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We prove that Hall polynomial exists for each triple of decomposition sequences associated with tame quivers with automorphism (or, equivalently, for all tame hereditary algebras over finite fields). This generalizes the results of Hubery (2010) [18] and Deng and Ruan (2017) [9]. [ABSTRACT FROM AUTHOR]

Let |$\textrm{coh}\ \mathbb{X}$| be the category of coherent sheaves over a weighted projective line |$\mathbb{X}$| and let |$D^b(\textrm{coh}\ \mathbb{X})$| be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in |$D^b(\textrm{coh}\ \mathbb{X})$| attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star-shaped quiver |${Q}$| associated with |$\mathbb{X}$|⁠. By further dealing with the Ringel–Hall algebra of |$\mathbb{X}$|⁠ , we show that these functors provide a realization for Tits' automorphisms of the Kac–Moody algebra |${\mathfrak g}_{Q}$| associated with |${Q}$|⁠ , as well as for Lusztig's symmetries of the quantum enveloping algebra of |${\mathfrak g}_{Q}$|⁠. [ABSTRACT FROM AUTHOR]

We construct a new basis for a slim cyclotomic q -Schur algebra S m (n , r) via symmetric polynomials in Jucys–Murphy operators of the cyclotomic Hecke algebra H m (r). We show that this basis, labelled by matrices, is not the double coset basis when H m (r) is the Hecke algebra of a Coxeter group, but coincides with the double coset basis for the corresponding group algebra, the Hecke algebra at q = 1. As further applications, we then discuss the cyclotomic Schur–Weyl duality at the integral level. This also includes a category equivalence and a classification of simple objects. [ABSTRACT FROM AUTHOR]

Slim cyclotomic q -Schur algebras are certain centralizer subalgebras of cyclotomic q -Schur algebras in the sense of Dipper, James and Mathas, including q -Schur algebras of type A and C as examples. In the present paper we provide a sufficient condition for a slim cyclotomic q -Schur algebra S m (n , r) to be quasi-hereditary. More precisely, it is shown that S m (n , r) is quasi-hereditary if the cyclotomic Hecke algebra has a semisimple bottom. Moreover, we prove that S m (1 , r) is quasi-hereditary if and only if the cyclotomic Hecke algebra has a semisimple bottom. In the case where m = n = r = 2 and q = ± 1 , we prove that this condition is also necessary. [ABSTRACT FROM AUTHOR]

Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel–Hall algebra D ( Q ) of a quiver Q as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation L ( Λ 0 ) of D ( Δ n ) of the cyclic quiver Δ n provides a realization of the q -deformed Fock space ⋀ ∞ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of Δ n , we obtain a decomposition of the basic representation L ( Λ 0 ) which induces the Kashiwara–Miwa–Stern decomposition of ⋀ ∞ and a construction of the canonical basis of ⋀ ∞ defined by Leclerc and Thibon in terms of certain monomial basis elements in D ( Δ n ) . [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, we show that the Ringel–Hall algebra of a finitary algebra over a finite field satisfies fundamental relations in a more general setting. By twisting the multiplication, we obtain the quantum Serre relations. Finally, certain relations between Ringel–Hall algebras of an algebra and its factor algebras are discussed. [Copyright &y& Elsevier]

Let D(Λ) be the double Ringel–Hall algebra of a finite dimensional hereditary algebra Λ. The present paper shows that the BGP-reflection operators of D(Λ) coincide with Lusztig''s symmetries (up to canonical isomorphisms) and satisfy the braid group relations on the whole double Ringel–Hall algebra D(Λ). This answers a question of Sevenhant and Van den Bergh [J. Algebra 221 (1999) 135–160]. [Copyright &y& Elsevier]