scholarly journals Up- and Down-Operators on Young's Lattice

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Ricky Liu ◽  
Christian Smith
Keyword(s):  
Plactic Monoid ◽  
Young’S Lattice

The up-operators $u_i$ and down-operators $d_i$ (introduced as Schur operators by Fomin) act on partitions by adding/removing a box to/from the $i$th column if possible. It is well known that the $u_i$ alone satisfy the relations of the (local) plactic monoid, and the present authors recently showed that relations of degree at most 4 suffice to describe all relations between the up-operators. Here we characterize the algebra generated by the up- and down-operators together, showing that it can be presented using only quadratic relations.

scholarly journals Unimodality and Young's lattice

1990 ◽  
Vol 54 (1) ◽  
pp. 41-53 ◽  
Author(s):  
Dennis Stanton
Keyword(s):  
Young’S Lattice

scholarly journals Identities of the plactic monoid

2014 ◽  
Vol 90 (1) ◽  
pp. 100-112 ◽  
Author(s):  
Łukasz Kubat ◽  
Jan Okniński
Keyword(s):  
Plactic Monoid

scholarly journals Dual filtered graphs

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Rebecca Patrias ◽  
Pavlo Pylyavskyy
Keyword(s):  
Hopf Algebra ◽  
Binary Tree ◽  
International Audience ◽  
Dual Graded Graphs ◽  
Young’S Lattice

International audience We define a $K$ -theoretic analogue of Fomin’s dual graded graphs, which we call dual filtered graphs. The key formula in the definition is $DU - UD = D + I$. Our major examples are $K$ -theoretic analogues of Young’s lattice, the binary tree, and the graph determined by the Poirier-Reutenauer Hopf algebra. Most of our examples arise via two constructions, which we call the Pieri construction and the Möbius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra, as described in Bergeron-Lam-Li, Nzeutchap, and Lam-Shimozono. The Möbius construction is more mysterious but also potentially more important, as it corresponds to natural insertion algorithms. Nous définissons un analogue $K$ -théorique aux graphes gradués en dualité de Fomin que nous appelons les graphes filtrés en dualité. La formule importante pour la définition est $DU - UD = D + I$. Nos principaux exemples sont un analogue $K$ -théorique aux graphe de Young, l’arbre binaire, et un graphe déterminé par l’algèbre de Hopf de Poirier-Reutenauer. La plupart de nos exemples surviennent de deux constructions que nous appelons la construction de Pieri et la construction de Möbius. La construction de Pieri est étroitement liée à la construction des graphes gradués en dualité d’une algèbre graduée de Hopf à la Bergeron-Lam-Li, Nzeutchap, et Lam-Shimozono. La construction de Möbius est plus mystérieuse, mais aussi peut-être plus importante car cette construction correspond aux algorithmes d’insertion naturelles.

scholarly journals Signature Verification Using Young's Lattice Grid Modeling

2016 ◽  
Vol 9 (6) ◽  
pp. 185-188
Author(s):  
A. Hadjipanteli ◽  
◽  
E. N. Zois ◽  
A. Nassiopoulos ◽  
◽  
...  
Keyword(s):  
Signature Verification ◽  
Young’S Lattice

scholarly journals An Order on Circular Permutations

10.37236/9982 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Antoine Abram ◽  
Nathan Chapelier-Laget ◽  
Christophe Reutenauer
Keyword(s):  
Symmetric Group ◽  
Weak Order ◽  
Rank Function ◽  
Weyl Groups ◽  
Eulerian Numbers ◽  
Affine Weyl Groups ◽  
Special Formula ◽  
Circular Permutations ◽  
Semidistributive Lattice ◽  
Young’S Lattice

Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

scholarly journals Symmetries of the k-bounded partition lattice

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Chris Berg ◽  
Mike Zabrocki
Keyword(s):  
Partition Lattice ◽  
International Audience ◽  
Young’S Lattice

International audience We generalize the symmetry on Young's lattice, found by Suter, to a symmetry on the $k$-bounded partition lattice of Lapointe, Lascoux and Morse. Nous généralisons la symétrie sur le treillis de Young, découvert par Suter, à une symétrie sur le treillis des partages bornés par $k$ et étudié par Lapointe, Lascoux and Morse.

scholarly journals New approaches to plactic monoid via Gröbner–Shirshov bases

2015 ◽  
Vol 423 ◽  
pp. 301-317 ◽  
Author(s):  
L.A. Bokut ◽  
Yuqun Chen ◽  
Weiping Chen ◽  
Jing Li
Keyword(s):  
New Approaches ◽  
Plactic Monoid
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