to (a translation of) a marked poset polytope. Additionally, we show that the face fan $\mathcal{F}_Q$ of $\operatorname{Root}(Q)$ refines the normal fan $\mathcal{N}(\mathcal{O}(P))$ of the order polytope $\mathcal{O}(P)$, and when $P$ is graded, these fans coincide. We moreover give a combinatorial description of the Picard group of the toric variety associated to $\mathcal{F}_Q$ for any ranked poset $P$, in terms of a new ``canonical extension'' of $P$. These results have applications to mirror symmetry.
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arXiv:2312.10012 (Published 2023-12-15)
The determinant of the Laplacian matrix of a quaternion unit gain graph
Comments: 16 pages, 1 figureA quaternion unit gain graph is a graph where each orientation of an edge is given a quaternion unit, and the opposite orientation is assigned the inverse of this quaternion unit. In this paper, we provide a combinatorial description of the determinant of the Laplacian matrix of a quaternion unit gain graph by using row-column noncommutative determinants recently introduced by one of the authors. A numerical example is presented for illustrating our results.
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arXiv:2312.01417 (Published 2023-12-03)
Lascoux polynomials and subdivisions of Gelfand-Zetlin polytopes
Comments: 21 pages, color picturesWe give a new combinatorial description for stable Grothendieck polynomials in terms of subdivisions of Gelfand-Zetlin polytopes. Moreover, these subdivisions also provide a description of Lascoux polynomials. This generalizes a similar result on key polynomials by Kiritchenko, Smirnov, and Timorin.
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arXiv:2311.08332 (Published 2023-11-14)
Graph Curve Matroids
Comments: 12 pages, 3 figures, comments are welcomeSubjects: 05E14We introduce a new class of matroids, called graph curve matroids. A graph curve matroid is associated to a graph and defined on the vertices of the graph as a ground set. We prove that these matroids provide a combinatorial description of hyperplane sections of degenerate canonical curves in algebraic geometry. Our focus lies on graphs that are 2-connected and trivalent, which define identically self-dual graph curve matroids, but we also develop generalizations. Finally, we provide an algorithm to compute the graph curve matroid associated to a given graph, as well as an implementation and data of examples that can be used in Macaulay2.
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arXiv:2309.01626 (Published 2023-09-04)
Order and chain polytopes of maximal ranked posets
Comments: 17 pagesCategories: math.COFor a class of posets we establish that the f-vector of the chain polytope dominates the f-vector of the order polytope. This supports a conjecture of Hibi and Li. Our proof reveals a combinatorial description of the faces of the corresponding chain polytopes.
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arXiv:2307.04553 (Published 2023-07-10)
Combinatorial generators for the cohomology of toric arrangements
Comments: 35 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1910.13836We give a new combinatorial description of the cohomology ring structure of $H^*(M(\mathcal{A});\mathbb{Z})$ of the complement $M(\mathcal{A})$ of a real complexified toric arrangement $\mathcal{A}$ in $(\mathbb{C}^*)^d$. In particular, we correct an error in the paper ``The integer cohomology algebra of toric arrangements'', Adv. Math., 2017.
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arXiv:2305.04535 (Published 2023-05-08)
On Cohen-Macaulay posets of dimension two and permutation graphs
Comments: 7 pages. Comments are welcomeCategories: math.COWe characterize Cohen-Macaulay posets of dimension two; they are precisely the shellable and strongly connected posets of dimension two. We also give a combinatorial description of these posets. Using the fact that co-comparability graph of a 2-dimensional poset is a permutation graph, we characterize Cohen-Macaulay permutation graphs.
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arXiv:2301.01972 (Published 2023-01-05)
A splitting property of the chromatic homology
Comments: 20 pages, 5 figuresKhovanov introduced a bigraded cohomology theory of links whose graded Euler characteristic is the Jones polynomial. The theory was subsequently applied to the chromatic polynomial of graph \cite{HY}, resulting in a categorification known as the ``chromatic homology''. Much as in the Khovanov homology, in the chromatic homology the chromatic polynomial can be obtained by taking the Euler characteristic of the chromatic homology. In the present paper, we introduce a combinatorial description of enhanced states that can be applied to analysis of the homology in an explicit way by hand. Using the new combinatorial description, we show a splitting property of the chromatic homology. Finally, as an application of the description, we compute the chromatic homology of the complete graph.
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arXiv:2212.11020 (Published 2022-12-21)
A combinatorial description of stability for toric vector bundles
The aim of this paper is to give a necessary and sufficient condition for the stability of a toric vector bundle in the combinatorial terms of its parliament of polytopes, a generalization of Newton polytopes for toric vector bundles by Di Rocco, Jabbusch and Smith. We also define subparliaments of polytopes and identify them with parliaments of equivariant subbundles.
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arXiv:2208.04296 (Published 2022-08-08)
The partial Temperley-Lieb algebra and its representations
Comments: 34 pagesWe give a combinatorial description of a new diagram algebra, the partial Temperley--Lieb algebra, arising as the generic centralizer algebra $\mathrm{End}_{\mathbf{U}_q(\mathfrak{gl}_2)}(V^{\otimes k})$, where $V = V(0) \oplus V(1)$ is the direct sum of the trivial and natural module for the quantized enveloping algebra $\mathbf{U}_q(\mathfrak{gl}_2)$. It is a proper subalgebra of the Motzkin algebra (the $\mathbf{U}_q(\mathfrak{sl}_2)$-centralizer) of Benkart and Halverson. We prove a version of Schur--Weyl duality for the new algebras, and describe their generic representation theory.
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arXiv:2205.03034 (Published 2022-05-06)
A combinatorial description of shape theory
Comments: 20 pages, 7 figuresWe give a combinatorial description of shape theory using finite topological $T_0$-spaces (finite partially ordered sets). This description may lead to a sort of computational shape theory. Then we introduce the notion of core for inverse sequences of finite spaces and prove some properties.
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arXiv:2107.10205 (Published 2021-07-21)
Quantum immanants, double Young-Capelli bitableaux and Schur shifted symmetric functions
Comments: arXiv admin note: substantial text overlap with arXiv:1608.06780In this paper are introduced two classes of elements in the enveloping algebra $\mathbf{U}(gl(n))$: the \emph{double Young-Capelli bitableaux} $[\ \fbox{$S \ | \ T$}\ ]$ and the \emph{central} \emph{Schur elements} $\mathbf{S}_{\lambda}(n)$, that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element $\mathbf{S}_{\lambda}(n)$ is the sum of all double Young-Capelli bitableaux $[\ \fbox{$S \ | \ S$}\ ]$, $S$ row (strictly) increasing Young tableaux of shape $\widetilde{\lambda}$. The Schur elements $\mathbf{S}_\lambda(n)$ are proved to be the preimages - with respect to the Harish-Chandra isomorphism - of the \emph{shifted Schur polynomials} $s_{\lambda|n}^* \in \Lambda^*(n)$. Hence, the Schur elements are the same as the Okounkov \textit{quantum immanants}, recently described by the present authors as linear combinations of \emph{Capelli immanants}. This new presentation of Schur elements/quantum immanants doesn't involve the irreducible characters of symmetric groups. The Capelli elements $\mathbf{H}_k(n)$ are column Schur elements and the Nazarov-Umeda elements $\mathbf{I}_k(n)$ are row Schur elements. The duality in $\boldsymbol{\zeta}(n)$ follows from a combinatorial description of the eigenvalues of the $\mathbf{H}_k(n)$ on irreducible modules that is {\it{dual}} (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the $\mathbf{I}_k(n)$. The passage $n \rightarrow \infty$ for the algebras $\boldsymbol{\zeta}(n)$ is obtained both as direct and inverse limit in the category of filtered algebras, via the \emph{Olshanski decomposition/projection}.
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arXiv:2104.04651 (Published 2021-04-10)
A combinatorial description of certain polynomials related to the XYZ spin chain. II. The polynomials $p_n$
Comments: 21 pages. arXiv admin note: text overlap with arXiv:2004.09924By specializing the parameters in the partition function of the 8VSOS model with domain wall boundary conditions and diagonal reflecting end, we find connections between the three-color model and certain polynomials $p_n(z)$ of Bazhanov and Mangazeev appearing in the eigenvectors of the Hamiltonian of the XYZ spin chain. This work is a continuation of a previous paper where we investigated the related polynomials $q_n(z)$ of Bazhanov and Mangazeev, also appearing in the eigenvectors of the XYZ spin chain.
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arXiv:2102.06316 (Published 2021-02-12)
Representations of Degenerate Affine Hecke Algebra of Type $C_n$ Under the Etingof-Freund-Ma Functor
We compute the image of a polynomial $GL_N$-module under the Etingof-Freund-Ma functor \cite{EFM}. We give a combinatorial description of the image in terms of standard tableaux on a collection of skew shapes and analyze weights of the image in terms of contents.
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arXiv:2004.14007 (Published 2020-04-29)
Quelques Éléments de Combinatoire des Matrices de $SL_2(\mathbb{Z})$
Categories: math.COA Theorem of V.Ovsienko characterizes sequences of positive integers $(a_{1},a_{2},\ldots,a_{n})$ such that the $(2\times2)$-matrix $\begin{pmatrix} a_{n} & -1 \\ 1 & 0 \end{pmatrix}\cdots \begin{pmatrix} a_{1} & -1 \\ 1 & 0 \end{pmatrix}$ is equal to $\pm Id$. In this paper, we study matrices $M$ such that some properties verified by the previous equation are still true when we replace $\pm Id$ by $\pm M$. We also give a combinatorial description of the solutions of this equation when $M=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ in terms of dissections of convex polygons.
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arXiv:2004.09924 (Published 2020-04-21)
A combinatorial description of certain polynomials related to the XYZ spin chain
Comments: 31 pagesWe study the connection between the three-color model and the polynomials $q_n(z)$ of Bazhanov and Mangazeev, which appear in the eigenvectors of the Hamiltonian of the XYZ spin chain. By specializing the parameters in the partition function of the 8VSOS model with DWBC and reflecting end, we find an explicit combinatorial expression for $q_n(z)$ in terms of the partition function of the three-color model with the same boundary conditions. Bazhanov and Mangazeev conjectured that $q_n(z)$ has positive integer coefficients. We prove the weaker statement that $q_n(z+1)$ and $(z+1)^{n(n+1)}q_n(1/(z+1))$ have positive integer coefficients. Furthermore, for the three-color model, we find some results on the number of states with a given number of faces of each color, and we compute strict bounds for the possible number of faces of each color.