Picture groups and maximal green sequences

Kiyoshi Igusa; Gordana Todorov; Kiyoshi Igusa; Gordana Todorov

doi:10.3934/era.2021025

Electronic Research Archive

2021, Volume 29, Issue 5: 3031-3068. doi: 10.3934/era.2021025

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Picture groups and maximal green sequences

Kiyoshi Igusa ^{1
,
,},
Gordana Todorov ^{2
,}

1.
Brandeis University, MS050, 415 South St, Waltham, MA 02454-9110, USA
2.
Northeastern University, 360 Huntington Ave, Boston, MA 02115, USA

Received: 01 July 2020 Revised: 01 February 2021 Published: 16 March 2021
Primary: 16G20; Secondary: 20F55

We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of finite "vertically and laterally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type).
Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.
- Real Schur roots,
- exceptional modules,
- wall-and-chamber structure,
- stability conditions,
- Coxeter element,
- atomic deformations
Citation: Kiyoshi Igusa, Gordana Todorov. Picture groups and maximal green sequences[J]. Electronic Research Archive, 2021, 29(5): 3031-3068. doi: 10.3934/era.2021025

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Abstract

We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of finite "vertically and laterally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type).

Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.

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