Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T03:11:31.192Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Hadamard Product of Hopf Monoids

Published online by Cambridge University Press:  20 November 2018

Marcelo Aguiar  and
Swapneel Mahajan
Marcelo Aguiar
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843. e-mail: maguiar@math.tamu.edu
Swapneel Mahajan
Affiliation:
Department of Mathematics, Indian Institute of Technology Mumbai, Powai, Mumbai 400 076, India. e-mail: swapneel@math.iitb.ac.in
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Combinatorial structures that compose and decompose give rise to Hopf monoids in Joyal's category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free.

The first main result is obtained by showing the existence of a one-parameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco that applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species that is intertwined by the free monoid functor with the Hadamard product. As an application of the first result, we deduce that the Boolean transform of the dimension sequence of a connected Hopf monoid is nonnegative.

Keywords

16T3018D3520B3018D1020F55speciesHopf monoidHadamard productgenerating functionBoolean transform
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 3 , 01 June 2014 , pp. 481 - 504
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Aguiar, M., Bergeron, N., and Thiem, N., Hopf monoids from class functions on unitriangular matrices. Algebra and Number Theory, to appear. arxiv:1203.1572v1Google Scholar
[2] Aguiar, M. and Lauve, A., Lagrange's Theorem for Hopf monoids in species. Canad. J. Math. 65(2013), no. 2, 241–265. http://dx.doi.org/10.4153/CJM-2011-098-9 Google Scholar
[3] Aguiar, M. and Mahajan, S., Coxeter groups and Hopf algebras. Fields Institute Monographs, 23, American Mathematical Society, Providence, RI; Fields Institute, Toronto, 2006.Google Scholar
[4] Aguiar, M., Monoidal functors, species and Hopf algebras. CRM Monograph Series, 29, American Mathematical Society, Providence, RI, 2010.Google Scholar
[5] Bergeron, F., Labelle, G., and Leroux, P., Combinatorial species and tree-like structures. Encyclopedia of Mathematics and its Applications, 67, Cambridge University Press, Cambridge, 1998.Google Scholar
[6] Bergeron, N., Reutenauer, C., Rosas, M., and Zabrocki, M., Invariants and coinvariants of the symmetric groups in noncommuting variables. Canad. J. Math. 60(2008), no. 2, 266–296. http://dx.doi.org/10.4153/CJM-2008-013-4 Google Scholar
[7] Bergeron, N. and Zabrocki, M., The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free. J. Algebra Appl. 8(2009), no. 4, 581–600. http://dx.doi.org/10.1142/S0219498809003485 Google Scholar
[8] Foissy, L., Free and cofree Hopf algebras. J. Pure Appl. Algebra 216(2012), no. 2, 480–494. http://dx.doi.org/10.1016/j.jpaa.2011.07.010 Google Scholar
[9] Joyal, A., Une théorie combinatoire des séries formelles. Adv. in Math. 42(1981), no. 1, 1–82. http://dx.doi.org/10.1016/0001-8708(81)90052-9 Google Scholar
[10] Livernet, M., From left modules to algebras over an operad: application to combinatorial Hopf algebras. Ann. Math. Blaise Pascal 17(2010), no. 1, 47–96. http://dx.doi.org/10.5802/ambp.278 Google Scholar
[11] Loday, J.-L. and Ronco, M., On the structure of cofree Hopf algebras. J. Reine Angew. Math. 592(2006), 123–155. http://dx.doi.org/10.1515/CRELLE.2006.025 Google Scholar
[12] Malvenuto, C. and Reutenauer, C., Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177(1995), no. 3, 967–982. http://dx.doi.org/10.1006/jabr.1995.1336Google Scholar
[13] Poirier, S. and Reutenauer, C., Algèbres de Hopf de tableaux. Ann. Sci. Math. Québec 19(1995), no. 1, 79–90.Google Scholar
[14] Popa, M., A new proof for the multiplicative property of the Boolean cumulants with applications to the operator-valued case. Colloq. Math. 117(2009), no. 1, 81–93. http://dx.doi.org/10.4064/cm117-1-5 Google Scholar
[15] Rosas, M. H. and Sagan, B. E., Symmetric functions in noncommuting variables. Trans. Amer. Math. Soc. 358(2006), no. 1, 215–232. http://dx.doi.org/10.1090/S0002-9947-04-03623-2 Google Scholar
[16] Sloane, N. J. A., The on-line encyclopedia of integer sequences. 2012. http://oeis.org. Google Scholar
[17] Speicher, R. and Woroudi, R., Boolean convolution. In: Free probability theory (Waterloo, ON, 1995), Fields Inst. Commun., 12, American Mathematical Society, Providence, RI, 1997, pp. 267–279.Google Scholar
[18] C.Wolf, M., Symmetric functions of non-commutative elements. Duke Math. J. 2(1936), no. 4, 626–637. http://dx.doi.org/10.1215/S0012-7094-36-00253-3 Google Scholar