2017/02/09 by Alan J. Cain, Cain, Alan J., António Malheiro +1
Mathematics · #05C05 #05E05 #05E15 (Primary) #20M05 (Secondary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1702.02998
openalex publication_date 2017/02/09 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
Kashiwara's crystal graphs have a natural monoid structure that arises by\nidentifying words labelling vertices that appear in the same position of\nisomorphic components. The celebrated plactic monoid (the monoid of Young\ntableaux), arises in this way from the crystal graph for the q-analogue of\nthe general linear Lie algebra mathfrakgln, and the so-called Kashiwara\noperators interact beautifully with the combinatorics of Young tableaux and\nwith the Robinson--Schensted--Knuth correspondence. The authors previously\nconstructed an analogous `quasi-crystal' structure for the related hypoplactic\nmonoid (the monoid of quasi-ribbon tableaux), which has similarly neat\ncombinatorial properties. This paper constructs an analogous `crystal-type'\nstructure for the sylvester and Baxter monoids (the monoids of binary search\ntrees and pairs of twin binary search trees, respectively). Both monoids are\nshown to arise from this structure just as the plactic monoid does from the\nusual crystal graph. The interaction of the structure with the sylvester and\nBaxter versions of the Robinson-Schensted-Knuth correspondence is studied. The\nstructure is then applied to prove results on the number of factorizations of\nelements of these monoids, and to prove that both monoids satisfy non-trivial\nidentities.\n