Geodesic
Commutation classes of the reduced words for the longest element of \(\mathfrak{S}_{n}\)
Gonçalo Gutierres1 ; Ricardo Mamede1 ; José Luis Santos
1University of Coimbra
The electronic journal of combinatorics, Tome 27 (2020) no. 2
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Using the standard Coxeter presentation for the symmetric group $\mathfrak{S}_{n}$, two reduced expressions for the same group element $\textsf{w}$ are said to be commutationally equivalent if one expression can be obtained from the other one by applying a finite sequence of commutations. The commutation classes can be seen as the vertices of a graph $\widehat{G}(\textsf{w})$, where two classes are connected by an edge if elements of those classes differ by a long braid relation. We compute the radius and diameter of the graph $\widehat{G}(\textsf{w}_{\bf 0})$, for the longest element $\textsf{w}_{\bf 0}$ in the symmetric group $\mathfrak{S}_{n}$, and show that it is not a planar graph for $n\geq 6$. We also describe a family of commutation classes which contains all atoms, that is classes with one single element, and a subfamily of commutation classes whose elements are in bijection with standard Young tableaux of certain moon-polyomino shapes.
DOI : 10.37236/9481
Classification : 05A05, 05A19, 05C12, 05E10, 20C30
Mots-clés : Young tableaux, moon-polyomino shapes

Gonçalo Gutierres  1   ; Ricardo Mamede  1   ; José Luis Santos 

1 University of Coimbra 
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     title = {Commutation classes of the reduced words for the longest element of {\(\mathfrak{S}_{n}\)}},
     journal = {The electronic journal of combinatorics},
     year = {2020},
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Gonçalo Gutierres; Ricardo Mamede; José Luis Santos. Commutation classes of the reduced words for the longest element of \(\mathfrak{S}_{n}\). The electronic journal of combinatorics, Tome 27 (2020) no. 2. doi: 10.37236/9481

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