We prove that the permutation representation of the finite orthogonal group $\Omega^{\varepsilon}(n,q)$, where $\varepsilon=+$ or $-$, on the set of anisotropic lines is multiplicity-free, if q is a power of 2 and $n\ge 6$ is even. This result is established by giving a description of orbitals of this action. The rank of this action is $(q^2+2q)/2$ if $\varepsilon=+$ and $n=6$, and $(q^2+2q+2)/2$ otherwise. Moreover, we compute the subdegrees of the orbitals of $\Omega^{\varepsilon}(n,q)$.

This paper follows the program of study initiated by S. Fomin and A. Zelevinsky, and demonstrates that the homogeneous coordinate ring of the Grassmannian $\mathbb{G}(k, n)$ is a {\it cluster algebra of geometric type}. Those Grassmannians that are of {\it finite cluster type} are identified and their cluster variables are interpreted geometrically in terms of configurations of points in $\mathbb{C}\mathbb{P}^2$.

In this paper, we investigate the structure of Ariki–Koike algebras and their Specht modules using Gröbner–Shirshov basis theory and combinatorics of Young tableaux. For a multipartition $\lambda$, we find a presentation of the Specht module $S^{\lambda}$ given by generators and relations, and determine its Gröbner–Shirshov pair. As a consequence, we obtain a linear basis of $S^{\lambda}$ consisting of standard monomials with respect to the Gröbner–Shirshov pair. We show that this monomial basis can be canonically identified with the set of cozy tableaux of shape $\lambda$.