scholarly journals The Skeleton of a Reduced Word and a Correspondence of Edelman and Greene

10.37236/1554 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Stefan Felsner
Keyword(s):  
Bruhat Order ◽  
Young Tableaux ◽  
Reduced Word ◽  
Reduced Decompositions ◽  
Bijective Proof ◽  
Identity Permutation ◽  
Weak Bruhat Order ◽  
Classical Correspondence ◽  
Standard Young Tableaux ◽  
Maximal Chains

Stanley conjectured that the number of maximal chains in the weak Bruhat order of $S_n$, or equivalently the number of reduced decompositions of the reverse of the identity permutation $ w_0 = n,n-1,n-2,\ldots,2,1$, equals the number of standard Young tableaux of staircase shape $s=\{n-1,n-2,\ldots,1\}$. Originating from this conjecture remarkable connections between standard Young tableaux and reduced words have been discovered. Stanley proved his conjecture algebraically, later Edelman and Greene found a bijective proof. We provide an extension of the Edelman and Greene bijection to a larger class of words. This extension is similar to the extension of the Robinson-Schensted correspondence to two line arrays. Our proof is inspired by Viennot's planarized proof of the Robinson-Schensted correspondence. As it is the case with the classical correspondence the planarized proofs have their own beauty and simplicity.

scholarly journals A direct bijective proof of the hook-length formula

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Jean-Christophe Novelli ◽  
Igor Pak ◽  
Alexander V. Stoyanovskii
Keyword(s):  
Young Tableaux ◽  
Hook Length ◽  
Bijective Proof ◽  
International Audience ◽  
Standard Young Tableaux

International audience This paper presents a new proof of the hook-length formula, which computes the number of standard Young tableaux of a given shape. After recalling the basic definitions, we present two inverse algorithms giving the desired bijection. The next part of the paper presents the proof of the bijectivity of our construction. The paper concludes with some examples.

scholarly journals Two Partial Orders for Standard Young Tableaux

10.37236/5967 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Justyna Kosakowska ◽  
Markus Schmidmeier ◽  
Hugh Thomas
Keyword(s):  
Symmetric Group ◽  
Invariant Subspaces ◽  
Bruhat Order ◽  
Partial Orders ◽  
Linear Operators ◽  
Dominance Order ◽  
Representation Space ◽  
Young Tableaux ◽  
Standard Young Tableaux

In this manuscript we show that two partial orders defined on the set of standard Young tableaux of shape $\alpha$ are equivalent. In fact, we give two proofs for the equivalence of  the box order and the dominance order for tableaux. Both are algorithmic. The first of these proofs emphasizes links to the Bruhat order for the symmetric group and the second provides a more straightforward construction of the cover relations. This work is motivated by the known result that the equivalence of the two combinatorial orders leads to a description of the geometry of the representation space of invariant subspaces of nilpotent linear operators.

scholarly journals Symmetry properties of the Novelli-Pak-Stoyanovskii algorithm

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Robin Sulzgruber
Keyword(s):  
Young Tableau ◽  
Sorting Algorithm ◽  
Young Tableaux ◽  
Hook Length ◽  
Jeu De Taquin ◽  
Bijective Proof ◽  
Symmetry Properties ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Standard Young Tableau

International audience The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin which transforms an arbitrary filling of a partition into a standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler and Müller defined the complexity of this algorithm as the average number of performed exchanges, and Neumann and the author proved it fulfils some nice symmetry properties. In this paper we recall and extend the previous results and provide new bijective proofs.

scholarly journals Higher Bruhat Orders in Type B

10.37236/5620 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Seth Shelley-Abrahamson ◽  
Suhas Vijaykumar
Keyword(s):  
Symmetric Group ◽  
Minimal Element ◽  
Partially Ordered Sets ◽  
Bruhat Order ◽  
Hyperplane Arrangements ◽  
Type B ◽  
Ordered Sets ◽  
Weak Bruhat Order ◽  
Positive Integers ◽  
Maximal Chains

Motivated by the geometry of hyperplane arrangements, Manin and Schechtman defined for each integer $n \geq 1$ a hierarchy of finite partially ordered sets $B(n, k),$ indexed by positive integers $k$, called the higher Bruhat orders.  The poset $B(n, 1)$ is naturally identified with the weak left Bruhat order on the symmetric group $S_n$, each $B(n, k)$ has a unique maximal and a unique minimal element, and the poset $B(n, k + 1)$ can be constructed from the set of maximal chains in $B(n, k)$.  Ben Elias has demonstrated a striking connection between the posets $B(n, k)$ for $k = 2$ and the diagrammatics of Bott-Samelson bimodules in type A, providing significant motivation for the development of an analogous theory of higher Bruhat orders in other Cartan-Killing types, particularly for $k = 2$.  In this paper we present a partial generalization to type B, complete up to $k = 2$, prove a direct analogue of the main theorem of Manin and Schechtman, and relate our construction to the weak Bruhat order and reduced expression graph for Weyl group $B_n$.

Standard Young tableaux in a (2,1)-hook and Motzkin paths

2021 ◽  
Vol 344 (7) ◽  
pp. 112395
Author(s):  
Rosena R.X. Du ◽  
Jingni Yu
Keyword(s):  
Young Tableaux ◽  
Standard Young Tableaux ◽  
Motzkin Paths

scholarly journals Symplectic Keys and Demazure Atoms in Type C

10.37236/9235 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
João Miguel Santos
Keyword(s):  
Type A ◽  
Bruhat Order ◽  
Young Tableaux ◽  
Hyperoctahedral Group ◽  
Similar Role ◽  
Type C ◽  
Left And Right ◽  
Demazure Atoms ◽  
The Right

We compute, mimicking the Lascoux-Schützenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More precisely, our symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type C. The right and the left symplectic keys are related through the Lusztig involution. A type C Schützenberger evacuation is defined to realize that involution.

scholarly journals Enumeration of Standard Young Tableaux of Shifted Strips with Constant Width

10.37236/6466 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Ping Sun
Keyword(s):  
Integral Method ◽  
Recurrence Relations ◽  
Young Tableau ◽  
Constant Width ◽  
Young Tableaux ◽  
Pell Number ◽  
Standard Young Tableaux ◽  
Standard Young Tableau

Let $g_{n_1,n_2}$ be the number of standard Young tableau of truncated shifted shape with $n_1$ rows and $n_2$ boxes in each row. By using the integral method this paper derives the recurrence relations of $g_{3,n}$, $g_{n,4}$ and $g_{n,5}$ respectively. Specifically, $g_{n,4}$ is the $(2n-1)$-st Pell number.

scholarly journals Evaluating the Numbers of some Skew Standard Young Tableaux of Truncated Shapes

10.37236/3890 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Ping Sun
Keyword(s):  
Young Tableaux ◽  
Multiple Integrals ◽  
Standard Young Tableaux ◽  
Product Formulas

In this paper the number of standard Young tableaux (SYT) is evaluated by the methods of multiple integrals and combinatorial summations. We obtain the product formulas of the numbers of skew SYT of certain truncated shapes, including the skew SYT $((n+k)^{r+1},n^{m-1}) / (n-1)^r $ truncated by a rectangle or nearly a rectangle, the skew SYT of truncated shape $((n+1)^3,n^{m-2}) / (n-2) \backslash \; (2^2)$, and the SYT of truncated shape $((n+1)^2,n^{m-2}) \backslash \; (2)$.

Sign in / Sign up

Export Citation Format

Share Document