Algebra & Number Theory

Transverse quiver Grassmannians and bases in affine cluster algebras

Grégoire Dupont

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Abstract

Sherman, Zelevinsky and Cerulli constructed canonically positive bases in cluster algebras associated to affine quivers having at most three vertices. Their constructions involve cluster monomials and normalized Chebyshev polynomials of the first kind evaluated at a certain “imaginary” element in the cluster algebra. Using this combinatorial description, it is possible to define for any affine quiver Q a set (Q), which is conjectured to be the canonically positive basis of the acyclic cluster algebra A(Q).

In this article, we provide a geometric realization of the elements in (Q) in terms of the representation theory of Q. This is done by introducing an analogue of the Caldero–Chapoton cluster character, where the usual quiver Grassmannian is replaced by a constructible subset called the transverse quiver Grassmannian.

Article information

Source
Algebra Number Theory, Volume 4, Number 5 (2010), 599-624.

Dates
Received: 26 October 2009
Revised: 17 March 2010
Accepted: 16 April 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729556

Digital Object Identifier
doi:10.2140/ant.2010.4.599

Mathematical Reviews number (MathSciNet)
MR2679100

Zentralblatt MATH identifier
1268.16019

Subjects
Primary: 16G99: None of the above, but in this section
Secondary: 13F99: None of the above, but in this section

Keywords
cluster algebras canonical bases Chebyshev polynomials cluster characters quiver Grassmannians

Citation

Dupont, Grégoire. Transverse quiver Grassmannians and bases in affine cluster algebras. Algebra Number Theory 4 (2010), no. 5, 599--624. doi:10.2140/ant.2010.4.599. https://projecteuclid.org/euclid.ant/1513729556


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