We study canonical basis elements in higher-level Fock spaces associated with the quantum group U_{q}(\mathfrak{gl}_{\infty}) , which are conjecturally related to Calogero–Moser theory for complex reflection groups. We generalize the Leclerc–Miyachi formula to arbitrary levels by introducing new explicit constructions based on symbols, including a column removal theorem and closed formulas in several cases. These results provide explicit descriptions of canonical basis elements with applications to Calogero–Moser cellular characters and to the decomposition matrices of Ariki–Koike algebras.
A nontrivial pseudovariety of semigroups is join irreducible if its inclusion in the join of a collection of pseudovarieties implies its inclusion in at least one member of that collection. The dimension of a finitely generated pseudovariety is defined as the order of its minimal generating semigroup. There are precisely 30 join irreducible pseudovarieties of dimension 5 or less. While those of dimension 6 have not been fully identified, 16 such examples are known so far.In this article, several pseudovarieties are examined to determine their join irreducible subpseudovarieties, achieving complete identification in some cases. By applying these results alongside existing sufficient conditions, join irreducible pseudovarieties of dimension 6 are completely identified. Notably, no additional examples exist beyond the 16 previously known. This is a surprising result, considering that there are 28{,}634 semigroups of order 6 up to isomorphism.
Let $\mathcal{A}$ be a cluster algebra of finite cluster type. We study the Gröbner cone $\mathcal{C}_{\mathcal{A}}$ parametrizing term orders inducing an initial degeneration of the ideal $I_{\mathcal{A}}$ of relations among the cluster variables of $\mathcal{A}$ to the ideal generated by products of incompatible cluster variables. We show that for any cluster variable $v$, the weight induced by taking compatibility degrees with $v$ belongs to $\mathcal{C}_{\mathcal{A}}$. This allows us to construct an explicit circular term order and prove a conjecture of Ilten, Nájera Chávez, and Treffinger. Furthermore, we give explicit descriptions of the rays and lineality spaces of $\mathcal{C}_{\mathcal{A}}$ in terms of combinatorial models for cluster algebras of types $A_n$, $B_n$, $C_n$, $D_n$ with a special choice of frozen variables, and in the case of no frozen variables.
The Ehrhart polynomial and Ehrhart series count lattice points in integer dilations of a lattice polytope. We introduce and study a q-deformation of the Ehrhart series, based on the notions of harmonic spaces and Macaulay's inverse systems for coordinate rings of finite point configurations. We conjecture that this q-Ehrhart series is a rational function, and introduce and study a bigraded algebra whose Hilbert series matches the q-Ehrhart series. Defining this algebra requires a new result on Macaulay inverse systems for Minkowski sums of point configurations.
We provide sufficient conditions for a free amalgamated product of torsionfree nilpotent groups to be residually nilpotent. We also characterise the residual nilpotence of certain higher-dimensional amalgams of unipotent groups over the rationals (known as KMS groups) in terms of their defining Cartan matrix. As an application, we give a normal form for the elements of a minimal Kac-Moody group over the rationals.
