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Determinants and perfect matchings

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Jan 2013Embargo end date: 01 Jan 2011 United States English Publisher:Elsevier BVJournal:Journal of Combinatorial Theory, Series A, volume 120, pages 304-314 (issn: 0097-3165, Copyright policy )
Authors: Arvind Ayyer;

doi: 10.1016/j.jcta.2012.08.007 , 10.48550/arxiv.1106.1465

arXiv: 1106.1465

Determinants and perfect matchings

Abstract

We give a combinatorial interpretation of the determinant of a matrix as a generating function over Brauer diagrams in two different but related ways. The sign of a permutation associated to its number of inversions in the Leibniz formula for the determinant is replaced by the number of crossings in the Brauer diagram. This interpretation naturally explains why the determinant of an even antisymmetric matrix is the square of a Pfaffian.

15 pages, terminology improved, exposition tightened, "deranged matchings" example removed

Country
United States
Related Organizations
Keywords

FOS: Computer and information sciences, Perfect matchings, Determinant expansion, Discrete Mathematics (cs.DM), Brauer diagrams, cs.DM, 05B20, 05C10, 05C70, 15B57, 15A15, Mathematics - Rings and Algebras, math.RT, Theoretical Computer Science, Computational Theory and Mathematics, Rings and Algebras (math.RA), FOS: Mathematics, Pfaffians, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, Combinatorics (math.CO), math.CO, Representation Theory (math.RT), math.RA, Mathematics - Representation Theory, Computer Science - Discrete Mathematics

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    popularity
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    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
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    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
4
Average
Average
Average
Green
hybrid