Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tropical representations and identities of plactic monoids   We exhibit a faithful representation of the plactic monoid of every finite
rank as a monoid of upper triangular matrices over the tropical semiring. This
answers a question first posed by Izhakian and subsequently studied by several
authors. A consequence is a proof of a conjecture of Kubat and Okni\'{n}ski
that every plactic monoid of finite rank satisfies a non-trivial semigroup
identity. In the converse direction, we show that every identity satisfied by
the plactic monoid of rank $n$ is satisfied by the monoid of $n \times n$ upper
triangular tropical matrices. In particular this implies that the variety
generated by the $3 \times 3$ upper triangular tropical matrices coincides with
that generated by the plactic monoid of rank $3$, answering another question of
Izhakian.

22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga