1606.03440v2.pdf (587.07 kB)
Download file

On Generalized Minors and Quiver Representations

Download (587.07 kB)
journal contribution
posted on 17.05.2019, 09:34 by Dylan Rupel, Salvatore Stella, Harold Williams
The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang-Zelevinsky in finite type. In type $A_n^{\!(1)}$ and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally.

Funding

This work was supported by an American Mathematical Society-Simons Travel Grant [to D.R.]; a Marie Curie - Istituto Nazionale di Alta Matematica Cofund Fellowship [to S.S.]; Israel Science Foundation grant [1144/16 to S.S.]; an National Science Foundation Postdoctoral Research Fellowship [DMS-1502845 to H.W.]; and National Science Foundation grant [DMS-1702489 to H.W.].

History

Citation

International Mathematics Research Notices, 2018, rny053

Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Mathematics

Version

AM (Accepted Manuscript)

Published in

International Mathematics Research Notices

Publisher

Oxford University Press (OUP)

eissn

1687-0247

Acceptance date

05/03/2018

Copyright date

2018

Available date

17/05/2019

Publisher version

https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rny053/4969402

Language

en