Quasisymmetric functions and Heisenberg doubles

Abstract

The ring of quasisymmetric functions is free over the ring of symmetric functions. This result was
previously proved by M. Hazewinkel combinatorially through constructing a polynomial basis for
quasisymmetric functions. The recent work by A. Savage and O. Yacobi on representation theory
provides a new proof to this result. In this paper, we proved that under certain conditions, the
positive part of a Heisenberg double is free over the positive part of the corresponding projective
Heisenberg double. Examples satisfying the above conditions are discussed.

References

1. N. Bergeron, H. Li, Algebraic structures on Grothendieck groups of a tower of algebras, J. Algebra 321(8) (2009) 2068–2084.
2. E. J. Ditters, Curves and formal (Co)groups, Invent. Math. 17(1) (1972) 1–20.
3. G. Duchamp, D. Krob, B. Leclerc, J-Y. Thibon, Fonctions quasi-symmétriques, fonctions symmétriques noncommutatives, et algèbres de Hecke à q = 0 (French) [Quasisymmetric functions, noncommutative symmetric functions and Hecke algebras at q = 0], C. R. Acad. Sci. Paris Sér. I Math. 322(2) (1996), 107–112.
4. L. Geissinger, Hopf algebras of symmetric functions and class functions. Combinatoire et représentation du group symétrique (Actes Table Ronde C.N.R.S., Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), pp. 168–181. Lecture Notes in Math., Vol. 579, Springer, Berlin, 1977.
5. R. S. González D’León, A family of symmetric functions associated with stirling permutations, preprint, 2015.
6. M. Hazewinkel, The algebra of quasi-symmetric functions is free over the integers, Adv. Math. 164(2) (2001) 283–300.
7. M. Hazewinkel, N. Gubareni, V. V. Kirichenko, Algebras, rings and modules. Lie algebras and Hopf algebras. Mathematical Surveys and Monographs, 168. American Mathematical Society, Providence, RI, 2010.
8. F. Hivert, N. M. Thiéry, Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras, Formal Power Series and Algebraic Combinatorics, San Diego, California, 2006.

9. D. Krob, J-Y. Thibon, Noncommutative symmetric functions. IV. Quantum linear groups and Hecke algebras at q = 0, J. Algebraic Combin. 6(4) (1997) 339–376.
10. Y. Li, Representation theory of 0-Hecke-Clifford algebras, J. Algebra 453 (2016) 189–220.
11. D. Rosso, A. Savage, Towers of graded superalgebras categorify the twisted Heisenberg double, J. Pure Appl. Algebra 219(11) (2015) 5040–5067.
12. A. Savage, O. Yacobi, Categorification and Heisenberg doubles arising from towers of algebras, J. Combin. Theory Ser. A 129 (2015) 19–56.
13. M. A. Semenov-Tian-Shansky, Poisson Lie groups, quantum duality principle, and the quantum double. Mathematical aspects of conformal and topological field theories and quantum groups (South Hadley, MA, 1992), 219–248, Contemp. Math., 175, Amer. Math. Soc., Providence, RI, 1994.
14. R. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, 1999.
15. J-Y. Thibon, B-C-V. Ung, Quantum quasi-symmetric functions and Hecke algebras, J. Phys. A: Math. Gen. 29(22) (1996) 7337–7348.