The canonical join complex. (English) Zbl 07032096

Summary: A canonical join representation is a certain minimal “factorization” of an element in a finite lattice \(L\) analogous to the prime factorization of an integer from number theory. The expression \(\bigvee A =w\) is the canonical join representation of \(w\) if \(A\) is the unique lowest subset of \(L\) satisfying \(\bigvee A=w\) (where “lowest” is made precise by comparing order ideals under containment). Canonical join representations appear in many familiar guises, with connections to comparability graphs and noncrossing partitions. When each element in \(L\) has a canonical join representation, we define the canonical join complex to be the abstract simplicial complex of subsets \(A\) such that \(\bigvee A\) is a canonical join representation. We characterize the class of finite lattices whose canonical join complex is flag, and show how the canonical join complex is related to the topology of \(L\).


06B15 Representation theory of lattices
06A07 Combinatorics of partially ordered sets


SF; posets; Maple; coxeter
Full Text: arXiv Link


[1] K. Adaricheva and J.B. Nation, Classes of Semidistributive Lattices. in Lattice Theory: Special Topics and Applications, ed. G. Gr¨atzer and F. Wehrung. Chapter 3 of Lattice Theory: Special Topics and Applications, Volume 2, Editors: George Gr¨atzer and Friedrich Wehrung, Springer 2016.
[2] D. Armstrong, C. Stump, H. Thomas, A uniform bijection between nonnesting and noncrossing partitions. Trans. Amer. Math. Soc. 365(8), 2013 · Zbl 1271.05011
[3] E. Barnard, The canonical join representation in algebraic combinatorics. Ph.D. Thesis, North Carolina State University, 2017.
[4] E. Barnard, A. Carrol, and S. Zhu, Minimal inclusions of torsion classes. arXiv:1710.08837
[5] A. Bj¨orner, Topological methods in combinatorics, Handbook of Combinatorics, Vol. 2, 1819–1872, Elsevier, Amsterdam, 1995.
[6] A. Brouwer and A. Schrijver, On the period of an operator, defined on antichains. Math Centrum report ZW 24/74 (1974). · Zbl 0282.06003
[7] P. Cameron and D. Fon-Der-Flaass, Orbits of antichains revisited. European J. Combin. 16 (1995), no. 6, 545–554. · Zbl 0831.06001
[8] C. Le Conte de Poly-Barbut, Sur les Treillis de Coxeter Finis (French). Math. Inf. Sci.hum. 32 no. 125 (1994), 41–57. · Zbl 0802.06016
[9] A. Day, A Simple Solution to the Word Problem for Lattices. Canad. Math. Bull. 13 (1970), 253–254. · Zbl 0206.29702
[10] R. Freese, J. Jezek and J. Nation, Free lattices. Mathematical Surveys and Monographs, 42, American Mathematical Society, 1995. · Zbl 0839.06005
[11] S. Fomin and N. Reading, Root Systems and Generalized Associahedra. Geometric combinatorics, 63-131, IAS/Park City Math. Ser., 13, Amer. Math. Soc., Providence, RI, 2007. · Zbl 1147.52005
[12] D. Fon-Der-Flaass, Orbits of antichains in ranked posets. European J. Combin. 14 (1993), no. 1, 17–22. · Zbl 0777.06002
[13] A. Garver and T. McConville, Lattice properties of oriented exchange graphs and torsion classes. Algebras and Representation Theory, (2015), 1–36. the electronic journal of combinatorics 26(1) (2019), #P1.2424
[14] G. Gr¨atzer, General Lattice Theory. Birkhauser Verlag, 1998.
[15] G. Gr¨atzer and F. Wehrung, A new lattice construction: the box product. J. Algebra 221 (1999), no. 1, 315–344. · Zbl 0961.06005
[16] W. Geyer, On Tamari lattices. Discrete Math. 133 (1994), no. 1-3, 99–122. · Zbl 0811.06005
[17] P. Hersh and K. M´esz´aros SB-labelings and posets with each interval homotopy equivalent to a sphere or a ball. J. of Combinatorial Theory, Series A, 152 (2017), 104–120. · Zbl 1369.05180
[18] O. Iyama, N. Reading, I. Reiten, and H. Thomas, Lattice structure of Weyl groups via representation theory of preprojective algebras. Compositio Mathematica 154, no. 6 (2018) 1269-1305. · Zbl 1443.16016
[19] T. McConville Crosscut-Simplicial Lattices. Order, 34 (2017), 465–477. · Zbl 1432.06003
[20] D. Panyushev, On orbits of antichains of positive roots. European J. Combin. 30 (2009), no. 2, 586–594. · Zbl 1165.06001
[21] N. Reading, Cambrian lattices. Adv. Math. 205 (2006) no. 2, 313–353 · Zbl 1106.20033
[22] N. Reading Lattice Theory of the Poset of Regions. in Lattice Theory: Special Topics and Applications, ed. G. Gr¨atzer and F. Wehrung. Chapter 9 of Lattice Theory: Special Topics and Applications, Volume 2, Editors: George Gr¨atzer and Friedrich Wehrung, Springer 2016.
[23] N. Reading, Noncrossing diagrams and canonical join representations. SIAM J. Discrete Math. 29 (2015), no. 2, 736–750. · Zbl 1314.05015
[24] N. Reading, Noncrossing partitions and the shard intersection order. J. Algebraic Combin. 33 (2011), no. 4, 483–530. · Zbl 1290.05163
[25] N. Reading and D. E. Speyer, Sortable elements in infinite Coxeter groups. Trans. Amer. Math. Soc. 363 (2011) no. 2, 699–761. · Zbl 1231.20036
[26] R. Stanley, Promotion and evacuation, Electron. J. Combin. 16 (2009), no. 2, R9. · Zbl 1169.06002
[27] R. P. Stanley, Enumerative combinatorics. Vol. 1, second edition. Cambridge Studies in Advanced Mathematics 49. Cambridge University Press, Cambridge, 2012.
[28] J. Stembridge, Maple packages for symmetric functions, posets, root systems, and finite Coxeter groups. Available athttp://www.math.lsa.umich.edu/ jrs/maple. html
[29] J. Striker and N. Williams, Promotion and Rowmotion. European J. Of Combinatorics, 33, (2012) no. 8, 1919–1942. · Zbl 1260.06004
[30] W. Trotter, Combinatorics and Partially Ordered Sets: Dimension Theory. The Johns Hopkins University Press, 1992. · Zbl 0764.05001
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