Combinatorial Hopf algebras from representations of families of wreath products
Algebraic Combinatorics, Volume 5 (2022) no. 1, pp. 93-120.

We construct Hopf algebras whose elements are representations of combinatorial automorphism groups, by generalising a theorem of Zelevinsky on Hopf algebras of representations of wreath products. As an application we attach symmetric functions to representations of graph automorphism groups, generalising and refining Stanley’s chromatic symmetric function.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.198
Classification: 05E10,  16T30,  20C15
Keywords: Combinatorial Hopf algebra, PSH algebra, wreath product, chromatic symmetric function
Crisp, Tyrone 1; Hill, Caleb Kennedy 1

1 Department of Mathematics & Statistics University of Maine 5752 Neville Hall, Room 333 Orono ME 04469 USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ALCO_2022__5_1_93_0,
     author = {Crisp, Tyrone and Hill, Caleb Kennedy},
     title = {Combinatorial {Hopf} algebras from representations of families of wreath products},
     journal = {Algebraic Combinatorics},
     pages = {93--120},
     publisher = {MathOA foundation},
     volume = {5},
     number = {1},
     year = {2022},
     doi = {10.5802/alco.198},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.198/}
}
TY  - JOUR
AU  - Crisp, Tyrone
AU  - Hill, Caleb Kennedy
TI  - Combinatorial Hopf algebras from representations of families of wreath products
JO  - Algebraic Combinatorics
PY  - 2022
DA  - 2022///
SP  - 93
EP  - 120
VL  - 5
IS  - 1
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.198/
UR  - https://doi.org/10.5802/alco.198
DO  - 10.5802/alco.198
LA  - en
ID  - ALCO_2022__5_1_93_0
ER  - 
%0 Journal Article
%A Crisp, Tyrone
%A Hill, Caleb Kennedy
%T Combinatorial Hopf algebras from representations of families of wreath products
%J Algebraic Combinatorics
%D 2022
%P 93-120
%V 5
%N 1
%I MathOA foundation
%U https://doi.org/10.5802/alco.198
%R 10.5802/alco.198
%G en
%F ALCO_2022__5_1_93_0
Crisp, Tyrone; Hill, Caleb Kennedy. Combinatorial Hopf algebras from representations of families of wreath products. Algebraic Combinatorics, Volume 5 (2022) no. 1, pp. 93-120. doi : 10.5802/alco.198. https://alco.centre-mersenne.org/articles/10.5802/alco.198/

[1] Aguiar, Marcelo; André, Carlos; Benedetti, Carolina; Bergeron, Nantel; Chen, Zhi; Diaconis, Persi; Hendrickson, Anders; Hsiao, Samuel; Isaacs, I. Martin; Jedwab, Andrea; Johnson, Kenneth; Karaali, Gizem; Lauve, Aaron; Le, Tung; Lewis, Stephen; Li, Huilan; Magaard, Kay; Marberg, Eric; Novelli, Jean-Christophe; Pang, Amy; Saliola, Franco; Tevlin, Lenny; Thibon, Jean-Yves; Thiem, Nathaniel; Venkateswaran, Vidya; Vinroot, C. Ryan; Yan, Ning; Zabrocki, Mike Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras, Adv. Math., Volume 229 (2012) no. 4, pp. 2310-2337 | DOI | MR | Zbl

[2] Aguiar, Marcelo; Bergeron, Nantel; Sottile, Frank Combinatorial Hopf algebras and generalized Dehn–Sommerville relations, Compos. Math., Volume 142 (2006) no. 1, pp. 1-30 | DOI | MR | Zbl

[3] Aguiar, Marcelo; Mahajan, Swapneel Monoidal functors, species and Hopf algebras, CRM Monograph Series, 29, American Mathematical Society, Providence, RI, 2010, lii+784 pages | DOI | MR | Zbl

[4] Birkhoff, George D. A determinant formula for the number of ways of coloring a map, Ann. of Math. (2), Volume 14 (1912/13) no. 1-4, pp. 42-46 | DOI | MR | Zbl

[5] Cameron, Peter J.; Jackson, Bill; Rudd, Jason D. Orbit-counting polynomials for graphs and codes, Discrete Math., Volume 308 (2008) no. 5-6, pp. 920-930 | DOI | MR | Zbl

[6] Cameron, Peter J.; Kayibi, Koko K. Orbital chromatic and flow roots, Combin. Probab. Comput., Volume 16 (2007) no. 3, pp. 401-407 | DOI | MR | Zbl

[7] Crisp, Tyrone; Meir, Ehud; Onn, Uri A variant of Harish-Chandra functors, J. Inst. Math. Jussieu, Volume 18 (2019) no. 5, pp. 993-1049 | DOI | MR | Zbl

[8] Crisp, Tyrone; Meir, Ehud; Onn, Uri An inductive approach to representations of general linear groups over compact discrete valuation rings (2020) (https://arxiv.org/abs/2005.05553)

[9] Crisp, Tyrone; Meir, Ehud; Onn, Uri Principal series for general linear groups over finite commutative rings, Comm. Algebra, Volume 49 (2021) no. 11, pp. 4857-4868 | DOI | MR | Zbl

[10] Frucht, Roberto Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Math., Volume 6 (1939), pp. 239-250 | Numdam | MR | Zbl

[11] Geissinger, Ladnor Hopf algebras of symmetric functions and class functions, Combinatoire et représentation du groupe symétrique (Actes Table Ronde C.N.R.S., Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) (Lecture Notes in Math.), Volume 579, Springer, Berlin, Heidelberg, 1977, pp. 168-181 | MR | Zbl

[12] Grinberg, Darij; Reiner, Victor Hopf Algebras in Combinatorics (2014) (https://arxiv.org/abs/1409.8356)

[13] Hanlon, Phil The chromatic polynomial of an unlabeled graph, J. Combin. Theory Ser. B, Volume 38 (1985) no. 3, pp. 226-239 | DOI | MR | Zbl

[14] James, Gordon; Kerber, Adalbert The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, Mass., 1981, xxviii+510 pages | MR

[15] Joni, S. A.; Rota, Gian-Carlo Coalgebras and bialgebras in combinatorics, Stud. Appl. Math., Volume 61 (1979) no. 2, pp. 93-139 | DOI | MR | Zbl

[16] Mackey, George W. On induced representations of groups, Amer. J. Math., Volume 73 (1951), pp. 576-592 | DOI | MR | Zbl

[17] Schmitt, William R. Hopf algebras of combinatorial structures, Canad. J. Math., Volume 45 (1993) no. 2, pp. 412-428 | DOI | MR | Zbl

[18] Schmitt, William R. Incidence Hopf algebras, J. Pure Appl. Algebra, Volume 96 (1994) no. 3, pp. 299-330 | DOI | MR | Zbl

[19] Shelley-Abrahamson, Seth Hopf modules and representations of finite wreath products, Algebr. Represent. Theory, Volume 20 (2017) no. 1, pp. 123-145 | DOI | MR | Zbl

[20] Stanley, Richard P. A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math., Volume 111 (1995) no. 1, pp. 166-194 | DOI | MR | Zbl

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

[22] van Leeuwen, Marc A. A. An application of Hopf-algebra techniques to representations of finite classical groups, J. Algebra, Volume 140 (1991) no. 1, pp. 210-246 | DOI | MR | Zbl

[23] Young, Alfred On Quantitative Substitutional Analysis (Fifth Paper), Proc. London Math. Soc. (2), Volume 31 (1930) no. 4, pp. 273-288 | DOI | MR | Zbl

[24] Zelevinsky, Andrey V. Representations of finite classical groups. A Hopf algebra approach, Lecture Notes in Mathematics, 869, Springer-Verlag, Berlin-New York, 1981, iv+184 pages | MR

Cited by Sources: