Related papers: Lower bound cluster algebras: presentations, Cohen…

We present a generic way to obtain level lower bounds for (promise) CSP hierarchies from degree lower bounds for algebraic proof systems. More specifically, we show that pseudo-reduction operators in the sense of Alekhnovich and Razborov…

We study the regularity and the projective dimension of the Stanley-Reisner ring of a $k$-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of $k$-decomposable…

The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use…

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…

In this paper we use refiltering methods to prove that certain types of multi-filtered algebras are Auslander-regular and Cohen-Macaulay. This is applied to obtain that the quantized enveloping algebra associated to a Cartan matrix is…

In the first purpose, we concentrate on the theory of quantum integrable systems underlying the Connes-Kreimer approach. We introduce a new family of Hamiltonian systems depended on the perturbative renormalization process in renormalizable…

Let $\Gamma$ be the fundamental group of the exterior of a knot in the three-sphere. We study deformations of representations of $\Gamma$ into $\mathrm{SL}_n(\mathbf{C})$ which are the sum of two irreducible representations. For such…

For a reductive group G, the products of projective rational varieties homogeneous under G that are spherical for G have been classified by Stembridge. We consider the B-orbit closures in these spherical varieties and prove that under some…

This is a survey on recent developments in Cohen-Macaulay representations via tilting and cluster tilting theory. We explain triangle equivalences between the singularity categories of Gorenstein rings and the derived (or cluster)…

In this work, we introduce a new class of algebras called skew-Brauer graph algebras, which generalize the well-known Brauer graph algebras. We establish that skew-Brauer graph algebras are symmetric and can be defined using a Brauer graph…

A simple method is proposed for deforming $A_\infty$-algebras by means of the resolution technique. The method is then applied to the associative algebras of polynomial functions on quantum superspaces. Specifically, by introducing suitable…

We prove the full Fock--Goncharov conjecture for $\mathcal{A}_{SL_2,\Sigma_{g,p}}$, the $\mathcal{A}$-cluster variety of the moduli of decorated twisted $SL_2$-local systems on triangulable surfaces $\Sigma_{g,p}$ with at least 2 punctures.…

We show that most arithmetic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique…

For any $d\ge 4$, by deformation theory of schemes, we present examples of (complete or excellent) $d$-dimensional mixed characteristic normal local domains admitting no small Cohen-Macaulay algebra, but admitting instances of small…

Given a pair of nilpotent orbits in a simple Lie algebra, one can associate a pair of vertex algebras called affine W-algebras. Under some compatibility conditions on these orbits, we prove that one of these W-algebras can be obtained as…

Let $K$ be a knot in $S^3$ and $X$ its complement. We study deformations of non-abelian, metabelian, reducible representations of the knot group $\pi\_1(X)$ into $\mathrm{SL}(n,\mathbf{C})$ which are associated to a simple root of the…

We show how, modulo the distinction between the antipode and the "twisted" or "renormalized" antipode, Connes and Kreimer's algebraic paradigm trivializes the proofs of equivalence of the (corrected) Dyson-Salam, Bogoliubov-Parasiuk-Hepp…

We present a fast version of the algorithm of Lascoux, Leclerc, and Thibon for the lower global crystal base for the Fock representation of quantum affine sl_n. We also show that the coefficients of the lower global crystal base coincide…

We present a short proof of Reisner's Theorem, characterizing which simplicial complexes have a Cohen-Macaulay face ring. In some cases, we can also express some homological invariants of the face ring in terms of the reduced homology of…

We give a geometric proof of inverse Hamiltonian reduction for all finite W-algebras in type $A$, a certain embedding of the finite W-algebra corresponding to an arbitrary nilpotent in $\mathfrak{gl}_N$ into that corresponding to a larger…