## Algebra & Number Theory

### The characteristic polynomial of the Adams operators on graded connected Hopf algebras

#### Abstract

The Adams operators $Ψn$ on a Hopf algebra $H$ are the convolution powers of the identity of $H$. They are also called Hopf powers or Sweedler powers. We study the Adams operators when $H$ is graded connected. The main result is a complete description of the characteristic polynomial — both eigenvalues and their multiplicities — for the action of the operator $Ψn$ on each homogeneous component of $H$. The eigenvalues are powers of $n$. The multiplicities are independent of $n$, and in fact only depend on the dimension sequence of $H$. These results apply in particular to the antipode of $H$, as the case $n = −1$. We obtain closed forms for the generating function of the sequence of traces of the Adams operators. In the case of the antipode, the generating function bears a particularly simple relationship to the one for the dimension sequence. In the case where $H$ is cofree, we give an alternative description for the characteristic polynomial and the trace of the antipode in terms of certain palindromic words. We discuss parallel results that hold for Hopf monoids in species and for $q$-Hopf algebras.

#### Article information

Source
Algebra Number Theory, Volume 9, Number 3 (2015), 547-583.

Dates
Received: 27 March 2014
Revised: 23 October 2014
Accepted: 1 December 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842311

Digital Object Identifier
doi:10.2140/ant.2015.9.547

Mathematical Reviews number (MathSciNet)
MR3340544

Zentralblatt MATH identifier
1330.16020

Subjects
Primary: 16T05: Hopf algebras and their applications [See also 16S40, 57T05]
Secondary: 16T30: Connections with combinatorics

#### Citation

Aguiar, Marcelo; Lauve, Aaron. The characteristic polynomial of the Adams operators on graded connected Hopf algebras. Algebra Number Theory 9 (2015), no. 3, 547--583. doi:10.2140/ant.2015.9.547. https://projecteuclid.org/euclid.ant/1510842311

#### References

• M. Aguiar and A. Lauve, “Antipode and convolution powers of the identity in graded connected Hopf algebras”, pp. 1053–1064 in 25th International Conference on Formal Power Series and Algebraic Combinatorics (Paris, 2013), Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2013.
• M. Aguiar and S. Mahajan, Monoidal functors, species and Hopf algebras, CRM Monograph Series 29, Amer. Math. Soc., Providence, RI, 2010.
• M. Aguiar and S. Mahajan, “Hopf monoids in the category of species”, pp. 17–124 in Hopf algebras and tensor categories, edited by N. Andruskiewitsch et al., Contemp. Math. 585, Amer. Math. Soc., Providence, RI, 2013.
• M. Aguiar and F. Sottile, “Cocommutative Hopf algebras of permutations and trees”, J. Algebraic Combin. 22:4 (2005), 451–470.
• M. Aguiar and F. Sottile, “Structure of the Malvenuto–Reutenauer Hopf algebra of permutations”, Adv. Math. 191:2 (2005), 225–275.
• M. Aguiar, N. Bergeron, and F. Sottile, “Combinatorial Hopf algebras and generalized Dehn–Sommerville relations”, Compos. Math. 142:1 (2006), 1–30.
• M. Aigner, A course in enumeration, Graduate Texts in Mathematics 238, Springer, Berlin, 2007.
• F. Bergeron, G. Labelle, and P. Leroux, Combinatorial species and tree-like structures, Encyclopedia of Mathematics and its Applications 67, Cambridge University Press, 1998.
• L. J. Billera, S. K. Hsiao, and S. van Willigenburg, “Peak quasisymmetric functions and Eulerian enumeration”, Adv. Math. 176:2 (2003), 248–276.
• C. Bonnafé and C. Hohlweg, “Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups”, Ann. Inst. Fourier $($Grenoble$)$ 56:1 (2006), 131–181.
• P. Cartier, “A primer of Hopf algebras”, pp. 537–615 in Frontiers in number theory, physics, and geometry, II, edited by P. M. Pierre Cartier, Bernard Julia and P. Vanhove, Springer, Berlin, 2007.
• P. Diaconis, C. Y. A. Pang, and A. Ram, “Hopf algebras and Markov chains: two examples and a theory”, J. Algebraic Combin. 39:3 (2014), 527–585.
• P. Flajolet and R. Sedgewick, Analytic combinatorics, Cambridge University Press, 2009.
• M. Gerstenhaber and S. D. Schack, “The shuffle bialgebra and the cohomology of commutative algebras”, J. Pure Appl. Algebra 70:3 (1991), 263–272.
• S.-J. Kang and M.-H. Kim, “Free Lie algebras, generalized Witt formula, and the denominator identity”, J. Algebra 183:2 (1996), 560–594.
• Y. Kashina, G. Mason, and S. Montgomery, “Computing the Frobenius–Schur indicator for abelian extensions of Hopf algebras”, J. Algebra 251:2 (2002), 888–913.
• Y. Kashina, Y. Sommerhäuser, and Y. Zhu, “On higher Frobenius–Schur indicators”, pp. viii+65 Mem. Amer. Math. Soc. 855, Amer. Math. Soc., Providence, RI, 2006.
• Y. Kashina, S. Montgomery, and S.-H. Ng, “On the trace of the antipode and higher indicators”, Israel J. Math. 188 (2012), 57–89.
• C. Kassel, Quantum groups, Graduate Texts in Mathematics 155, Springer, New York, 1995.
• V. Linchenko and S. Montgomery, “A Frobenius–Schur theorem for Hopf algebras”, Algebr. Represent. Theory 3:4 (2000), 347–355.
• J.-L. Loday, Cyclic homology, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 301, Springer, Berlin, 1992.
• J.-L. Loday and M. Ronco, “On the structure of cofree Hopf algebras”, J. Reine Angew. Math. 592 (2006), 123–155.
• M. Lothaire, Combinatorics on words, Cambridge University Press, 1997. http://msp.org/idx/mr/98g:68134MR 98g:68134
• I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, New York, 1995.
• C. Malvenuto and C. Reutenauer, “Duality between quasi-symmetric functions and the Solomon descent algebra”, J. Algebra 177:3 (1995), 967–982. http://msp.org/idx/mr/97d:05277MR 97d:05277
• J. W. Milnor and J. C. Moore, “On the structure of Hopf algebras”, Ann. of Math. $(2)$ 81 (1965), 211–264.
• S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics 82, Amer. Math. Soc., Providence, RI, 1993.
• S.-H. Ng and P. Schauenburg, “Central invariants and higher indicators for semisimple quasi-Hopf algebras”, Trans. Amer. Math. Soc. 360:4 (2008), 1839–1860.
• J.-C. Novelli, F. Patras, and J.-Y. Thibon, “Natural endomorphisms of quasi-shuffle Hopf algebras”, Bull. Soc. Math. France 141 (2013), 107–130.
• OEIS, “The On–Line Encyclopedia of Integer Sequences”, hook http://oeis.org \posturlhook.
• A. Pang, Hopf algebras and Markov chains, Ph.D. thesis, Stanford University, 2014, hook https://stacks.stanford.edu/file/druid:vy459jk2393/thesis_submit-augmented.pdf \posturlhook.
• F. Patras, “La décomposition en poids des algèbres de Hopf”, Ann. Inst. Fourier $($Grenoble$)$ 43:4 (1993), 1067–1087.
• F. Patras and M. Schocker, “Twisted descent algebras and the Solomon–Tits algebra”, Adv. Math. 199:1 (2006), 151–184.
• D. Quillen, “Rational homotopy theory”, Ann. of Math. $(2)$ 90 (1969), 205–295.
• D. E. Radford, Hopf algebras, Series on Knots and Everything 49, World Scientific, Hackensack, NJ, 2012.
• C. Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New Series 7, The Clarendon Press, Oxford University Press, New York, 1993.
• D. S. Sage and M. D. Vega, “Twisted Frobenius–Schur indicators for Hopf algebras”, J. Algebra 354 (2012), 136–147.
• W. R. Schmitt, “Incidence Hopf algebras”, J. Pure Appl. Algebra 96:3 (1994), 299–330.
• J.-P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer, New York, 1977.
• K. Shimizu, “Some computations of Frobenius–Schur indicators of the regular representations of Hopf algebras”, Algebr. Represent. Theory 15:2 (2012), 325–357.
• N. J. A. Sloane and S. Plouffe, The encyclopedia of integer sequences, Academic Press, San Diego, CA, 1995.
• R. P. Stanley, Enumerative combinatorics, vol. 1, 2nd ed., Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, 2012.
• J. R. Stembridge, “Enriched $P$-partitions”, Trans. Amer. Math. Soc. 349:2 (1997), 763–788.
• M. E. Sweedler, Hopf algebras, W. A. Benjamin, New York, 1969.
• E. J. Taft, “The order of the antipode of finite-dimensional Hopf algebra”, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 2631–2633.
• M. Takeuchi, “Free Hopf algebras generated by coalgebras”, J. Math. Soc. Japan 23 (1971), 561–582.
• H. S. Wilf, generatingfunctionology, 3rd ed., A K Peters, Wellesley, MA, 2006. http://msp.org/idx/mr/2006i:05014MR 2006i:05014