TY - JOUR
T1 - Quivers with subadditive labelings
T2 - classification and integrability
AU - Galashin, Pavel
AU - Pylyavskyy, Pavlo
N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2020/8/1
Y1 - 2020/8/1
N2 - Strictly subadditive, subadditive and weakly subadditive labelings of quivers were introduced by the second author, generalizing Vinberg’s definition for undirected graphs. In our previous work we have shown that quivers with strictly subadditive labelings are exactly the quivers exhibiting Zamolodchikov periodicity. In this paper, we classify all quivers with subadditive labelings. We conjecture them to exhibit a certain form of integrability, namely, as the T-system dynamics proceeds, the values at each vertex satisfy a linear recurrence. Conversely, we show that every quiver integrable in this sense is necessarily one of the 19 items in our classification. For the quivers of type A^ ⊗ A we express the coefficients of the recurrences in terms of the partition functions for domino tilings of a cylinder, called Goncharov–Kenyon Hamiltonians. We also consider tropical T-systems of type A^ ⊗ A and explain how affine slices exhibit solitonic behavior, i.e. soliton resolution and speed conservation. Throughout, we conjecture how the results in the paper are expected to generalize from A^ ⊗ A to all other quivers in our classification.
AB - Strictly subadditive, subadditive and weakly subadditive labelings of quivers were introduced by the second author, generalizing Vinberg’s definition for undirected graphs. In our previous work we have shown that quivers with strictly subadditive labelings are exactly the quivers exhibiting Zamolodchikov periodicity. In this paper, we classify all quivers with subadditive labelings. We conjecture them to exhibit a certain form of integrability, namely, as the T-system dynamics proceeds, the values at each vertex satisfy a linear recurrence. Conversely, we show that every quiver integrable in this sense is necessarily one of the 19 items in our classification. For the quivers of type A^ ⊗ A we express the coefficients of the recurrences in terms of the partition functions for domino tilings of a cylinder, called Goncharov–Kenyon Hamiltonians. We also consider tropical T-systems of type A^ ⊗ A and explain how affine slices exhibit solitonic behavior, i.e. soliton resolution and speed conservation. Throughout, we conjecture how the results in the paper are expected to generalize from A^ ⊗ A to all other quivers in our classification.
KW - Cluster algebras
KW - Discrete solitons
KW - Domino tilings
KW - Linear recurrence
KW - Zamolodchikov integrability
UR - http://www.scopus.com/inward/record.url?scp=85070271110&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85070271110&partnerID=8YFLogxK
U2 - 10.1007/s00209-019-02374-x
DO - 10.1007/s00209-019-02374-x
M3 - Article
AN - SCOPUS:85070271110
VL - 295
SP - 945
EP - 992
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
SN - 0025-5874
IS - 3-4
ER -