scholarly journals Toric matrix Schubert varieties and root polytopes (extended abstract)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Laura Escobar ◽  
Karola Mészáros
Keyword(s):  
Schubert Variety ◽  
Schubert Varieties ◽  
Dimensional Torus ◽  
Special Cases ◽  
Ehrhart Series ◽  
The Matrix ◽  
International Audience ◽  
Reduced Forms ◽  
The Ideal ◽  
Grothendieck Polynomials

International audience Start with a permutation matrix π and consider all matrices that can be obtained from π by taking downward row operations and rightward column operations; the closure of this set gives the matrix Schubert variety Xπ. We characterize when the ideal defining Xπ is toric (with respect to a 2n − 1-dimensional torus) and study the associated polytope of its projectivization. We construct regular triangulations of these polytopes which we show are geometric realizations of a family of subword complexes. We also show that these complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of β-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We also write the volume and Ehrhart series of root polytopes in terms of β-Grothendieck polynomials. Subword complexes were introduced by Knutson and Miller in 2004, who showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.

scholarly journals Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Suho Oh ◽  
Hwanchul Yoo
Keyword(s):  
Hyperplane Arrangement ◽  
Schubert Variety ◽  
Bruhat Order ◽  
Hyperplane Arrangements ◽  
Schubert Varieties ◽  
Flag Manifolds ◽  
Poincaré Polynomial ◽  
Generalized Flag Manifolds ◽  
Nous Montrons ◽  
International Audience

International audience We link Schubert varieties in the generalized flag manifolds with hyperplane arrangements. For an element of a Weyl group, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth. Nous relions des variétés de Schubert dans le variété flag généralisée avec des arrangements des hyperplans. Pour un élément dún groupe de Weyl, nous construisons un certain arrangement graphique des hyperplans. Nous montrons que la fonction génératrice pour les régions de cet arrangement coincide avec le polynome de Poincaré de la variété de Schubert correspondante si et seulement si la variété de Schubert est rationnellement lisse.

scholarly journals Rational smoothness and affine Schubert varieties of type A

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Sara Billey ◽  
Andrew Crites
Keyword(s):  
Weyl Group ◽  
Schubert Variety ◽  
Type A ◽  
Algebraic Combinatorics ◽  
Schubert Varieties ◽  
Topological Properties ◽  
Affine Weyl Group ◽  
International Audience ◽  
Affine Permutations

International audience The study of Schubert varieties in G/B has led to numerous advances in algebraic combinatorics and algebraic geometry. These varieties are indexed by elements of the corresponding Weyl group, an affine Weyl group, or one of their parabolic quotients. Often times, the goal is to determine which of the algebraic and topological properties of the Schubert variety can be described in terms of the combinatorics of its corresponding Weyl group element. A celebrated example of this occurs when G/B is of type A, due to Lakshmibai and Sandhya. They showed that the smooth Schubert varieties are precisely those indexed by permutations that avoid the patterns 3412 and 4231. Our main result is a characterization of the rationally smooth Schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations. L'étude des variétés de Schubert dans G/B a mené à plusieurs avancées en combinatoire algébrique. Ces variétés sont indexées soit par l'élément du groupe de Weyl correspondant, soit par un groupe de Weyl affine, soit par un de leurs quotients paraboliques. Souvent, le but est de déterminer quelles propriétés algébriques et topologiques des variétés de Schubert peuvent être décrites en termes des propriétés combinatoires des éléments du groupe de Weyl correspondant. Un exemple bien connu, dû à Lakshmibai et Sandhya, concerne le cas où G/B est de type A. Ils ont montré que les variétés de Schubert lisses sont exactement celles qui sont indexées par les permutations qui évitent les motifs 3412 et 4231. Notre résultat principal est une caractérisation des variétés de Schubert lisses et rationnelles qui correspondent à des permutations affines pour les motifs 4231 et 3412 et les permutations spirales tordues.

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

scholarly journals Polynomial identities related to special Schubert varieties

2021 ◽  
Author(s):  
Francesca Cioffi ◽  
Davide Franco ◽  
Carmine Sessa
Keyword(s):  
Arbitrary Number ◽  
Schubert Variety ◽  
Decomposition Theorem ◽  
Point Of View ◽  
Explicit Description ◽  
Intersection Cohomology ◽  
Schubert Varieties ◽  
Polynomial Identities ◽  
Poincaré Polynomial

AbstractLet $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

scholarly journals Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 870
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Functional Analysis ◽  
Computer Algebra ◽  
Toeplitz Matrices ◽  
Test Cases ◽  
Recursive Procedure ◽  
Tridiagonal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Eigenvalues ◽  
Complex Valued

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

scholarly journals Poincaré polynomials of generic torus orbit closures in Schubert varieties

2021 ◽  
pp. 189-208
Author(s):  
Eunjeong Lee ◽  
Mikiya Masuda ◽  
Seonjeong Park ◽  
Jongbaek Song
Keyword(s):  
Schubert Variety ◽  
Flag Variety ◽  
Schubert Varieties ◽  
Orbit Closure ◽  
Poincaré Polynomial ◽  
Content Type ◽  
Eulerian Polynomial ◽  
Orbit Closures

The closure of a generic torus orbit in the flag variety G / B G/B of type  A A is known to be a permutohedral variety, and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in  G / B G/B . When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.

scholarly journals Algebraically Solvable Problems: Describing Polynomials as Equivalent to Explicit Solutions

10.37236/734 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Combinatorial Nullstellensatz ◽  
Algebraic Solution ◽  
Explicit Solutions ◽  
Polynomial Map ◽  
Combinatorial Number Theory ◽  
Special Cases ◽  
The Matrix ◽  
Grid Points ◽  
Algebraic Solutions ◽  
Solvable Problems

The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz. On its own, it is a result about polynomials, providing some information about the polynomial map $P|_{\mathfrak{X}_1\times\cdots\times\mathfrak{X}_n}$ when only incomplete information about the polynomial $P(X_1,\dots,X_n)$ is given.In a very general working frame, the grid points $x\in \mathfrak{X}_1\times\cdots\times\mathfrak{X}_n$ which do not vanish under an algebraic solution – a certain describing polynomial $P(X_1,\dots,X_n)$ – correspond to the explicit solutions of a problem. As a consequence of the coefficient formula, we prove that the existence of an algebraic solution is equivalent to the existence of a nontrivial solution to a problem. By a problem, we mean everything that "owns" both, a set ${\cal S}$, which may be called the set of solutions; and a subset ${\cal S}_{\rm triv}\subseteq{\cal S}$, the set of trivial solutions.We give several examples of how to find algebraic solutions, and how to apply our coefficient formula. These examples are mainly from graph theory and combinatorial number theory, but we also prove several versions of Chevalley and Warning's Theorem, including a generalization of Olson's Theorem, as examples and useful corollaries.We obtain a permanent formula by applying our coefficient formula to the matrix polynomial, which is a generalization of the graph polynomial. This formula is an integrative generalization and sharpening of:1. Ryser's permanent formula.2. Alon's Permanent Lemma.3. Alon and Tarsi's Theorem about orientations and colorings of graphs.Furthermore, in combination with the Vigneron-Ellingham-Goddyn property of planar $n$-regular graphs, the formula contains as very special cases:4. Scheim's formula for the number of edge $n$-colorings of such graphs.5. Ellingham and Goddyn's partial answer to the list coloring conjecture.

scholarly journals Operations on partially ordered sets and rational identities of type A

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Adrien Boussicault
Keyword(s):  
Partially Ordered Sets ◽  
Rational Functions ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Closed Expression ◽  
Schubert Polynomials ◽  
Special Cases ◽  
The Family ◽  
International Audience

Combinatorics International audience We consider the family of rational functions ψw= ∏( xwi - xwi+1 )-1 indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P. In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP. We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P, and we compute its numerator in some special cases. We show that the computation of ΨP can be reduced to the case of bipartite posets. Finally, we compute the numerators associated to some special bipartite graphs as Schubert polynomials.

scholarly journals Placozoa and Cnidaria are sister taxa

2017 ◽  
Author(s):  
Christopher E. Laumer ◽  
Harald Gruber-Vodicka ◽  
Michael G. Hadfield ◽  
Vicki B. Pearse ◽  
Ana Riesgo ◽  
...  
Keyword(s):  
Strong Support ◽  
Sister Group ◽  
Sister Group Relationship ◽  
Large Dataset ◽  
Animal Evolution ◽  
Phylogenetic Placement ◽  
The Matrix ◽  
Show Evidence ◽  
Animal Phylogeny ◽  
Reduced Forms

AbstractThe phylogenetic placement of the morphologically simple placozoans is crucial to understanding the evolution of complex animal traits. Here, we examine the influence of adding new genomes from placozoans to a large dataset designed to study the deepest splits in the animal phylogeny. Using site-heterogeneous substitution models, we show that it is possible to obtain strong support, in both amino acid and reduced-alphabet matrices, for either a sister-group relationship between Cnidaria and Placozoa, or for Cnidaria and Bilateria (=Planulozoa), also seen in most published work to date, depending on the orthologues selected to construct the matrix. We demonstrate that a majority of genes show evidence of compositional heterogeneity, and that the support for Planulozoa can be assigned to this source of systematic error. In interpreting this placozoan-cnidarian clade, we caution against a peremptory reading of placozoans as secondarily reduced forms of little relevance to broader discussions of early animal evolution.

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