×

zbMATH — the first resource for mathematics

Poset edge densities, nearly reduced words, and barely set-valued tableaux. (English) Zbl 1391.05269
Summary: In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it. We show that this coincidence of expectations holds for Cartesian products of chains, connected minuscule posets, weak Bruhat orders on finite Coxeter groups, certain lower intervals in Young’s lattice, and certain lower intervals in the weak Bruhat order below dominant permutations. Our tools involve formulas for counting nearly reduced factorizations in 0-Hecke algebras; that is, factorizations that are one letter longer than the Coxeter group length.

MSC:
05E10 Combinatorial aspects of representation theory
06A07 Combinatorics of partially ordered sets
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20C08 Hecke algebras and their representations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bergeron, F.; Préville-Ratelle, L.-F., Higher trivariate diagonal harmonics via generalized Tamari posets, J. Comb., 3, 317-341, (2012) · Zbl 1291.05213
[2] Björner, A.; Brenti, F., Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, (2005), Springer New York · Zbl 1110.05001
[3] Björner, A.; Las Vergnas, M.; Sturmfels, B.; White, N.; Ziegler, G. M., Oriented matroids, Encyclopedia of Mathematics and its Applications, vol. 46, (1999), Cambridge University Press Cambridge · Zbl 0944.52006
[4] Boussicault, A.; Féray, V.; Lascoux, A.; Reiner, V., Linear extension sums as valuations on cones, J. Algebraic Combin., 35, 573-610, (2012) · Zbl 1242.05274
[5] Brüstle, T.; Yang, Dong, Ordered exchange graphs, (2013), preprint · Zbl 1353.16012
[6] Buch, A., A Littlewood-Richardson rule for the K-theory of Grassmannians, Acta Math., 189, 37-78, (2002) · Zbl 1090.14015
[7] Buch, A.; Samuel, M., K-theory of minuscule varieties, J. Reine Angew. Math., 719, 133-171, (2016) · Zbl 1431.19001
[8] Buch, A.; Kresch, A.; Shimozono, M.; Tamvakis, H.; Yong, A., Stable Grothendieck polynomials and K-theoretic factor sequences, Math. Ann., 340, 359-382, (2008) · Zbl 1157.14036
[9] Chan, M.; Haddadan, S.; Hopkins, S.; Moci, L., The expected jaggedness of order ideals, (2015), preprint
[10] Chan, M.; Martín, A. L.; Pflueger, N.; Teixidor i Bigas, M., Genera of brill-Noether curves and staircase paths in Young tableaux, (2015), preprint
[11] Derksen, H.; Weyman, J.; Zelevinsky, A., Quivers with potentials and their representations II: applications to cluster algebras, J. Amer. Math. Soc., 23, 749-790, (2010) · Zbl 1208.16017
[12] Fomin, S.; Greene, C., Noncommutative Schur functions and their applications, Selected Papers in Honor of Adriano Garsia, Taormina, 1994, Discrete Math., 193, 179-200, (1998) · Zbl 1011.05062
[13] Fomin, S.; Kirillov, A. N., Grothendieck polynomials and the Yang-Baxter equation, (Proc. 6th Intern. Conf. on Formal Power Series and Algebraic Combinatorics, (1994), DIMACS), 183-190
[14] Fomin, S.; Kirillov, A. N., Universal exponential solution of the Yang-Baxter equation, Lett. Math. Phys., 37, 273-284, (1996) · Zbl 0867.17009
[15] Fomin, S.; Kirillov, A. N., Reduced words and plane partitions, J. Algebraic Combin., 6, 311-319, (1997) · Zbl 0882.05010
[16] Fomin, S.; Stanley, R. P., Schubert polynomials and the nilcoxeter algebra, Adv. Math., 103, 196-207, (1994) · Zbl 0809.05091
[17] Garver, A.; McConville, T., Lattice properties of oriented exchange graphs and torsion classes, (2015), preprint
[18] Green, R. M., Combinatorics of minuscule representations, Cambridge Tracts in Mathematics, vol. 199, (2013), Cambridge University Press Cambridge · Zbl 1320.17005
[19] Haglund, J., The \(q, t\)-Catalan numbers and the space of diagonal harmonics, University Lecture Series, vol. 41, (2008), American Mathematical Society Providence, RI · Zbl 1142.05074
[20] Hopkins, S., The CDE property for minuscule lattices, J. Combin. Theory Ser. A, 152, 45-103, (2017) · Zbl 1369.05209
[21] Iyama, O.; Reiten, I.; Thomas, H.; Todorv, G., Lattice structure of torsion classes for path algebras, Bull. Lond. Math. Soc., 47, 639-650, (2015) · Zbl 1397.16011
[22] Kapranov, M. M.; Voevodsky, V. A., Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (List of results), Cah. Topol. Géom. Différ. Catég., 32, 11-27, (1991) · Zbl 0748.18010
[23] Kerov, S. V., Transition probabilities of continual Young diagrams and the Markov moment problem, Funct. Anal. Appl., 27, 104-117, (1993) · Zbl 0808.05098
[24] Kerov, S. V., The boundary of Young lattice and random Young tableaux, (Proc. 8th Intern. Conf. on Formal Power Series and Algebraic Combinatorics, (1996), DIMACS), 133-158 · Zbl 0856.05008
[25] Knuth, D. E., Sorting and searching, The Art of Computer Programming, vol. 3, (1973), Addison-Wesley Reading, MA · Zbl 0302.68010
[26] Knutson, A.; Miller, E.; Yong, A., Gröbner geometry of vertex decompositions and of flagged tableaux, J. Reine Angew. Math., 630, 1-31, (2009) · Zbl 1169.14033
[27] Lascoux, A., Transition on Grothendieck polynomials, (Physics and Combinatorics, Nagoya, 2000, (2001), World Sci. Publ. River Edge, NJ), 164-179 · Zbl 1052.14059
[28] Lascoux, A., Schubert & Grothendieck: un bilan bidécennal, Sém. Lothar. Combin., 50, (2003/04) · Zbl 1061.05099
[29] Lascoux, A.; Schützenberger, M. P., Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math., 295, 629-633, (1982) · Zbl 0542.14030
[30] Law, S.; Reading, N., The Hopf algebra of diagonal rectangulations, J. Combin. Theory Ser. A, 119, 788-824, (2012) · Zbl 1246.16027
[31] Lenart, C., Noncommutative Schubert calculus and Grothendieck polynomials, Adv. Math., 143, 159-183, (1999) · Zbl 0978.05074
[32] Macdonald, I. G., Notes on Schubert polynomials, (1991), Univ. du Québec à Montréal, Publications du LaCIM · Zbl 0784.05061
[33] Manin, Y. I.; Schechtman, V. V., Arrangements of hyperplanes, higher braid groups, and higher Bruhat orders, (Adv. Stud. in Pure Math., vol. 17, (1989)), 289-308 · Zbl 0759.20002
[34] Manivel, L., Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, vol. 6, (2001), American Mathematical Society/Société Mathématique de France Providence, RI/Paris · Zbl 0998.14023
[35] McConville, T., Lattice structure of grid-Tamari orders, (2015), preprint · Zbl 1335.05225
[36] (Müller-Hoissen, F.; Pallo, J.-M.; Stasheff, J., Associahedra, Tamari Lattices and Related Structures, Tamari Memorial Festschrift, Progress in Mathematical Physics, vol. 299, (2012), Birkhäuser/Springer Basel) · Zbl 1253.00013
[37] Norton, P. N., 0-Hecke algebras, J. Aust. Math. Soc. A, 27, 337-357, (1979) · Zbl 0407.16019
[38] Pilaud, V., Brick polytopes, lattice quotients, and Hopf algebras, (2015), preprint · Zbl 1377.05024
[39] Proctor, R. A., Bruhat lattices, plane partition generating functions, and minuscule representations, European J. Combin., 5, 331-350, (1984) · Zbl 0562.05003
[40] R.A. Proctor, unpublished research announcement, 1984.
[41] Reading, N., Cambrian lattices, Adv. Math., 205, 313-353, (2006) · Zbl 1106.20033
[42] Rush, D. B., On order ideals of minuscule posets III: the CDE property, (2016), preprint
[43] Santos, F.; Stump, C.; Welker, V., Noncrossing sets and a Grassmann associahedron, (2014), preprint · Zbl 1362.52007
[44] Stanley, R. P., Theory and applications of plane partitions: part 2, Stud. Appl. Math., 50, 259-279, (1971) · Zbl 0225.05012
[45] Stanley, R. P., On the number of reduced decompositions of elements of Coxeter groups, European J. Combin., 5, 359-372, (1984) · Zbl 0587.20002
[46] Stanley, R. P., Enumerative combinatorics, vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, (1999), Cambridge University Press Cambridge · Zbl 0928.05001
[47] Stanley, R. P., Enumerative combinatorics, vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, (2012), Cambridge University Press Cambridge · Zbl 1247.05003
[48] Thomas, H.; Yong, A., A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus, Algebra Number Theory, 3, 121-148, (2009) · Zbl 1229.05285
[49] Thomas, H.; Yong, A., Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm, Adv. in Appl. Math., 46, 610-642, (2011) · Zbl 1227.05262
[50] Ziegler, G. M., Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, (1995), Springer-Verlag New York · Zbl 0823.52002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.