Crystal Structure on the Category of Modules over Colored Planar Rook Algebra

TL;DR

This paper demonstrates that the category of modules over the colored planar rook algebra admits a crystal structure that categorifies the crystal bases of polynomial representations of the quantum group U_q(gl_{n+1}).

Contribution

It introduces a new categorical framework linking colored planar rook algebra modules with crystal bases of quantum group representations.

Findings

The module category is completely reducible.

Suitable functors induce a crystal structure.

Categorification of crystal bases for polynomial representations.

Abstract

Colored planar rook algebra is a semigroup algebra in which the basis element has a diagrammatic description. The category of finite dimensional modules over this algebra is completely reducible and suitable functors are defined on this category so that it admits a crystal structure in the sense of Kashiwara. We show that the category and functors categorify the crystal bases for the polynomial representations of quantized enveloping algebra $U_q(gl_{n+1})$.