scholarly journals Some Quotients of Chain Products are Symmetric Chain Orders

10.37236/2430 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Dwight Duffus ◽  
Jeremy McKibben-Sanders ◽  
Kyle Thayer
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Boolean Lattice ◽  
Disjoint Cycles ◽  
Chain Order ◽  
Symmetric Chain ◽  
Straightforward Proof ◽  
Symmetric Chain Order

Canfield and Mason have conjectured that for all subgroups $G$ of the automorphism group of the Boolean lattice $B_n$ (which can be regarded as the symmetric group $S_n$), the quotient order $B_n/G$ is a symmetric chain order.  We provide a straightforward proof of a generalization of a result of K. K. Jordan: namely, $B_n/G$ is an SCO whenever $G$ is generated by powers of disjoint cycles. In addition, the Boolean lattice $B_n$ can be replaced by any product of finite chains. The symmetric chain decompositions of Greene and Kleitman provide the basis for partitions of these quotients.

scholarly journals Symmetric Chain Decompositions of Quotients by Wreath Products

10.37236/5073 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Dwight Duffus ◽  
Kyle Thayer
Keyword(s):  
Symmetric Group ◽  
Elementary Proof ◽  
Boolean Lattice ◽  
Wreath Products ◽  
Cyclic Groups ◽  
Chain Order ◽  
A Chain ◽  
Symmetric Chain ◽  
Symmetric Chain Order

Subgroups of the symmetric group $S_n$ act on $C^n$ (the $n$-fold product $C \times \cdots \times C$ of a chain $C$) by permuting coordinates, and induce automorphisms of the power $C^n$. For certain families of subgroups of $S_n$, the quotients defined by these groups can be shown to have symmetric chain decompositions (SCDs). These SCDs allow us to enlarge the collection of subgroups $G$ of $S_n$ for which the quotient $\mathbf{2}^n/G$ on the Boolean lattice $\mathbf{2}^n$ is a symmetric chain order (SCO). The methods are also used to provide an elementary proof that quotients of powers of SCOs by cyclic groups are SCOs.

scholarly journals Symmetric Chain Decomposition of Necklace Posets

10.37236/1178 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Vivek Dhand
Keyword(s):  
Disjoint Union ◽  
Cyclic Permutation ◽  
Chain Decomposition ◽  
Chain Order ◽  
Symmetric Chain Decomposition ◽  
Symmetric Chain ◽  
Symmetric Chain Order

A finite ranked poset is called a symmetric chain order if it can be written as a disjoint union of rank-symmetric, saturated chains. If $\mathcal{P}$ is any symmetric chain order, we prove that $\mathcal{P}^n/\mathbb{Z}_n$ is also a symmetric chain order, where $\mathbb{Z}_n$ acts on $\mathcal{P}^n$ by cyclic permutation of the factors.

scholarly journals Analysis of Some Properties from Construction of Finite Projective Planes of Order 2 and 3

CAUCHY ◽  
2016 ◽  
Vol 4 (3) ◽  
pp. 131
Author(s):  
Vira Hari Krisnawati ◽  
Corina Karim
Keyword(s):  
Automorphism Group ◽  
Projective Plane ◽  
Symmetric Group ◽  
Block Design ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Steiner System ◽  
Incidence Relation ◽  
Finite Projective Planes ◽  
Set Of Points

<p class="abstract"><span lang="IN">In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system <em>S</em>(<em>t</em>, <em>k</em>, <em>v</em>) is a set of <em>v</em> points and <em>k</em> blocks which satisfy that every <em>t</em>-subset of <em>v</em>-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with <em>t</em> = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.</span></p><p class="abstract"><span lang="IN">In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.</span></p>

scholarly journals $${\mathfrak {S}}_5$$-equivariant syzygies for the Del Pezzo surface of degree 5

2020 ◽  
Author(s):  
Ingrid Bauer ◽  
Fabrizio Catanese
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Blow Up ◽  
Pezzo Surface ◽  
Irreducible Representations ◽  
Del Pezzo Surface ◽  
Del Pezzo ◽  
Arithmetically Gorenstein ◽  
Pfaffian Equations

Abstract The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in $${\mathbb {P}}^5$$P5 by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $${\mathfrak {S}}_5$$S5. We give canonical explicit $${\mathfrak {S}}_5$$S5-invariant Pfaffian equations through a 6$$\times $$×6 antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of $${\mathfrak {S}}_5$$S5. Finally, we give $${\mathfrak {S}}_5$$S5-invariant equations for the embedding of Y inside $$({\mathbb {P}}^1)^5$$(P1)5, and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.

scholarly journals On the existence of symmetric chain decompositions in a quotient of the Boolean lattice

2009 ◽  
Vol 309 (17) ◽  
pp. 5278-5283 ◽  
Author(s):  
Zongliang Jiang ◽  
Carla D. Savage
Keyword(s):  
Boolean Lattice ◽  
Symmetric Chain

The Automorphism Group of the Strong Order of the Symmetric Group

1988 ◽  
Vol s2-37 (2) ◽  
pp. 193-202 ◽  
Author(s):  
Joseph P. S. Kung ◽  
David C. Sutherland
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Strong Order

scholarly journals Decompositions of the Boolean Lattice into Rank-symmetric Chains

10.37236/5328 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
István Tomon
Keyword(s):  
Lower Bound ◽  
Additional Condition ◽  
Boolean Lattice ◽  
Partial Ordering ◽  
Symmetric Chains ◽  
Power Set ◽  
A Chain ◽  
Symmetric Chain

The Boolean lattice $2^{[n]}$ is the power set of $[n]$ ordered by inclusion. A chain $c_{0}\subset\cdots\subset c_{k}$ in $2^{[n]}$ is rank-symmetric, if $|c_{i}|+|c_{k-i}|=n$ for $i=0,\ldots,k$; and it is symmetric, if $|c_{i}|=(n-k)/2+i$. We show that there exist a bijection $$p: [n]^{(\geq n/2)}\rightarrow [n]^{(\leq n/2)}$$ and a partial ordering $<$ on $[n]^{(\geq n/2)}$ satisfying the following properties:$\subset$ is an extension of $<$ on $[n]^{(\geq n/2)}$;if $C\subset [n]^{(\geq n/2)}$ is a chain with respect to $<$, then $p(C)\cup C$ is a rank-symmetric chain in $2^{[n]}$, where $p(C)=\{p(x): x\in C\}$;the poset $([n]^{(\geq n/2)},<)$ has the so called normalized matching property.We show two applications of this result.A conjecture of  Füredi asks if $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains such that the size of any two chains differ by at most 1. We prove an asymptotic version of this conjecture with the additional condition that every chain in the partition is rank-symmetric: $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ rank-symmetric chains, each of size $\Theta(\sqrt{n})$.Our second application gives a lower bound for the number of symmetric chain partitions of $2^{[n]}$. We show that $2^{[n]}$ has at least $2^{\Omega(2^{n}\log n/\sqrt{n})}$ symmetric chain partitions.

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