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Séminaire Lotharingien de Combinatoire, 80B.92 (2018), 12 pp.

# Anastasia Chavez and John Guo

# Dual Equivalence Graphs and CAT(0) Combinatorics

**Abstract.**
In this paper we explore the combinatorial structure of dual
equivalence graphs *G*_{λ}. The vertices are Standard Young
tableaux of a fixed shape λ that allows us to further
understand the combinatorial structure of *G*_{λ}, and the edges
are given by dual Knuth equivalences. The graph *G*_{λ} is the
1-skeleton of a cubical complex *C*_{λ}, and one can ask
whether the cubical complex is CAT(0); this is a desirable metric
property that allows us to describe the combinatorial structure of
*G*_{λ} very explicitly. We prove that *C*_{λ} is CAT(0)
if and only if λ is a hook.

Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.

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