Related papers: Regular cell complexes in total positivity
We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms…
The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) $J$-total positivity on the full flag variety of an arbitrary…
We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite…
We show that, in a highest weight category with duality, the endomorphism algebra of a tilting object is naturally a cellular algebra. Our proof generalizes a recent construction of Andersen, Stroppel, and Tubbenhauer. This result raises…
We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common…
These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians $Gr_2(\mathbb{C})$, and the appearance of…
Double Bruhat cells in a connected complex semisimple Lie group $G$ emerged as a crucial concept in the work of S. Fomin and A. Zelevinsky on total positivity and cluster algebras. These cells are special instances of a broader class of…
This thesis proposes a framework based on a notion of combinatorial cell complex (cc) whose cells are defined simply as finite sets of vertices. The cells of a cc are subject to four axioms involving a rank function that assigns a rank (or…
In 2013, Lee, Li, and Zelevinsky introduced combinatorial objects called compatible pairs to construct the greedy bases for rank-2 cluster algebras, consisting of indecomposable positive elements including the cluster monomials.…
This note introduces the superunitary region of a cluster algebra, the subspace of the totally positive region on which each cluster variable is at least 1. Our main result is that the superunitary region of a finite type cluster algebra is…
We show that the commutator subgroup of the group of unitaries connected to the identity in a simple unital C*-algebra is simple modulo its center. We then go on to investigate the role of regularity properties in the structure of the…
We consider Kazhdan-Lusztig cells of the symmetric group $S_n$ containing the longest element of a standard parabolic subgroup of $S_n$. Extending some of the ideas in [Beitr{\"a}ge zur Algebra und Geometrie, 59 (2018), no. 3, 523-547] and…
One can iteratively obtain a free resolution of any monomial ideal $I$ by considering the mapping cone of the map of complexes associated to adding one generator at a time. Herzog and Takayama have shown that this procedure yields a minimal…
In the seminal paper of Borel and Tits about reductive groups, they show some fundamental results about Bruhat cells with respect to a minimal parabolic subgroup, e.g., relative Bruhat decomposition and its geometrization, relative Bruhat…
An interval in a combinatorial structure S is a set I of points which relate to every point from S I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a…
We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…
In this paper, we establish the rationality conjecture raised in \cite{FKS} for any $(r-1)$-connected ($r\geq 2$) $kr$-dimensional CW-complex $X$ ($k\geq 2$) having a unique spherical cohomology class $u\in \tilde{H}^r(X, \mathbb{Z})$ such…
We construct a family of integrable Hamiltonian systems generalizing the relativistic periodic Toda lattice, which is recovered as a special case. The phase spaces of these systems are double Bruhat cells corresponding to pairs of Coxeter…
Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely one where finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence.…
The classical Theorem of Mumford states that a topologically regular complex algebraic surface in $\mathbb{C}^3$ with an isolated singular point is smooth. We proof that any Lipschitz regular complex algebraic set is smooth. No restriction…