Browse issues
no. 1
Some applications of Rees products of posets to equivariant gamma-positivity
Athanasiadis, Christos A.1
1 Department of Mathematics National and Kapodistrian University of Athens Panepistimioupolis Athens 15784, Greece
Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 281-300.

The Rees product of partially ordered sets was introduced by Björner and Welker. Using the theory of lexicographic shellability, Linusson, Shareshian and Wachs proved formulas, of significance in the theory of gamma-positivity, for the dimension of the homology of the Rees product of a graded poset P with a certain t-analogue of the chain of the same length as P. Equivariant generalizations of these formulas are proven in this paper, when a group of automorphisms acts on P, and are applied to establish the Schur gamma-positivity of certain symmetric functions arising in algebraic and geometric combinatorics.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.85
Classification: 05E05, 05E18, 05E45, 06A07
Keywords: Rees product, poset homology, group action, Schur gamma-positivity, local face module

Athanasiadis, Christos A. 1

1 Department of Mathematics National and Kapodistrian University of Athens Panepistimioupolis Athens 15784, Greece
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2020__3_1_281_0,
     author = {Athanasiadis, Christos A.},
     title = {Some applications of {Rees} products of posets to equivariant gamma-positivity},
     journal = {Algebraic Combinatorics},
     pages = {281--300},
     publisher = {MathOA foundation},
     volume = {3},
     number = {1},
     year = {2020},
     doi = {10.5802/alco.85},
     mrnumber = {3144389},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.85/}
}
TY  - JOUR
AU  - Athanasiadis, Christos A.
TI  - Some applications of Rees products of posets to equivariant gamma-positivity
JO  - Algebraic Combinatorics
PY  - 2020
SP  - 281
EP  - 300
VL  - 3
IS  - 1
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.85/
DO  - 10.5802/alco.85
LA  - en
ID  - ALCO_2020__3_1_281_0
ER  - 
%0 Journal Article
%A Athanasiadis, Christos A.
%T Some applications of Rees products of posets to equivariant gamma-positivity
%J Algebraic Combinatorics
%D 2020
%P 281-300
%V 3
%N 1
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.85/
%R 10.5802/alco.85
%G en
%F ALCO_2020__3_1_281_0
Athanasiadis, Christos A. Some applications of Rees products of posets to equivariant gamma-positivity. Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 281-300. doi : 10.5802/alco.85. https://alco.centre-mersenne.org/articles/10.5802/alco.85/

[1] Adin, Ron M.; Athanasiadis, Christos A.; Elizalde, Sergi; Roichman, Yuval Character formulas and descents for the hyperoctahedral group, Adv. in Appl. Math., Volume 87 (2017), pp. 128-169 | DOI | MR | Zbl

[2] Athanasiadis, Christos A. Flag subdivisions and γ-vectors, Pacific J. Math., Volume 259 (2012) no. 2, pp. 257-278 | DOI | MR | Zbl

[3] Athanasiadis, Christos A. Edgewise subdivisions, local h-polynomials, and excedances in the wreath product ℤ r ≀𝔖 n , SIAM J. Discrete Math., Volume 28 (2014) no. 3, pp. 1479-1492 | DOI | MR | Zbl

[4] Athanasiadis, Christos A. A survey of subdivisions and local h-vectors, The mathematical legacy of Richard P. Stanley, Amer. Math. Soc., Providence, RI, 2016, pp. 39-51 | DOI | Zbl

[5] Athanasiadis, Christos A. Gamma-positivity in combinatorics and geometry, Sém. Lothar. Combin., Volume 77 ([2016-2018]), Paper no. B77i, 64 pages | MR | Zbl

[6] Athanasiadis, Christos A. Binomial Eulerian polynomials for colored permutations (2018) (https://arxiv.org/abs/1812.00434)

[7] Björner, Anders Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Adv. Math., Volume 52 (1984) no. 3, pp. 173-212 | DOI | MR | Zbl

[8] Björner, Anders; Welker, Volkmar Segre and Rees products of posets, with ring-theoretic applications, J. Pure Appl. Algebra, Volume 198 (2005) no. 1-3, pp. 43-55 | DOI | MR | Zbl

[9] Dolgachev, Igor; Lunts, Valery A character formula for the representation of a Weyl group in the cohomology of the associated toric variety, J. Algebra, Volume 168 (1994) no. 3, pp. 741-772 | DOI | MR | Zbl

[10] Gal, Światosław R. Real root conjecture fails for five- and higher-dimensional spheres, Discrete Comput. Geom., Volume 34 (2005) no. 2, pp. 269-284 | MR | Zbl

[11] Lehrer, Gustav I. Rational points and Coxeter group actions on the cohomology of toric varieties, Ann. Inst. Fourier (Grenoble), Volume 58 (2008) no. 2, pp. 671-688 | DOI | Numdam | MR | Zbl

[12] Linusson, Svante; Shareshian, John; Wachs, Michelle L. Rees products and lexicographic shellability, J. Comb., Volume 3 (2012) no. 3, pp. 243-276 | MR | Zbl

[13] Poirier, Stéphane Cycle type and descent set in wreath products, Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995) (Discrete Math.), Volume 180 (1998) no. 1-3, pp. 315-343 | MR | Zbl

[14] Procesi, Claudio The toric variety associated to Weyl chambers, Mots (Lang. Raison. Calc.), Hermès, Paris, 1990, pp. 153-161 | Zbl

[15] Sagan, Bruce E. The symmetric group, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001, xvi+238 pages | MR | Zbl

[16] Savvidou, Christina Barycentric subdivisions, clusters and permutation enumeration (in Greek), Ph. D. Thesis, University of Athens (Greece) (2013)

[17] Shareshian, John; Wachs, Michelle L. Poset homology of Rees products, and q-Eulerian polynomials, Electron. J. Combin., Volume 16 (2009) no. 2, Paper no. Research Paper R20, 29 pages (Special volume in honor of Anders Björner) | MR | Zbl

[18] Shareshian, John; Wachs, Michelle L. Eulerian quasisymmetric functions, Adv. Math., Volume 225 (2010) no. 6, pp. 2921-2966 | DOI | MR | Zbl

[19] Shareshian, John; Wachs, Michelle L. From poset topology to q-Eulerian polynomials to Stanley’s chromatic symmetric functions, The mathematical legacy of Richard P. Stanley, Amer. Math. Soc., Providence, RI, 2016, pp. 301-321 | DOI | Zbl

[20] Shareshian, John; Wachs, Michelle L. Gamma-positivity of variations of Eulerian polynomials (2018) (https://arxiv.org/abs/1702.06666v3) | Zbl

[21] Stanley, Richard P. Some aspects of groups acting on finite posets, J. Combin. Theory Ser. A, Volume 32 (1982) no. 2, pp. 132-161 | DOI | MR | Zbl

[22] Stanley, Richard P. Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986) (Ann. New York Acad. Sci.), Volume 576, New York Acad. Sci., New York, 1989, pp. 500-535 | MR | Zbl

[23] Stanley, Richard P. Subdivisions and local h-vectors, J. Amer. Math. Soc., Volume 5 (1992) no. 4, pp. 805-851 | MR | Zbl

[24] Stanley, Richard P. Combinatorics and commutative algebra, Progress in Mathematics, 41, Birkhäuser Boston, Inc., Boston, MA, 1996, x+164 pages | MR | Zbl

[25] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | MR

[26] Stanley, Richard P. Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | MR | Zbl

[27] Stembridge, John R. The projective representations of the hyperoctahedral group, J. Algebra, Volume 145 (1992) no. 2, pp. 396-453 | DOI | MR | Zbl

[28] Stembridge, John R. Some permutation representations of Weyl groups associated with the cohomology of toric varieties, Adv. Math., Volume 106 (1994) no. 2, pp. 244-301 | DOI | MR | Zbl

[29] Sundaram, Sheila The homology representations of the symmetric group on Cohen–Macaulay subposets of the partition lattice, Adv. Math., Volume 104 (1994) no. 2, pp. 225-296 | DOI | MR | Zbl

[30] Wachs, Michelle L. Poset topology: tools and applications, Geometric combinatorics (IAS/Park City Math. Ser.), Volume 13, Amer. Math. Soc., 2007, pp. 497-615 | DOI | MR | Zbl

Cited by Sources: