Related papers: An expansion formula for type A and Kronecker quan…
We compare the integral category O of shifted affine quantum groups of symmetric and non symmetric types. To do so we compute the K-theoretic analog of the Coulomb branches with symmetrizers introduced by Nakajima and Weekes. This yields an…
We introduce and study the twisted adapted $r$-cluster point and its combinatorial Auslander-Reiten quivers, called twisted AR-quivers and folded AR-quivers, of type $A_{2n+1}$ which are closely related to twisted Coxeter elements and the…
We prove refined generating series formulae for characters of (virtual) cohomology representations of external products of suitable coefficients, e.g., (complexes of) constructible or coherent sheaves, or (complexes of) mixed Hodge modules…
We prove a general result that relates certain pushouts of $E_k$-algebras to relative tensors over $E_{k+1}$-algebras. Specializations include a number of established results on classifying spaces, resolutions of modules, and (co)homology…
We consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the…
We use the Hecke algebras of affine symmetric groups and their associated Schur algebras to construct a new algebra through a basis, and a set of generators and explicit multiplication formulas of basis elements by generators. We prove that…
In this paper we study general quantum affinizations $\U_q(\hat{\Glie})$ of symmetrizable quantum Kac-Moody algebras and we develop their representation theory. We prove a triangular decomposition and we give a classication of (type 1)…
In this paper, we study the quantum virtual Grothendieck ring, denoted by $\frakK_q(\g)$, which was introduced in [39], and further investigated in [26, 25]. Our approach involves examining this ring from two perspectives: first, by…
We generalise the expansion formulae of Musiker, Schiffler and Williams, obtained for cluster algebras from orientable surfaces, to a larger class of coefficients which we call principal laminations. In doing so, for any quasi-cluster…
We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials,…
Let $Q$ be a finite acyclic valued quiver. We give the cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$. These formulas can…
We extend the the definition of Kumjian-Pask algebras to include algebras associated to finitely aligned higher-rank graphs. We show that these Kumjian-Pask algebras are universally defined and have a graded uniqueness theorem. We also…
Motivated by the definition of super-Teichm\"uller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super-Teichm\"uller space, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain…
The complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits. We study two families of polynomials introduced by Wyser and Yong representing the $K$-theory classes of the closures of these…
In a k-linear triangulated category (where k is a field) we show that the existence of Auslander-Reiten triangles implies that objects are determined, up to shift, by knowing dimensions of homomorphisms between them. In most cases the…
We provide explicit formulae for highest-weight to highest-weight correlation functions of perfect vertex operators of $U_q(\hat{\mathfrak{sl}(2)})$ at arbitrary integer level $\ell$. They are given in terms of certain Macdonald…
We present new proofs and generalizations of unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker…
We express cluster variables of type $B_n$ and $C_n$ in terms of cluster variables of type $A_n$. Then we associate a cluster tilted bound symmetric quiver $Q$ of type $A_{2n-1}$ to any seed of a cluster algebra of type $B_n$ and $C_n$.…
We quantize the regularity properties of classical graphs that determine spin models for singly-generated Yang-Baxter planar algebras, including the Kauffman polynomial, and construct explicit examples. A source of examples comes from…
The reflection equations (RE) are a consistent extension of the Yang-Baxter equations (YBE) with an addition of one element, the so-called reflection matrix or $K$-matrix. For example, they describe the conditions for factorizable…