English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

The vertices of any (combinatorial) Kashiwara crystal graph carry a naturalmonoid structure given by identifying words labelling vertices that appear inthe same position of isomorphic components of the crystal. Working on a purelycombinatorial and monoid-theoretical level, we prove some foundational resultsfor these crystal monoids, including the observation that they have decidableword problem when their weight monoid is a finite rank free abelian group. Theproblem of constructing finite complete rewriting systems, and biautomaticstructures, for crystal monoids is then investigated. In the case of Kashiwaracrystals of types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$ (corresponding to the$q$-analogues of the Lie algebras of these types) these monoids are preciselythe generalised plactic monoids investigated in work of Lecouvey. We constructpresentations via finite complete rewriting systems for all of these typesusing a unified proof strategy that depends on Kashiwara's crystal bases andanalogies of Young tableaux, and on Lecouvey's presentations for these monoids.As corollaries, we deduce that plactic monoids of these types have finitederivation type and satisfy the homological finiteness properties left andright $\mathrm{FP}_\infty$. These rewriting systems are then applied to showthat plactic monoids of these types are biautomatic and thus have word problemsoluble in quadratic time.

journal of combinatorial theory series a

Crystal monoids \& crystal bases: rewriting systems and biautomatic  structures for plactic monoids of types $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$,  and $G_2$