Related papers: Hernandez-Leclerc modules and snake graphs
We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that are not the 'usual' nil-Coxeter algebras:…
In this survey, we review some of the recent connections between the representation theory of (untwisted) quantum affine algebras and the representation theory of current algebras. We mainly focus on the finite-dimensional representations…
We consider skew-symmetrizable (upper) cluster algebras with a compatible Poisson structure, called $\mathsf{\Lambda}$-(upper) cluster algebras. For any two good elements (e.g., cluster monomials) in a $\mathsf{\Lambda}$-upper cluster…
We address two fundamental questions in the representation theory of affine Hecke algebras of classical types. One is an inductive algorithm to compute characters of tempered modules, and the other is the determination of the constants in…
In this paper, we apply the theory of cluster algebras to study minimal affinizations for the quantum affine algebra of type $F_4$. We show that the $q$-characters of a large family of minimal affinizations of type $F_4$ satisfy a system of…
Let $F$ be a local field of mixed characteristic, let $k$ be a finite extension of its residue field, let ${\mathcal H}$ be the pro-$p$-Iwahori Hecke $k$-algebra attached to ${\rm GL}_{d+1}(F)$ for some $d\ge1$. We construct an exact and…
Snake graphs are a class of planar graphs that are important in the theory of cluster algebras. Indeed, the Laurent expansions of the cluster variables in cluster algebras from surfaces are given as weight generating functions for 1-dimer…
In this paper, we introduce higher rank generalizations of Macdonald polynomials. The higher rank non-symmetric Macdonald polynomials are Laurent polynomials in several sets of variables which form weight bases for higher rank polynomial…
In this paper we revisit Ribenboim's notion of higher derivations of modules and relate it to the recent work of De Fernex and Docampo on the sheaf of differentials of the arc space. In particular, we derive their formula for the K\"ahler…
The endomorphism algebras of the permutation modules for transitive permutation groups, known as Hecke algebras, are fundamental objects in representation theory. While group algebras are known to be symmetric over any field, it is natural…
We consider the quantum affine vertex algebra $\mathcal{V}_{c}(\mathfrak{gl}_N)$ associated with the rational $R$-matrix, as defined by Etingof and Kazhdan. We introduce certain subalgebras $\textrm{A}_c (\mathfrak{gl}_N)$ of the completed…
We develop a functorial approach to the study of $n$-abelian categories by reformulating their axioms in terms of their categories of finitely presented functors. Such an approach allows the use of classical homological algebra and…
We develop an explicit theory of formal modular forms over arbitrary number fields $K$, as functions of modular points. We define modular points for $\Gamma_0({\mathfrak n})$ and $\Gamma_1({\mathfrak n})$, where the level ${\mathfrak n}$ is…
In this note we are interested in labelling the irreducible representations of non-semisimple specialisations of Hecke algebras of complex reflection groups. We will use category O for the rational Cherednik algebra and the KZ functor…
We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on…
A refined notion of curvature for a linear system of Hermitian vector spaces, in the sense of Grothendieck, leads to the unitary classification of a large class of analytic Hilbert modules. Specifically, we study Hilbert sub-modules, for…
Let $W$ be a finite Weyl group of classical type which may not be irreducible, $F$ an algebraically closed field, $q$ an invertible element of $F$. We denote by $\mathcal H_W(q)$ the associated Hecke algebra. If $q=1$ then it is $FW$ and we…
We compare the context of Hodge structures with that of vertex algebras of conformal field theory. Vertex algebras appear as the highest weight representations of infinite dimensional Lie algebras. A correspondence between Higgs bundles and…
We define and investigate a new three-parameter family of graphs that further generalizes the Fibonacci and metallic cubes. Namely, the number of vertices in this family of graphs satisfies Horadam recurrence, a linear recurrence of second…
The nilpotence variety for extended supersymmetric quantum mechanics is a cone over a quadric in projective space. The pure spinor correspondence, which relates the description of off-shell supermultiplets to the classification of modules…