33 research outputs found
T-Path Formula and Atomic Bases for Cluster Algebras of Type D
We extend a T-path expansion formula for arcs on an unpunctured surface to
the case of arcs on a once-punctured polygon and use this formula to give a
combinatorial proof that cluster monomials form the atomic basis of a cluster
algebra of type D
Kirillov-Reshetikhin crystals for using Nakajima monomials
We give a realization of the Kirillov--Reshetikhin crystal using
Nakajima monomials for using the crystal structure
given by Kashiwara. We describe the tensor product in terms of a shift of indices, allowing us to recover the Kyoto
path model. Additionally, we give a model for the KR crystals using
Nakajima monomials.Comment: 24 pages, 6 figures; v2 improved introduction, added more figures,
and other misc improvements; v3 changes from referee report
Cambrian combinatorics on quiver representations (type A)
This paper presents a geometric model of the Auslander-Reiten quiver of a
type A quiver together with a stability function for which all indecomposable
modules are stable. We also introduce a new Catalan object which we call a
maximal almost rigid representation. We show that its endomorphism algebra is a
tilted algebra of type A. We define a partial order on the set of maximal
almost rigid representations and use our new geometric model to show that this
partial order is a Cambrian lattice.Comment: 27 pages, 15 figures. Comments are welcome. Revised July 2020: New
section (Section 5) on stability function, references added, typos correcte
Atomic Bases and -path Formula for Cluster Algebras of Type
International audienceWe extend a -path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type .Nous généralisons une formule de développement en -chemins pour les arcs sur une surface non-perforée aux arcs sur un polygone à une perforation. Nous utilisons cette formule pour donner une preuve combinatoire du fait que les monômes amassées constituent la base atomique d’une algèbre amassée de type
Atomic Bases and -path Formula for Cluster Algebras of Type
We extend a -path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type
Box-ball systems and RSK recording tableaux
A box-ball system (BBS) is a discrete dynamical system consisting of n balls
in an infinite strip of boxes. During each BBS move, the balls take turns
jumping to the first empty box, beginning with the smallest-numbered ball. The
one-line notation of a permutation can be used to define a BBS state. This
paper proves that the Robinson-Schensted (RS) recording tableau of a
permutation completely determines the dynamics of the box-ball system
containing the permutation.
Every box-ball system eventually reaches steady state, decomposing into
solitons. We prove that the rightmost soliton is equal to the first row of the
RS insertion tableau and it is formed after at most one BBS move. This fact
helps us compute the number of BBS moves required to form the rest of the
solitons. First, we prove that if a permutation has an L-shaped soliton
decomposition then it reaches steady state after at most one BBS move.
Permutations with L-shaped soliton decompositions include noncrossing
involutions and column reading words. Second, we make partial progress on the
conjecture that every permutation on n objects reaches steady state after at
most n-3 BBS moves. Furthermore, we study the permutations whose soliton
decompositions are standard; we conjecture that they are closed under
consecutive pattern containment and that the RS recording tableaux belonging to
such permutations are counted by the Motzkin numbers.Comment: 24 pages, 2 figure
RSK tableaux and the weak order on fully commutative permutations
For each fully commutative permutation, we construct a "boolean core," which
is the maximal boolean permutation in its principal order ideal under the right
weak order. We partition the set of fully commutative permutations into the
recently defined crowded and uncrowded elements, distinguished by whether or
not their RSK insertion tableaux satisfy a sparsity condition. We show that a
fully commutative element is uncrowded exactly when it shares the RSK insertion
tableau with its boolean core. We present the dynamics of the right weak order
on fully commutative permutations, with particular interest in when they change
from uncrowded to crowded. In particular, we use consecutive permutation
patterns and descents to characterize the minimal crowded elements under the
right weak order.Comment: 20 pages, 2 figure