The bigraded Hilbert function and the minimal free resolutions for the diagonal coinvariants of the dihedral groups are exhibited, as well as for all their bigraded invariant Gorenstein quotients.

The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind:\ first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in $\mathbb{Z}[\tau]$ (with $\tau$ the golden ratio), to construct new regular $4$-polytopes of hyperbolic types $\{3,5,3\}$ and $\{5,3,5\}$ with automorphism groups given by finite orthogonal groups.

If $A$ is a subset of the set of reflections of a finite Coxeter group $W$, we define a sub-$\ZM$-module $\DC_A(W)$ of the group algebra $\ZM W$. We discuss cases where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra, the group algebra and, if $W$ is of type $B$, the Mantaci--Reutenauer algebra.

We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also count the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.

No abstract.

The growth series of compact hyperbolic Coxeter groups with 4 and 5 generators are explicitly calculated. The assertions of J.~Cannon and Ph.~Wagreich for the 4-generated groups, that the poles of the growth series lie on the unit circle, with the exception of a single real reciprocal pair of poles, are verified. We also verify that for the 5-generated groups, this phenomenon fails.