Generic bases are dual semicanonical bases for unipotent cells

This is a report on joint work with B. Leclerc and J. Schröer.

Recall that Derksen, Weyman and Zelvinsky managed to prove many combinatorial conjectures about cluster algebras with symmetric exchange matrix by studying certain representations of the corresponding quiver together with a non-degenerate potential.

We proposed in [GLS2], that in this situation the generic values of Palu's cluster character on the strongly reduced components of the corresponding representation varieties should be a good candidate for a basis of the cluster algebra. It is not hard to see that these elements have the following properties:

(a) contain all cluster monomials,

(b) they are independent of the mutation class,

(c) belong to the upper cluster algebra.

On the other hand, we showed in [GLS2] that the dual of Lusztig's semicanonical basis (which is defined for the enveloping algebra of the positive  part of any Kac-Moody Lie algebra with symmetric Cartan matrix) restricts to a basis of the coordinate ring of corresponding unipotent cell associated to an Weyl group element.

This dual semicanonical basis contains all cluster monomials.

In [GLS2] we showed, that in  the above mentioned candidate for a "generic basis" and the dual semicanonical basis coincide.

In this talk I will try to explain this development in the special case of a symmetric Cartan matrix of finite type and the longest element in the Weyl group.

References

[GLS1] Ch. Geiss, B Leclerc, J Schröer:

Kac–Moody groups and cluster algebras

Advances in Mathematics 228 (2011), 329-433

[GLS2] Ch. Geiss, B Leclerc, J Schröer:

Generic bases for cluster algebras and the Chamber Ansatz Journal of the American Mathematical Society 25 (2012), 21-76

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