Related papers: Coxeter Transformations, the McKay correspondence,…

In this Colloqium Lecture (by one of the authors (D.S)) a thorough presentation of the authors' research on the subjects, stated in the title, is given. By quite laborious mathematics it is explained how one can handle systems in which each…

The standard representation of c*-algebra is used to describe fields in compactified space-time dimensions characterized by topologies of the type $ \Gamma_{D}^{d}=(\mathbb{S}^{1})^{d}\times \mathbb{M}^{D-d}$. The modular operator is…

A transformation is obtained which completes the unification of quadrirational Yang-Baxter maps and known integrable multi-quadratic quad equations. By combining theory from these two classes of quad-graph models we find an extension of the…

Let $C(T)$ be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either $B_n$ or $D_n$. Let $C_Y(T)$ be a natural quotient of $C(T)$, and if $C(T)$ is simply-laced (which means all the relations…

A celebrated theorem of Pimsner states that a covariant representation $T$ of a $C^*$-correspondence $E$ extends to a $C^*$-representation of the Toeplitz algebra of $E$ if and only if $T$ is isometric. This paper is mainly concerned with…

This paper generalizes in the affine symmetric group the notion of Coxeter sortable (or c-sortable for short) elements, as well as the classical bijection between c-sortable elements and c-noncrossing partitions defined by Reading in finite…

A diagram obtained from the Carter diagram $\Gamma$ by adding one root together with its bonds such that the resulting subset of roots is linearly independent is said to be the {\it linkage diagram}. Given a linkage diagram, we associate…

If Y is a diagram of spectra indexed by an arbitrary poset C together with a specified sub-poset D, we define the total cofibre \Gamma (Y) of Y as the strict cofibre of the map from hocolim_D (Y) to hocolim_C (Y). We construct a comparison…

We extend the short presentation due to [Borel '53] of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its Schubert varieties. Our result is stated in a root-system uniform…

We introduce a family of cluster algebras of infinite rank associated with root systems of type $A$, $D$, $E$. We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories…

We study associative multiplications in semi-simple associative algebras over C compatible with the usual one. An interesting class of such multiplications is related to the affine Dynkin diagrams of A, D, E-type. In this paper we…

In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, we show that the $\Gamma$-equivariant Hilbert scheme $X^{\Gamma[n]}$ and the Hilbert scheme $X_\Gamma^{[n]}$ (where $X=\C^2$,…

We derive a formula for the Coxeter polynomial of the s-fold tensor product F[A_{n_1-1}] x ... x F[A_{n_s-1}] of path algebras of linearly oriented quivers of Dynkin type A_{n_i-1}, in terms of the weights n_1, ..., n_s > 1, and show that…

A Coxeter system is an ordered pair (W,S) where S is the generating set in a particular type of presentation for the Coxeter group W. A subgroup of W is called special if it is generated by a subset of S. Amalgamated product decompositions…

Motivated by work of Coxeter (1957), we study a class of algebras associated to Coxeter groups, which we term 'generalized nil-Coxeter algebras'. We construct the first finite-dimensional examples other than usual nil-Coxeter algebras;…

We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v,w$ -- biaxial double $(\beta,q)$-Grothendieck polynomials -- which specialize at $q=0$ and $v=1$ to double $\beta$-Grothendieck polynomials…

Recently, L.Rozansky and E.Witten (hep-th/9612216) associated to any hyperKaehler manifold X a system of "weights" (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a very…

Alternating-sign Hopf plumbing along a tree yields fibered alternating links whose homological monodromy is, up to a sign, conjugate to some alternating-sign Coxeter transformation. Exploiting this tie, we obtain results about the location…

Let $G$ be a regular graph of degree $d$ and let $A\subset V(G)$. Say that $A$ is $\eta$-closed if the average degree of the subgraph induced by $A$ is at least $\eta d$. This says that if we choose a random vertex $x\in A$ and a random…

Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This…