Plethysms and operads

• Cebrian, Alex ^[1]
  1. [1] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 75, Fasc. 1, 2024, págs. 247-303
• Idioma: inglés
• DOI: 10.1007/s13348-022-00386-1
• Enlaces
  • Texto completo
• Resumen

  • We introduce the {\mathcal {T}}-construction, an endofunctor on the category of generalized operads, as a general mechanism by which various notions of plethystic substitution arise from more ordinary notions of substitution. In the special case of one-object unary operads, i.e. monoids, we recover the T-construction of Giraudo. We realize several kinds of plethysm as convolution products arising from the homotopy cardinality of the incidence bialgebra of the bar construction of various operads obtained from the {\mathcal {T}}-construction. The bar constructions are simplicial groupoids, and in the special case of the terminal reduced operad \textsf {Sym}, we recover the simplicial groupoid of Cebrian (Algebraic Geom Topol 21(1):421–446, 2021), a combinatorial model for ordinary plethysm in the sense of Pólya, given in the spirit of Waldhausen S and Quillen Q constructions. In some of the cases of the {\mathcal {T}}-construction, an analogous interpretation is possible.

• Referencias bibliográficas
  • Baez, J.C., Dolan, J.: From finite sets to Feynman diagrams. In: Mathematics Unlimited—2001 and Beyond, pp. 29–50. Springer, Berlin (2001)
  • Bauer, T.: Formal plethories. Adv. Math. 254, 497–569 (2014). arXiv:1107.5745
  • Bergeron, F.: Une combinatoire du pléthysme. J. Combin. Theory Ser. A 46, 291–305 (1987)
  • Bergeron, F.: A combinatorial outlook on symmetric functions. J. Combin. Theory Ser. A 50, 226–234 (1989)
  • Borger, J., Wieland, B.: Plethystic algebra. Adv. Math. 194, 246–283 (2005). arXiv:math/0407227
  • Brouder, C., Frabetti, A., Krattenthaler, C.: noncommutative Hopf algebra of formal diffeomorphisms. Adv. Math. 200, 479–524 (2006). arXiv:math/0406117
  • Burroni, A.: T-catégories (catégories dans un triple). Cah. Topol. Géom. Différ. Catég. 12, 215–321 (1971)
  • Cartier, P., Foata, D.: Problèmes combinatoires de commutation et réarrangements. No. 85 in Lecture Notes in Mathematics. Springer-Verlag,...
  • Cebrian, A.: A simplicial groupoid for plethysm. Algebr. Geom. Topol. 21(1), 421–446 (2021)
  • Cebrian, A.: Combinatorics of plethysm via Segal groupoids and operads. Ph.D. thesis. Universitat Autònoma de Barcelona (2020)
  • Chapoton, F., Livernet, M.: Relating two Hopf algebras built from an operad. Int. Math. Res. Notices 2007 (2007). arXiv:0707.3725
  • Doubilet, P.: A Hopf algebra arising from the lattice of partitions of a set. J. Algebra 28, 127–132 (1974)
  • Dyckerhoff , T., Kapranov, M.: Higher Segal Spaces. Lecture Notes in Mathematics 2244 (2019). arXiv:1212.3563
  • Ebrahimi-Fard, K., Lundervold, A., Manchon, D.: Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras. Int. J. Algebra...
  • Gálvez-Carrillo, I., Kock, J., Tonks, A.: Groupoids and Faà di Bruno formulae for Green functions in bialgebras of trees. Adv. Math. 254,...
  • Gálvez-Carrillo, I., Kock, J., Tonks, A.: Homotopy linear algebra. Proc. R. Soc. Edinb. A. 148, 293–325 (2018)
  • Gálvez-Carrillo, I., Kock, J., Tonks, A.: Decomposition spaces, incidence algebras and Möbius inversion I: basic theory. Adv. Math. 331, 952–105...
  • Gálvez-Carrillo, I., Kock, J., Tonks, A.: Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration,...
  • Gálvez-Carrillo, I., Kock, J., Tonks, A., Corrigendum to “Decomposition spaces, incidence algebras and Möbius inversion II: completeness,...
  • Giraudo, S.: Combinatorial operads from monoids. J. Algebra Combin. 41, 493–538 (2015)
  • Giraudo, S.: Nonsymmetric Operads in Combinatorics. Springer Nature Switzerland AG (2018)
  • Joyal, A.: Une théorie combinatoire des séries formelles. Adv. Math. 42, 1–82 (1981)
  • Kock, A.: Strong functors and monoidal monads. Arch. Math. 23, 113–120 (1972)
  • Kock, J., Weber, M.: Faà di Bruno for operads and internal algebras. J. Lond. Math. Soc. 99, 919–944 (2019)
  • Lawvere, F.W., Menni, M.: The Hopf algebra of Möbius intervals. Theory Appl. Categ. 24, 221–265 (2010)
  • Leinster, T.: Higher Operads. Higher Categories. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2004)
  • Littlewood, D.E.: Invariant theory, tensors and group characters. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 239, 305–365 (1944)
  • Loday, J.-L., Vallette, B.: Algebraic Operads. Grundlehren der mathematischen Wissenschaften 346. Springer-Verlag, Berlin (2012)
  • Lundervold, A., Munthe-Kaas, H.: Hopf algebras of formal diffeomorphisms and numerical integration on manifolds. Contemp. Math. 539, 295–324...
  • Ian, G.: Macdonald. Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press,...
  • May, J.P.: The Geometry of Iterated Loop Spaces. Lecture Notes in Mathematics, vol. 271. Springer-Verlag, Berlin (1972)
  • Méndez, M.: Set Operads in Combinatorics and Computer Science. SpringerBriefs in Mathematics. Springer, Cham (2015)
  • Méndez, M., Nava, O.: Colored species, c-monoids, and plethysm. J. Combin. Theory Ser. A 64, 102–129 (1993)
  • Moggi, E.: Notions of computation and monads. Inform. Comput. 93, 155–92 (1991)
  • Nava, O.: On the combinatorics of plethysm. J. Combin. Theory Ser. A 46, 212–251 (1987)
  • Nava, O., Rota, G.-C.: Plethysm, categories, and combinatorics. Adv. Math. 58, 61–88 (1985)
  • Pólya, G.: Kombinatorische Anzahlbestimmungen für Gruppen. Graphen und chemische Verbindungen. Acta Math. 68, 145–254 (1937)
  • Rota, G.-C.: On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2,...
  • Schmitt, W.R.: Incidence Hopf algebras. J. Pure Appl. Algebra 96, 299–330 (1994)
  • Stanley, R.P.: Enumerative Combinarorics. Cambridge Studies in Advanced Mathematics, vol. 2. Cambridge University Press, Cambridge (1999)
  • van der Laan, P.: Operads and the Hopf Algebras of Renormalisation. arXiv:math-ph/0311013
  • Van der Laan, P., Moerdijk, I.: The Renormalisation Bialgebra and Operads. arXiv:hep-th/0210226
  • Wadler, P.: Comprehending Monads. Special issue of selected papers from 6’th Conference on Lisp and Functional Programming, 2, 461–493 (1992)
  • Weber, M.: Operads as polynomial 2-monads. Theory Appl. Categ. 30, 1659–1712 (2015). arXiv:1412.7599
  • Weber, M.: Internal algebra classifiers as codescent objects of crossed internal categories. Theory Appl. Categ. 30, 1713–1792 (2015)