In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables. This ring generalizes the ring of symmetric functions. This ring can be realized as a specific limit of the rings of quasisymmetric polynomials in n variables, as n goes to infinity. This ring serves as universal structure in which relations between quasisymmetric polynomials can be expressed in a way independent of the number n of variables (but its elements are neither polynomials nor functions).

Definitions

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The ring of quasisymmetric functions, denoted QSym, can be defined over any commutative ring R such as the integers. Quasisymmetric functions are power series of bounded degree in variables   with coefficients in R, which are shift invariant in the sense that the coefficient of the monomial   is equal to the coefficient of the monomial   for any strictly increasing sequence of positive integers   indexing the variables and any positive integer sequence   of exponents.[1] Much of the study of quasisymmetric functions is based on that of symmetric functions.

A quasisymmetric function in finitely many variables is a quasisymmetric polynomial. Both symmetric and quasisymmetric polynomials may be characterized in terms of actions of the symmetric group   on a polynomial ring in   variables  . One such action of   permutes variables, changing a polynomial   by iteratively swapping pairs   of variables having consecutive indices. Those polynomials unchanged by all such swaps form the subring of symmetric polynomials. A second action of   conditionally permutes variables, changing a polynomial   by swapping pairs   of variables except in monomials containing both variables.[2][3] Those polynomials unchanged by all such conditional swaps form the subring of quasisymmetric polynomials. One quasisymmetric polynomial in four variables   is the polynomial

 

The simplest symmetric polynomial containing these monomials is

 

Important bases

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QSym is a graded R-algebra, decomposing as

 

where   is the  -span of all quasisymmetric functions that are homogeneous of degree  . Two natural bases for   are the monomial basis   and the fundamental basis   indexed by compositions   of  , denoted  . The monomial basis consists of   and all formal power series

 

The fundamental basis consists   and all formal power series

 

where   means we can obtain   by adding together adjacent parts of  , for example, (3,2,4,2)   (3,1,1,1,2,1,2). Thus, when the ring   is the ring of rational numbers, one has

 

Then one can define the algebra of symmetric functions   as the subalgebra of QSym spanned by the monomial symmetric functions   and all formal power series   where the sum is over all compositions   which rearrange to the integer partition  . Moreover, we have  . For example,   and  

Other important bases for quasisymmetric functions include the basis of quasisymmetric Schur functions,[4] the "type I" and "type II" quasisymmetric power sums,[5] and bases related to enumeration in matroids.[6][7]

Applications

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Quasisymmetric functions have been applied in enumerative combinatorics, symmetric function theory, representation theory, and number theory. Applications of quasisymmetric functions include enumeration of P-partitions,[8][9] permutations,[10][11][12][13] tableaux,[14] chains of posets,[14][15] reduced decompositions in finite Coxeter groups (via Stanley symmetric functions),[14] and parking functions.[16] In symmetric function theory and representation theory, applications include the study of Schubert polynomials,[17][18] Macdonald polynomials,[19] Hecke algebras,[20] and Kazhdan–Lusztig polynomials.[21] Often quasisymmetric functions provide a powerful bridge between combinatorial structures and symmetric functions.

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As a graded Hopf algebra, the dual of the ring of quasisymmetric functions is the ring of noncommutative symmetric functions. Every symmetric function is also a quasisymmetric function, and hence the ring of symmetric functions is a subalgebra of the ring of quasisymmetric functions.

The ring of quasisymmetric functions is the terminal object in category of graded Hopf algebras with a single character.[22] Hence any such Hopf algebra has a morphism to the ring of quasisymmetric functions.

One example of this is the peak algebra.[23]

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The Malvenuto–Reutenauer algebra[24] is a Hopf algebra based on permutations that relates the rings of symmetric functions, quasisymmetric functions, and noncommutative symmetric functions, (denoted Sym, QSym, and NSym respectively), as depicted the following commutative diagram. The duality between QSym and NSym mentioned above is reflected in the main diagonal of this diagram.

 

Many related Hopf algebras were constructed from Hopf monoids in the category of species by Aguiar and Majahan.[25]

One can also construct the ring of quasisymmetric functions in noncommuting variables.[3][26]

References

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  1. ^ Stanley, Richard P. Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999. ISBN 0-521-56069-1 (hardback) ISBN 0-521-78987-7 (paperback).
  2. ^ Hivert, Florent (2000), "Hecke Algebras, Difference Operators, and Quasi-Symmetric Functions", Advances in Mathematics, 155 (2): 181–238, doi:10.1006/aima.1999.1901
  3. ^ a b Hivert, Florent, Ph.D. Thesis, Marne-la-Vallée
  4. ^ Haglund, J.; Luoto, K.; Mason, S.; van Willigenburg, S. (2011), "Quasisymmetric Schur functions", Journal of Combinatorial Theory, Series A, 118 (2): 463–490, arXiv:0810.2489, doi:10.1016/j.jcta.2009.11.002
  5. ^ Ballantine, Christina; Daughtery, Zajj; Hicks, Angela; Mason, Sarah; Niese, Elizabeth (2020), "On Quasisymmetric Power Sums", Journal of Combinatorial Theory, Series A, 175: 105273, arXiv:1710.11613, doi:10.1016/j.jcta.2020.105273, S2CID 51775423
  6. ^ Luoto, K. (2008), "A matroid-friendly basis for the quasisymmetric functions", Journal of Combinatorial Theory, Series A, 115 (5): 777–798, arXiv:0704.0836, Bibcode:2007arXiv0704.0836L, doi:10.1016/j.jcta.2007.10.003
  7. ^ Billera, L.; Jia, N.; Reiner, V. (2009), "A quasisymmetric function for matroids", European Journal of Combinatorics, 30 (8): 1727–1757, arXiv:math/0606646, Bibcode:2006math......6646B, doi:10.1016/j.ejc.2008.12.007
  8. ^ Stanley, Richard P. Ordered structures and partitions, Memoirs of the American Mathematical Society, No. 119, American Mathematical Society, 1972.
  9. ^ Gessel, Ira. Multipartite P-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983), 289–317, Contemp. Math., 34, Amer. Math. Soc., Providence, RI, 1984.
  10. ^ Gessel, Ira; Reutenauer, Christophe (1993), "Counting permutations with given cycle structure and descent set", Journal of Combinatorial Theory, Series A, 64 (2): 189–215, doi:10.1016/0097-3165(93)90095-P
  11. ^ Shareshian, John; Wachs, Michelle L. (2007), " -Eulerian polynomials: excedance number and major index", Electron. Res. Announc. Amer. Math. Soc., 13 (4): 33–45, arXiv:math/0608274, doi:10.1090/S1079-6762-07-00172-2, S2CID 15394306
  12. ^ Shareshian, John; Wachs, Michelle L. (2010), "Eulerian quasisymmetric functions", Advances in Mathematics, 225 (6): 2921–2966, arXiv:0812.0764, doi:10.1016/j.aim.2010.05.009
  13. ^ Hyatt, Matthew (2012), "Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups", Advances in Applied Mathematics, 48 (3): 465–505, arXiv:1007.0459, Bibcode:2010arXiv1007.0459H, doi:10.1016/j.aam.2011.11.005, S2CID 119118644
  14. ^ a b c Stanley, Richard P. (1984), "On the number of reduced decompositions of elements of Coxeter groups", European Journal of Combinatorics, 5 (4): 359–372, doi:10.1016/s0195-6698(84)80039-6
  15. ^ Ehrenborg, Richard (1996), "On posets and Hopf algebras", Advances in Mathematics, 119 (1): 1–25, doi:10.1006/aima.1996.0026
  16. ^ Haglund, James; The q,t-Catalan numbers and the space of diagonal harmonics. University Lecture Series, 41. American Mathematical Society, Providence, RI, 2008. viii+167 pp. ISBN 978-0-8218-4411-3; 0-8218-4411-3
  17. ^ Billey, Sara C.; Jockusch, William; Stanley, Richard P. (1993), "Some combinatorial properties of Schubert polynomials" (PDF), Journal of Algebraic Combinatorics, 2 (4): 345–374, doi:10.1023/A:1022419800503
  18. ^ Fomin, Sergey; Stanley, Richard P. (1994), "Schubert polynomials and the nil-Coxeter algebra", Advances in Mathematics, 103 (2): 196–207, doi:10.1006/aima.1994.1009
  19. ^ Assaf, Sami (2010), Dual Equivalence Graphs I: A combinatorial proof of LLT and Macdonald positivity, arXiv:1005.3759, Bibcode:2010arXiv1005.3759A
  20. ^ Duchamp, Gérard; Krob, Daniel; Leclerc, Bernard; Thibon, Jean-Yves (1996), "Fonctions quasi-symétriques, fonctions symétriques non commutatives et algèbres de Hecke à  ", C. R. Acad. Sci. Paris, Sér. I Math., 322 (2): 107–112
  21. ^ Billera, Louis J.; Brenti, Francesco (2011), "Quasisymmetric functions and Kazhdan–Lusztig polynomials", Israel Journal of Mathematics, 184: 317–348, arXiv:0710.3965, doi:10.1007/s11856-011-0070-0
  22. ^ Aguiar, Marcelo; Bergeron, Nantel; Sottile, Frank (2006), "Combinatorial Hopf algebras and generalized Dehn–Sommerville relations", Compositio Mathematica, 142 (1): 1–30, arXiv:math/0310016, Bibcode:2003math.....10016A, doi:10.1112/S0010437X0500165X, S2CID 2635356
  23. ^ Stembridge, John R. (1997), "Enriched P-partitions", Trans. Amer. Math. Soc., 349 (2): 763–788, doi:10.1090/S0002-9947-97-01804-7
  24. ^ Malvenuto, Clauda; Reutenauer, Christophe (1995), "Duality between quasi-symmetric functions and the Solomon descent algebra", Journal of Algebra, 177 (3): 967–982, doi:10.1006/jabr.1995.1336
  25. ^ Aguiar, Marcelo; Mahajan, Swapneel Monoidal Functors, Species and Hopf Algebras CRM Monograph Series, no. 29. American Mathematical Society, Providence, RI, 2010.
  26. ^ Bergeron, Nantel; Zabrocki, Mike (2009), "The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free", Journal of Algebra and Its Applications, 8 (4): 581–600, arXiv:math/0509265, doi:10.1142/S0219498809003485, S2CID 18601994
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