Before revision:  Assuming the grid cells form a vector representation of self-position, we elucidate a minimally simple recurrent model for grid cells' path integration based on two coupled matrix Lie algebras that underlie two coupled rotation systems that mirror the agent's self-motion : (1) When the agent moves along a certain direction, the vectoris rotated by a generator matrix. (2) When the agent changes direction, the generator matrix is rotated by another generator matrix. Our experiments show that our model learns hexagonal grid response patterns that resemble the firing patterns observed from the grid cells in the  brain.
After revision:  Assuming the grid cells form a vector representation of self-position, we elucidate a minimally simple recurrent model for grid cells' path integration based on two coupled matrix Lie algebras that underlie two coupled rotation systems that mirror the agent's self-motion is represented by the rotation of the vector. We connect the isotropic condition under the linear model to the hexagon grid patterns of the response maps of grid cells. Our numerical experiments demonstrate that our model learns hexagonal grid response patterns that resemble the firing patterns observed from the grid cells in the brain.