|
Devadoss, Fehrman, Heath, Vashist の [Dev+12] によると, Stasheff とは独立に, Dov Tamari
は 1951年の thesis [Tam51] で assosiahedron \(K_n\) の記述を得ている。 \(n\)文字の括弧付けの成す lattice の実現として,
である。その lattice が Tamari lattice \(T_{n}\) である。
文献としては, Tamari memorial Festschrift [MPS12] がある。
\(T_{n}\) の元の個数は Catalan number \(C_{n}\) であり, よって様々なものと関係している。例えば以下のようなもの。
一般化や変種も色々考えられている。
- Cambrian lattice [RS11]
-
\(m\)-Tamari lattice [BP12]
- higher Stasheff-Tamari order [KV91]
-
\(\nu \)-Tamari lattice [PV]
- flip order [BM21]
References
-
[BK04]
-
Aslak Bakke Buan and Henning Krause. “Tilting and cotilting
for quivers and type \(Ã_n\)”. In: J. Pure Appl. Algebra 190.1-3 (2004),
pp. 1–21. url: https://doi.org/10.1016/j.jpaa.2003.11.004.
-
[BM21]
-
Emily
Barnard and Thomas McConville. “Lattices from graph associahedra
and subalgebras of the Malvenuto-Reutenauer algebra”. In: Algebra
Universalis 82.1 (2021), Paper No. 2, 53. arXiv: 1808.05670. url:
https://doi.org/10.1007/s00012-020-00689-z.
-
[BP12]
-
François Bergeron and Louis-François Préville-Ratelle. “Higher
trivariate diagonal harmonics via generalized Tamari posets”.
In: J. Comb. 3.3 (2012), pp. 317–341. arXiv: 1105.3738. url:
https://doi.org/10.4310/JOC.2012.v3.n3.a4.
-
[Dev+12]
-
Satyan L. Devadoss, Benjamin Fehrman, Timothy Heath, and Aditi
Vashist.
“Moduli spaces of punctured Poincaré disks”. In: Associahedra,
Tamari lattices and related structures. Vol. 299. Progr. Math.
Birkhäuser/Springer, Basel, 2012, pp. 99–117. arXiv: 1109.2830.
url: https://doi.org/10.1007/978-3-0348-0405-9_6.
-
[HU05]
-
Dieter Happel and Luise Unger. “On a partial order of tilting
modules”. In: Algebr. Represent. Theory 8.2 (2005), pp. 147–156.
url: http://dx.doi.org/10.1007/s10468-005-3595-2.
-
[KV91]
-
M. M. Kapranov and V. A. Voevodsky. “Combinatorial-geometric
aspects of polycategory theory: pasting schemes and higher Bruhat
orders (list of results)”. In: Cahiers Topologie Géom. Différentielle
Catég. 32.1 (1991). International Category Theory Meeting (Bangor,
1989 and Cambridge, 1990), pp. 11–27.
-
[MPS12]
-
Folkert Müller-Hoissen, Jean Marcel Pallo, and Jim Stasheff, eds.
Associahedra, Tamari
lattices and related structures. Vol. 299. Progress in Mathematical
Physics. Tamari memorial Festschrift. Birkhäuser/Springer, Basel,
2012, pp. xx+433. isbn: 978-3-0348-0404-2; 978-3-0348-0405-9. url:
https://doi.org/10.1007/978-3-0348-0405-9.
-
[PV]
-
Louis-François Préville-Ratelle and Xavier Viennot. An extension
of Tamari lattices. arXiv: 1406.3787.
-
[RS11]
-
Nathan Reading and David E. Speyer. “Sortable elements in infinite
Coxeter groups”. In: Trans.
Amer. Math. Soc. 363.2 (2011), pp. 699–761. arXiv: 0803.2722.
url: http://dx.doi.org/10.1090/S0002-9947-2010-05050-0.
-
[Tam51]
-
Dov Tamari. Monoı̈des préordonnés et chaı̂nes de Malcev. Thèse,
Université de Paris, 1951, iv+81 pp. (mimeographed).
|
