Algebras Of Finite Type Research Articles

Sortable elements in infinite Coxeter groups

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group W. We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection \pidown^c mapping each element of W to the unique maximal c-sortable element below it in the weak order. The fibers of \pidown^c essentially define the c-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of \pidown^c. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.

Some applications of higher commutators in Mal’cev algebras

We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras, we can check in polynomial time whether two given polynomial terms induce the same function.

Homology of graded Hecke algebras

Let H be a graded Hecke algebra with complex deformation parameters and Weyl group W . We show that the Hochschild, cyclic and periodic cyclic homologies of H are all independent of the parameters, and compute them explicitly. We use this to prove that, if the deformation parameters are real, the collection of irreducible tempered H -modules with real central character forms a Q -basis of the representation ring of W . Our method involves a new interpretation of the periodic cyclic homology of finite type algebras, in terms of the cohomology of a sheaf over the underlying complex affine variety.

Periodic cyclic homology of reductive $p$-adic groups

Let G be a reductive p -adic group, \mathcal{H}(G) its Hecke algebra and \mathcal{S}(G) its Schwartz algebra. We will show that these algebras have the same periodic cyclic homology. This provides an alternative proof of the Baum–Connes conjecture for G , modulo torsion. As preparation for our main theorem we prove two results that have independent interest. Firstly, a general comparison theorem for the periodic cyclic homology of finite type algebras and certain Fréchet completions thereof. Secondly, a refined form of the Langlands classification for G , which clarifies the relation between the smooth spectrum and the tempered spectrum.

Periodic resolutions and self-injective algebras of finite type

We say that an algebra A is periodic if it has a periodic projective resolution as an ( A , A ) -bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B → A , B is periodic if and only if A is. In addition, when A has finite representation type, we build upon results of Buchweitz to show that periodicity passes between A and its stable Auslander algebra. Finally, we use Asashiba’s classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi–Yau dimensions.

On Bass' question for finitely generated algebras over large fields

Recently Cortinas-Haesemayer-Walker-Weibel gave affirmative answer to Bass' 1972 question on NK-groups for algebras of essentially finite type over large fields of characteristic 0. Here we give an alternative short proof of this result for algebras of finite type over such fields. Our approach is based on classical techniques in higher K-theory of rings and a direct K_i-analogue of an old observation of Murthy-Pedrini, dating back from the same 1972.

James' conjecture for Hecke algebras of exceptional type, I

In this paper, and a second part to follow, we complete the programme (initiated more than 15 years ago) of determining the decomposition numbers and verifying James' conjecture for Iwahori–Hecke algebras of exceptional type. The new ingredients which allow us to achieve this aim are: • the fact, recently proved by the first author, that all Hecke algebras of finite type are cellular in the sense of Graham–Lehrer, and • the explicit determination of W-graphs for the irreducible (generic) representations of Hecke algebras of type E 7 and E 8 by Howlett and Yin. Thus, we can reduce the problem of computing decomposition numbers to a manageable size where standard techniques, e.g., Parker's MeatAxe and its variations, can be applied. In this part, we describe the theoretical foundations for this procedure.

Cluster Algebras of Finite Type via Coxeter Elements and Principal Minors

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 the simply connected semisimple algebraic group of the same Cartan–Killing type. In this realization, the cluster variables appear as certain (generalized) principal minors.

Rigid Dualizing Complexes Over Commutative Rings

In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper (Yekutieli and Zhang, Trans AMS 360:3211-3248, 2008). In the present paper we obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic rings. In the sequel paper (Yekutieli, Rigid dualizing complexes on schemes, in preparation) these results will be used to construct and study rigid dualizing complexes on schemes.

A Cluster Expansion Formula ($A_n$ case)

We consider the Ptolemy cluster algebras, which are cluster algebras of finite type $A$ (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed $\Sigma$ in a Ptolemy cluster algebra, we present a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed $\Sigma$. Our formula is given in a combinatorial way, using paths on a triangulation of the polygon that corresponds to the seed $\Sigma$.

Perfect Matchings and Cluster Algebras of Classical Type

In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their perfect matchings encodes the numerator of the associated Laurent polynomial while decompositions of the graphs correspond to the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the $A_n$ case while providing a novel interpretation for the $B_n$, $C_n$, and $D_n$ cases. Dans cet article nous donnons une interprétation combinatoire en termes de théorie des graphes pour les variables de clusters qui apparaissent dans la plupart des algèbres à clusters de type fini. En particulier, nous décrivons une famille de graphes tels qu'une énumération pondérée de leurs matchings parfaits encode le numérateur du polynôme de Laurent associé, tandis que les décompositions du graphe correspondent au dénominateur. Ceci complète les récents travaux de Schiffler et Carroll-Price qui donnent une formule pour le développement d'une variable de cluster dans le cas $A_n$, tout en fournissant une nouvelle interprétation dans les cas $B_n$, $C_n$ et $D_n$.

Hecke algebras of finite type are cellular

Let $\cH$ be the one-parameter Hecke algebra associated to a finite Weyl group $W$, defined over a ground ring in which ``bad'' primes for $W$ are invertible. Using deep properties of the Kazhdan--Lusztig basis of $\cH$ and Lusztig's $\ba$-function, we show that $\cH$ has a natural cellular structure in the sense of Graham and Lehrer. Thus, we obtain a general theory of ``Specht modules'' for Hecke algebras of finite type. Previously, a general cellular structure was only known to exist in types $A_n$ and $B_n$.

On cyclically orientable graphs

Graph G is called cyclically orientable (CO) if it admits an orientation in which every simple chordless cycle is cyclically oriented. This family of graphs was introduced by Barot et al. [Cluster algebras of finite type and positive symmetrizable matrices, J. London Math. Soc. (3) 73 (2006) 545–564]. The authors obtained several nice characterizations of CO-graphs, being motivated primarily by their applications in cluster algebras. Here we obtain several new characterizations that provide algorithms for recognizing CO-graphs and obtaining their cyclic orientations in linear time. We show that the edge maximal CO-graphs are 2-trees; that is, G = ( V , E ) is a 2-tree if and only if G is CO and G ′ = ( V , E ′ ) is not CO whenever E is a proper subset of E ′ .

The Birman–Schwinger principle in von Neumann algebras of finite type

We introduce a relative index for a pair of dissipative operators in a von Neumann algebra of finite type and prove an analog of the Birman–Schwinger principle in this setting. As an application of this result, revisiting the Birman–Krein formula in the abstract scattering theory, we represent the de la Harpe–Skandalis determinant of the characteristic function of dissipative operators in the algebra in terms of the relative index.

Recognizing Cluster Algebras of Finite Type

We compute the list of all minimal 2-infinite diagrams, which are cluster algebraic analogues of extended Dynkin graphs.

Cluster algebras IV: Coefficients

We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of ‘principal’ coefficients. We show that the exchange graph of a cluster algebra with principal coefficients covers the exchange graph of any cluster algebra with the same exchange matrix. We investigate two families of parameterizations of cluster monomials by lattice points, determined, respectively, by the denominators of their Laurent expansions and by certain multi-gradings in cluster algebras with principal coefficients. The properties of these parameterizations, some proven and some conjectural, suggest links to duality conjectures of Fock and Goncharov. The coefficient dynamics leads to a natural generalization of Zamolodchikov's -systems, previously known in finite type only, and sharpen the periodicity result from an earlier paper. For cluster algebras of finite type, we identify a canonical ‘universal’ choice of coefficients such that an arbitrary cluster algebra can be obtained from the universal one (of the same type) by an appropriate specialization of coefficients.

Invariant hypersurfaces for derivations in positive characteristic

Let A be an integral k -algebra of finite type over an algebraically closed field k of characteristic p > 0 . Given a collection D of k -derivations on A , that we interpret as algebraic vector fields on X = Spec ( A ) , we study the group spanned by the hypersurfaces V ( f ) of X invariant under D modulo the rational first integrals of D . We prove that this group is always a finite dimensional F p -vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant under a foliation of codimension 1. As a application, given a k -algebra B between A p and A , we show that the kernel of the pull-back morphism Pic ( B ) → Pic ( A ) is a finite F p -vector space. In particular, if A is a UFD, then the Picard group of B is finite.

Cluster-tilted algebras of finite representation type

We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) cluster-tilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of cluster-tilted algebras of finite representation type are obtained along the way.

Tight closure test exponents for certain parameter ideals

This paper is concerned with the tight closure of an ideal $I$ in a commutative Noetherian ring $R$ of prime characteristic $p$. The formal definition requires, on the face of things, an infinite number of checks to determine whether or not an element of $R$ belongs to the tight closure of $I$. The situation in this respect is much improved by Hochster's and Huneke's test elements for tight closure, which exist when $R$ is a reduced algebra of finite type over an excellent local ring of characteristic $p$. More recently, Hochster and Huneke have introduced the concept of test exponent for tight closure: existence of these test exponents would mean that one would have to perform just one single check to determine whether or not an element of $R$ belongs to the tight closure of $I$. However, to quote Hochster and Huneke, 'it is not at all clear whether to expect test exponents to exist; roughly speaking, test exponents exist if and only if tight closure commutes with localization'. The main purpose of this paper is to provide a short direct proof that test exponents exist for parameter ideals in a reduced excellent equidimensional local ring of characteristic $p$.

Deep Matrix Algebras of Finite Type

Deep matrix algebras \(\varepsilon {\left( {X,A} \right)}\) based on a set \(X\) over a ring \(A\) are introduced and studied by McCrimmon when \(X\) has infinite cardinality. Here, we construct a new \(\varepsilon {\left( {X,A} \right)}\)-module that is faithful for \(X\) of any cardinality. For \(A=\mathbb{K}\) a field of arbitrary characteristic and \(2\le |X\,|<\infty\), \(\varepsilon {\left( {X,\mathbb{K}} \right)}\) is studied in depth. The algebra \(\varepsilon {\left( {X,\mathbb{K}} \right)}\) is shown to possess a unique proper non-zero ideal, isomorphic to \(Mat_{\infty}(\mathbb{K})\). This leads to a new and interesting simple algebra, \({\user1{\mathcal{A}}}{\left( {X,\,\mathbb{K}} \right)}\), the quotient of \(\varepsilon {\left( {X,\mathbb{K}} \right)}\) by its unique ideal. Both \(\varepsilon {\left( {X,\mathbb{K}} \right)}\) and the quotient algebra are shown to have centers isomorphic to \(\mathbb{K}\). Via the new faithful representation, all automorphisms of \(\varepsilon {\left( {X,\mathbb{K}} \right)}\) are shown to be inner in the sense of Definition 18.