On the Hall algebra of an elliptic curve, I

Burban, Igor; Schiffmann, Olivier

doi:10.1215/00127094-1593263

Cited by 149 publications

(331 citation statements)

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“…Here I vec r,d is the set of isomorphism classes of vector bundles over X of rank r and degree d. The completion H X is still an algebra, see e.g., [BS,Section 2].…”

Section: Recall That We Have Fixed An Identificationmentioning

confidence: 99%

Hall algebras of curves, commuting varieties and Langlands duality

Schiffmann

Vasserot

2011

Math. Ann.

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Abstract. We construct an isomorphism between the (universal) spherical Hall algebra of a smooth projective curve of genus g and a convolution algebra in the (equivariant) K-theory of the genus g commuting varietiesWe can view this isomorphism as a version of the geometric Langlands duality in the formal neighborhood of the trivial local system, for the group GLr. We extend this to all reductive groups and we compute the image, under our correspondence, of the skyscraper sheaf supported on the trivial local system.

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“…Here I vec r,d is the set of isomorphism classes of vector bundles over X of rank r and degree d. The completion H X is still an algebra, see e.g., [BS,Section 2].…”

Section: Recall That We Have Fixed An Identificationmentioning

confidence: 99%

Hall algebras of curves, commuting varieties and Langlands duality

Schiffmann

Vasserot

2011

Math. Ann.

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“…However since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished. As a consequence of very recent publications [4], [11], [19], [20], [21], a truly remarkable expansion of this theory has taken place. However most of this work has appeared in a language that is virtually inaccessible to practitioners of Algebraic Combinatorics.…”

Section: Introductionmentioning

confidence: 95%

Some remarkable new plethystic operators in the theory of Macdonald polynomials

Bergeron

Garsia

Leven

et al. 2016

Journal of Combinatorics

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In the 90's a collection of Plethystic operators were introduced in [3], [7] and [8] to solve some Representation Theoretical problems arising from the Theory of Macdonald polynomials. This collection was enriched in the research that led to the results which appeared in [5], [6] and [9]. However since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished. As a consequence of very recent publications [4], [11], [19], [20], [21], a truly remarkable expansion of this theory has taken place. However most of this work has appeared in a language that is virtually inaccessible to practitioners of Algebraic Combinatorics. Yet, these developments have led to a variety of new conjectures in [2] in the Combinatorics and Symmetric function Theory of Macdonald Polynomials. The present work results from an effort to obtain in an elementary and accessible manner all the background necessary to construct the symmetric function side of some of these new conjectures. It turns out that the above mentioned unpublished results provide precisely the tools needed to carry out this project to its completion. IntroductionOur main actors in this development are the operators D k introduced in [8], whose action on a symmetric function F [X] is defined by setting Thus A is clearly a graded algebra. What is surprising is that A is in fact bi-graded by simply assigning the generators D k bi-degree (1, k).To make this more precise consider first where Π u,v denotes the portion of Π which is a linear combination of words in D of total bi-degree (u, v). To show that A is bi-graded it is necessary and sufficient to prove that Π, as an operator, acts by zero on Λ if and only if all the Π u,v act by zero. This is one of the very first things we will prove about A.The connection of A to the above mentioned developments is that it gives a concrete realization of a proper subspace of the Elliptic Hall Algebra studied by Schiffmann and Vasserot in [20], [21] and [19]. In particular it contains a distinguished family of operators {Q u,v } of bi-degree given by their index that play a central role in the above mentioned conjectures. For a co-prime bi-degree their construction is so simple that we need only illustrate it in a special case.For instance, to obtain Q 3,5 we start by drawing the 3 × 5 lattice square with its diagonal (the line (0, 0) → (3, 5), as shown in the adjacent figure), we then look for the lattice point (a, b) that is closest to and below the diagonal. In this case (a, b) = (2, 3). This yields the decomposition (3, 5) = (2, 3) + (1, 2) and we setWe must next work precisely in the same way with the 2 × 3 rectangle and, as indicated in the adjacent figure, obtain the decomposition (2, 3) = (1, 1) + (1, 2) and setNow, in this case, we are done, since it turns out that we may setIn particular by combining 0.2, 0.3 and 0.4 we obtainIn the general co-prime case (m, n), the precise definition is based on an elementary number theoretical Lemma that characterizes th...

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“…However, there are ample examples of Cohen-Macaulay local rings not having isolated singularities, including the local hypersurfaces of type (A ∞ ) and (D ∞ ) appearing above. Cohen-Macaulay representation theory for non-isolated singularities has been studied by many authors so far; see [2,10,14,19] for instance. It should be remarked that a Cohen-Macaulay local ring with a non-isolated singularity always admits maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum.…”

Section: Introductionmentioning

confidence: 99%

Maximal Cohen–Macaulay modules that are not locally free on the punctured spectrum

Kobayashi

Lyle

Takahashi

2020

Journal of Pure and Applied Algebra

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We say that a Cohen-Macaulay local ring has finite CM+-representation type if there exist only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum. In this paper, we consider finite CM+-representation type from various points of view, relating it with several conjectures on finite/countable Cohen-Macaulay representation type. We prove in dimension one that the Gorenstein local rings of finite CM+-representation type are exactly the local hypersurfaces of countable CM-representation type, that is, the hypersurfaces of type (A∞) and (D∞). We also discuss the closedness and dimension of the singular locus of a Cohen-Macaulay local ring of finite CM+-representation type.2010 Mathematics Subject Classification. 13C60, 13H10, 16G60.This theorem may look technical, but it actually gives rise to a lot of restrictions which having finite CM + -representation type produces, and is used in the later sections. One concrete example where Theorem 5.5 applies is when I = (x) and (0 : x) = (x); see Corollary 5.9. Here we introduce one of the applications of the above theorem. Denote by CM(R) the category of maximal Cohen-Macaulay R-modules, and by D sg (R) the singularity category of R. Theorem 1.5 (Theorem 5.8). Let R be a Cohen-Macaulay local ring of dimension d > 0. Let I be an ideal of R with V(I) ⊆ V(0 : I) such that R/I is maximal Cohen-Macaulay over R.Suppose that R has finite CM + -representation type. Then one must have d = 1. If I n = 0 for some integer n > 0, then CM(R) has dimension at most n − 1 in the sense of [18]. If R is Gorenstein, then R is a hypersurface and D sg (R) has dimension at most n − 1 in the sense of [23].There are folklore conjectures that a Gorenstein local ring of countable CM-representation type is a hypersurface, and that, for a Cohen-Macaulay local ring R of countable CM-representation type, CM(R) has dimension at most one. The above theorem gives partial answers to the variants of these folklore conjectures for finite CM + -representation type.

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On the Hall algebra of an elliptic curve, I

Cited by 149 publications

References 29 publications

Hall algebras of curves, commuting varieties and Langlands duality

Hall algebras of curves, commuting varieties and Langlands duality

Some remarkable new plethystic operators in the theory of Macdonald polynomials

Maximal Cohen–Macaulay modules that are not locally free on the punctured spectrum

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