The strong Lefschetz property of the coinvariant ring of the Coxeter group of type H4

We prove that the coinvariant ring of the irreducible Coxeter group of type H₄ has the strong Lefschetz property.     

Orbits for the adjoint coaction on quantum matrices

Conjugation coactions of the quantum general linear group on the algebra of quantum matrices have been introduced in an earlier paper and the coinvariants have been determined. In this paper the notion of orbit is considered via co-orbit maps associated with -points of the space of quantum matrices, mapping the coordinate ring of quantum matrices into the coordinate ring of the quantum general linear group. The co-orbit maps are calculated explicitly for 2×2 quantum matrices. For quantum matrices of arbitrary size, it is shown that when the deformation parameter is transcendental over the base field, then the kernel of the co-orbit map associated with a -point ξ is a right ideal generated by coinvariants, provided that the classical adjoint orbit of ξ is maximal. If ξ is diagonal with pairwise different eigenvalues, then the image of the co-orbit map coincides with the subalgebra of coinvariants with respect to the left coaction of the diagonal quantum subgroup of the quantum general linear group.     

Packed words and quotient rings

The coinvariant algebra is a quotient of the polynomial ring $\mathbb{Q}[x_1,\dots ,x_n]$ whose algebraic properties are governed by the combinatorics of permutations of length n. A word $w=w_1\dots w_n$ over the positive integers is packed if whenever $i>2$ appears as a letter of w, so does $i?1$. We introduce a quotient $S_n$ of $\mathbb{Q}[x_1,\dots ,x_n]$ which is governed by the combinatorics of packed words. We relate our quotient $S_n$ to the generalized coinvariant rings of Haglund, Rhoades, and Shimozono as well as the superspace coinvariant ring.