ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.02
https://doi.org/10.26493/1855-3974.2691.0b7
(Also available at http://amc-journal.eu)
Double generalized majorization and
diagrammatics*
Marija Dodig †
CEAFEL, Departamento de Matématica, Universidade de Lisboa, Edificio C6,
Campo Grande, 1749-016 Lisbon, Portugal, and
Mathematical Institute SANU, Knez Mihajlova 36, 11000 Belgrade, Serbia
Marko Stošić
CAMGSD, Departamento de Matemática, Instituto Superior Técnico,
Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal, and
Mathematical Institute SANU, Knez Mihajlova 36, 11000 Belgrade, Serbia
Received 7 September 2021, accepted 16 May 2022, published online 11 November 2022
Abstract
In this paper we show that the generalized majorization of partitions of integers has a
surprising completing-squares property. Together with the previously obtained transitivity-
like property, this enables a compelling diagrammatical interpretation. Apart from purely
combinatorial interest, the main result has applications in matrix completion problems, and
representation theory of quivers.
Keywords: Partitions, majorization, diagrammatics, inequalities.
Math. Subj. Class. (2020): 05A17
*We would like to thank the Referee for the valuable comments and suggestions. This work was done within
the activities of CEAFEL and was partially supported by FCT, project UIDB/04721/2020, and Exploratory Grant
EXPL/MAT-PUR/0584/2021. In the final stages, this work was also partially supported by the Science Fund of
the Republic of Serbia, Projects no. 7744592, MEGIC – “Integrability and Extremal Problems in Mechanics,
Geometry and Combinatorics” (M.D.) and no. 7749891, GWORDS – “Graphical Languages" (M.S.).
†Corresponding author.
E-mail addresses: msdodig@fc.ul.pt (Marija Dodig), mstosic@isr.ist.utl.pt (Marko Stošić)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.02
https://doi.org/10.26493/1855-3974.2691.0b7
(Dostopno tudi na http://amc-journal.eu)
Dvojna posplošena majorizacija in
diagramatika*
Marija Dodig †
CEAFEL, Departamento de Matématica, Universidade de Lisboa, Edificio C6,
Campo Grande, 1749-016 Lisbon, Portugal, and
Mathematical Institute SANU, Knez Mihajlova 36, 11000 Belgrade, Serbia
Marko Stošić
CAMGSD, Departamento de Matemática, Instituto Superior Técnico,
Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal, and
Mathematical Institute SANU, Knez Mihajlova 36, 11000 Belgrade, Serbia
Prejeto 7. septembra 2021, sprejeto 16. maja 2022, objavljeno na spletu 11. novembra 2022
Povzetek
V tem članku dokažemo, da ima posplošena majorizacija particij presenetljivo lastnost
dopolnitve kvadratov. Skupaj s predhodno dobljeno lastnostjo, podobno tranzitivnsti, to
omogoča prepričljivo diagramsko interpretacijo. Poleg čisto kombinatoričnega prispevka
je glavni rezultat uporaben pri problemih matrične dopolnitve ter v predstavitveni teoriji
tulcev.
Ključne besede: Particije, majorizacija, diagramatska predstavitev, neenakosti.
Math. Subj. Class. (2020): 05A17
*Rada bi se zahvalila recenzentu za dragocene pripombe in predloge. To delo je bilo opravljeno v okviru
dejavnosti CEAFEL in je bilo delno podprto s strani FCT s projektom UIDB/04721/2020 ter s strani dotacije za
raziskovanje EXPL/MAT-PUR/0584/2021. V zaključnih fazah je bilo to delo delno podprto tudi s strani Science
Fund of the Republic of Serbia, v okviru projektov št. 7744592, MEGIC – “Integrability and Extremal Problems
in Mechanics, Geometry and Combinatorics” (M.D.) in št. 7749891, GWORDS – “Graphical Languages" (M.S.).
†Kontaktni avtor.
E-poštna naslova: msdodig@fc.ul.pt (Marija Dodig), mstosic@isr.ist.utl.pt (Marko Stošić)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/