A Quiver Construction of Symmetric Crystals

Enomoto, Naoya

doi:10.1093/imrn/rnp014

Cited by 8 publications

(12 citation statements)

References 12 publications

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“…Returning to Hall modules, define operators E i , F i , T i ∈ End R (M Q ) as follows (see also [8]). Put…”

Section: Hall Modules From Quivers With Involutionmentioning

confidence: 99%

“…The right-hand side of this equation is equal to E(U ⊕ V ), proving that E descends to K(A). Finally, it is straightforward to verify equation (7) using equation (8).…”

Section: Introductionmentioning

confidence: 99%

“…The main application of this result is to the explicit decomposition of Hall modules of finite type quivers into irreducible highest weight B σ (g Q )-modules; see Theorems 4.4 and 4.6. The generators are written as alternating sums of the F q /F q -forms of self-dual indecomposables.In [8], Enomoto proved a result related to Theorems 3.5 and 3.10, showing that induction and restriction operators along [S i ] endow the Grothendieck group of a category of perverse sheaves on the moduli stack of complex orthogonal representations with the structure of a highest weight B σ (g Q )-module. In the terminology of the present paper, the weight module in [8] is generated by the trivial orthogonal representation, whereas Theorems 3.5 and 3.10 hold for arbitrary dualities and describe the decomposition of the entire Hall module.…”

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confidence: 99%

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The Hall module of an exact category with duality

Young

2016

Journal of Algebra

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We construct from a finitary exact category with duality A a module over its Hall algebra, called the Hall module, encoding the first order self-dual extension structure of A. We study in detail Hall modules arising from the representation theory of a quiver with involution. In this case we show that the Hall module is naturally a module over the specialized reduced σ-analogue of the quantum Kac-Moody algebra attached to the quiver. For finite type quivers, we explicitly determine the decomposition of the Hall module into irreducible highest weight modules.Date: November 9, 2018. 2010 Mathematics Subject Classification. Primary: 16G20 ; Secondary 17B37. Key words and phrases. Representations of quivers, Hall algebras, quantum groups. 1 2 M. B. YOUNGinterest. In Theorem 2.9, we prove an identity relating E, the Euler form and the stacky number of self-dual extensions in the case that A is hereditary. The proof develops some basic self-dual homological algebra and uses the combinatorics of self-dual analogues of Grothendieck's extensions panachées [12], [4].In Section 3 we study Hall modules arising from the representation theory of a quiver with contravariant involution (Q, σ). From the involution and a choice of signs we define a duality structure on Rep Fq (Q), with q odd. For particular signs, the self-dual objects coincide with the orthogonal and symplectic representations of Derksen and Weyman [7]. The module and comodule structures are incompatible in that M Q is not a Hopf module, even in a twisted sense. In Theorem 3.5, we instead show that the action and coaction of the simple representationsThe proof is combinatorial in nature and involves counting configurations of pairs of self-dual exact sequences, in the spirit of Green's proof of the bialgebra structure of the Hall algebra [11]. We describe in Theorem 3.10 the decomposition of M Q into irreducible highest weight B σ (g Q )-modules. The generators are cuspidal elements of M Q , i.e. elements that are annihilated by the coaction of each [S i ]. The proof relies on a canonically defined non-degenerate bilinear form on the Hall module and a characterization of irreducible highest weight modules due to Enomoto and Kashiwara [9].In Section 4 we restrict attention to finite type quivers. Unlike ordinary quiver representations, self-dual representations in general have non-trivial F q /F q -forms. We extend results of [7] (over algebraically closed fields) to explicitly describe all such forms and classify the indecomposable self-dual F q -representations. The classification is summarized in Theorem 4.2 where a partial root theoretic interpretation of the indecomposables is given. The main application of this result is to the explicit decomposition of Hall modules of finite type quivers into irreducible highest weight B σ (g Q )-modules; see Theorems 4.4 and 4.6. The generators are written as alternating sums of the F q /F q -forms of self-dual indecomposables.In [8], Enomoto proved a result related to Theorems 3.5 and 3.10, showing that induction and restr...

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“…Returning to Hall modules, define operators E i , F i , T i ∈ End R (M Q ) as follows (see also [8]). Put…”

Section: Hall Modules From Quivers With Involutionmentioning

confidence: 99%

“…The right-hand side of this equation is equal to E(U ⊕ V ), proving that E descends to K(A). Finally, it is straightforward to verify equation (7) using equation (8).…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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The Hall module of an exact category with duality

Young

2016

Journal of Algebra

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“…Similarly, the simplicial stack version of R • (C) is relative 2-Segal over S • (C) and recovers the perverse sheaf theoretic [10], motivic and cohomological [47] Hall algebra representations which appear in the representation theory of quantum enveloping algebras and orientifold Donaldson-Thomas theory. ⊳…”

Section: Remarksmentioning

confidence: 80%

“…The input for the R • -construction is a proto-exact category with duality which satisfies a reduction assumption. In the case of exact categories the R • -construction categorifies the Hall algebra representations of [43], [10], [46], [47] while for the proto-exact category Rep F1 (Q) of representations of a quiver over F 1 we obtain new modules over Szczesny's combinatorial Hall algebras [41]. The latter modules will be the subject of future work.…”

Section: Introductionmentioning

confidence: 99%

Relative 2–Segal spaces

Young

2018

Algebr. Geom. Topol.

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We introduce a relative version of the 2-Segal simplicial spaces defined by Dyckerhoff and Kapranov and Gálvez-Carrillo, Kock and Tonks. Examples of relative 2-Segal spaces include the categorified unoriented cyclic nerve, real pseudoholomorphic polygons in almost complex manifolds and the R•-construction from Grothendieck-Witt theory. We show that a relative 2-Segal space defines a categorical representation of the Hall algebra associated to the base 2-Segal space. In this way, after decategorification we recover a number of known constructions of Hall algebra representations. We also describe some higher categorical interpretations of relative 2-Segal spaces.

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