scholarly journals Sects and Lattice Paths over the Lagrangian Grassmannian

10.37236/8664 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Aram Bingham ◽  
Özlem Uğurlu
Keyword(s):  
Symmetric Space ◽  
Borel Subgroup ◽  
Cell Decomposition ◽  
Bruhat Order ◽  
Lattice Paths ◽  
Lagrangian Grassmannian ◽  
A Cell ◽  
Weighted Lattice Paths

We examine Borel subgroup orbits in the classical symmetric space of type $CI$, which are parametrized by skew symmetric $(n, n)$-clans. We describe bijections between such clans, certain weighted lattice paths, and pattern-avoiding signed involutions, and we give a cell decomposition of the symmetric space in terms of collections of clans called sects. The largest sect with a conjectural closure order is isomorphic (as a poset) to the Bruhat order on partial involutions.

scholarly journals Lattice Paths, Sampling Without Replacement, and Limiting Distributions

10.37236/156 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
M. Kuba ◽  
A. Panholzer ◽  
H. Prodinger
Keyword(s):  
Generating Functions ◽  
Probability Distributions ◽  
Random Variable ◽  
Phase Changes ◽  
Lattice Paths ◽  
Sampling Without Replacement ◽  
Sample Paths ◽  
Quarter Plane ◽  
Explicit Formulae ◽  
Weighted Lattice Paths

In this work we consider weighted lattice paths in the quarter plane ${\Bbb N}_0\times{\Bbb N}_0$. The steps are given by $(m,n)\to(m-1,n)$, $(m,n)\to(m,n-1)$ and are weighted as follows: $(m,n)\to(m-1,n)$ by $m/(m+n)$ and step $(m,n)\to(m,n-1)$ by $n/(m+n)$. The considered lattice paths are absorbed at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$. We provide explicit formulæ for the sum of the weights of paths, starting at $(m,n)$, which are absorbed at a certain height $k$ at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$, using a generating functions approach. Furthermore these weighted lattice paths can be interpreted as probability distributions arising in the context of Pólya-Eggenberger urn models, more precisely, the lattice paths are sample paths of the well known sampling without replacement urn. We provide limiting distribution results for the underlying random variable, obtaining a total of five phase changes.

On a cell decomposition of the Hilbert scheme of points in the plane

1988 ◽  
Vol 91 (2) ◽  
pp. 365-370 ◽  
Author(s):  
Geir Ellingsrud ◽  
Stein Arild Str�mme
Keyword(s):  
Hilbert Scheme ◽  
Cell Decomposition ◽  
Hilbert Scheme Of Points ◽  
A Cell

scholarly journals GENERALIZING DELANNOY NUMBERS VIA COUNTING WEIGHTED LATTICE PATHS

Integers ◽  
2014 ◽  
Author(s):  
M. Dziemianczuk
Keyword(s):  
Lattice Paths ◽  
Delannoy Numbers ◽  
Weighted Lattice Paths

scholarly journals On weighted lattice paths

1973 ◽  
Vol 14 (1) ◽  
pp. 21-29 ◽  
Author(s):  
R.D Fray ◽  
D.P Roselle
Keyword(s):  
Lattice Paths ◽  
Weighted Lattice Paths

scholarly journals On the cohomology of loop spaces of compact Lie groups

1985 ◽  
Vol 26 (1) ◽  
pp. 91-99 ◽  
Author(s):  
Howard Hiller
Keyword(s):  
Homogeneous Spaces ◽  
Cell Decomposition ◽  
Point Of View ◽  
Loop Spaces ◽  
Bruhat Decomposition ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Analogous Question ◽  
A Cell ◽  
Simply Connected

Let G be a compact, simply-connected Lie group. The cohomology of the loop space ΏG has been described by Bott, both in terms of a cell decomposition [1] and certain homogeneous spaces called generating varieties [2]. It is possible to view ΏG as an infinite dimensional “Grassmannian” associated to an appropriate infinite dimensional group, cf. [3], [7]. From this point of view the above cell-decomposition of Bott arises from a Bruhat decomposition of the associated group. We choose a generator H ∈ H2(ΏG, ℤ) and call it the hyperplane class. For a finite-dimensional Grassmannian the highest power of H carries geometric information about the variety, namely, its degree. An analogous question for ΏG is: What is the largest integer Nk = Nk(G) which divides Hk ∈ H2k(ΏG, ℤ)?Of course, if G = SU(2) = S3, one knows Nk = h!. In general, the deviation of Nk from k! measures the failure of H to generate a divided polynomial algebra in H*(ΏG, ℤ).

THE RENNER MONOIDS AND CELL DECOMPOSITIONS OF THE SYMPLECTIC ALGEBRAIC MONOIDS

2003 ◽  
Vol 13 (02) ◽  
pp. 111-132 ◽  
Author(s):  
ZHENHENG LI ◽  
LEX E. RENNER
Keyword(s):  
Cross Section ◽  
Borel Subgroup ◽  
Unit Group ◽  
Admissible Sets ◽  
Cell Decompositions ◽  
A Cell ◽  
Direct Definition ◽  
The Cross ◽  
Definition Of ◽  
Renner Monoid

In this paper we explicitly determine the Renner monoid ℛ and the cross section lattice Λ of the symplectic algebraic monoid MSpn in terms of the Weyl group and the concept of admissible sets; it turns out that ℛ is a submonoid of ℛn, the Renner monoid of the whole matrix monoid Mn, and that Λ is a sublattice of Λn, the cross section lattice of Mn. Cell decompositions in algebraic geometry are usually obtained by the method of [1]. We give a more direct definition of cells for MSpn in terms of the B × B-orbits, where B is a Borel subgroup of the unit group G of MSpn. Each cell turns out to be the intersection of MSpn with a cell of Mn. We also show how to obtain these cells using a carefully chosen one parameter subgroup.

scholarly journals CELL DECOMPOSITION AND CLASSIFICATION OF DEFINABLE SETS INp-OPTIMAL FIELDS

2017 ◽  
Vol 82 (1) ◽  
pp. 120-136 ◽  
Author(s):  
LUCK DARNIÈRE ◽  
IMMANUEL HALPUCZOK
Keyword(s):  
Decomposition Theorem ◽  
Cell Decomposition ◽  
Extreme Value ◽  
Property A ◽  
Definable Subset ◽  
Skolem Functions ◽  
Definable Sets ◽  
A Cell ◽  
Preparation Theorem ◽  
Definable Functions

AbstractWe prove that forp-optimal fields (a very large subclass ofp-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strongp-minimality. Then we turn to stronglyp-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to ap-adic one. For such fieldsK, we prove that every definable subset ofK×Kdwhose fibers overKare inverse images by the valuation of subsets of the value group is semialgebraic. Combining the two we get a preparation theorem for definable functions onp-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension.

On Computing a Cell Decomposition of a Real Surface Containing Infinitely Many Singularities

2014 ◽  
pp. 246-252 ◽  
Author(s):  
Daniel J. Bates ◽  
Daniel A. Brake ◽  
Jonathan D. Hauenstein ◽  
Andrew J. Sommese ◽  
Charles W. Wampler
Keyword(s):  
Cell Decomposition ◽  
Real Surface ◽  
A Cell ◽  
Infinitely Many Singularities

LIFTING FILLING DEHN SPHERES

2012 ◽  
Vol 21 (08) ◽  
pp. 1250082 ◽  
Author(s):  
RUBEN VIGARA
Keyword(s):  
Triple Point ◽  
Cell Decomposition ◽  
Double Curve ◽  
A Cell

A Dehn sphere Σ [C. D. Papakyriakopoulos, On Dehn's Lemma and the asphericity of knots, Ann. Math. (2) 66 (1957) 1–26] in a closed 3-manifold M is a sphere immersed in M with only double curve and triple point singularities. The Dehn sphere Σ ⊂ M lifts to M × I, where I is an interval, if there exists an embedded sphere in M × I that projects onto Σ. Every closed 3-manifold has a filling Dehn sphere [J. M. Montesinos-Amilibia, Representing 3-Manifold by Dehn Spheres, Contribuciones Matemáticas: Homenaje a Joaquín Arregui Fernández (Editorial Complutense, 2000), pp. 239–247], i.e. a Dehn sphere that defines a cell decomposition of M. In [R. Vigara, Representación de 3-variedades por esferas de Dehn rellenantes, Ph.D. Thesis, UNED, Madrid (2006)], it is shown that every closed 3-manifold M has a filling Dehn sphere that lifts to M × I. In this paper it is proved that every closed 3-manifold has a filling Dehn sphere that does not lift to M × I. These results solve a question of Roger Fenn.

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