Quasisymmetric function

For quasisymmetric functions in the theory of metric spaces or complex analysis, see quasisymmetric map.

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

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 x_1,x_2,x_3, \dots with coefficients in R, which are shift invariant in the sense that the coefficient of the monomial x_1^{\alpha_1}x_2^{\alpha_2} \cdots x_k^{\alpha_k} is equal to the coefficient of the monomial x_{i_1}^{\alpha_1} x_{i_2}^{\alpha_2}\cdots x_{i_k}^{\alpha_k} for any strictly increasing sequence of positive integers i_1< i_2< \cdots < i_k indexing the variables and any positive integer sequence (\alpha_1, \alpha_2,\ldots,\alpha_k) 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 S_n^{} on a polynomial ring in n^{} variables x_1^{},\dots, x_n. One such action of S_n permutes variables, changing a polynomial p(x_1^{},\dots,x_n) by iteratively swapping pairs (x_i^{}, x_{i+1}) of variables having consecutive indices. Those polynomials unchanged by all such swaps form the subring of symmetric polynomials. A second action of S_n conditionally permutes variables, changing a polynomial p(x_1,\ldots,x_n) by swapping pairs (x_i^{}, x_{i+1}) of variables except in monomials containing both variables. Those polynomials unchanged by all such conditional swaps form the subring of quasisymmetric polynomials. One quasisymmetric function in four variables is the polynomial

x_1^2 x_2 x_3 + x_1^2 x_2 x_4 + x_1^2 x_3 x_4 + x_2^2 x_3 x_4. \,

The simplest symmetric function containing all of these monomials is

\begin{align} x_1^2 x_2 x_3 + x_1^2 x_2 x_4 + x_1^2 x_3 x_4 + x_2^2 x_3 x_4 + x_1 x_2^2 x_3 + x_1 x_2^2 x_4 + x_1 x_3^2 x_4 + x_2 x_3^2 x_4 \\ {} + x_1 x_2 x_3^2 + x_1 x_2 x_4^2 + x_1 x_3 x_4^2 + x_2 x_3 x_4^2. \, \end{align}

Important bases

QSym is a graded R-algebra, decomposing as

\mathrm{QSym} = \bigoplus_{n \ge 0} \mathrm{QSym}_n, \,

where \mathrm{QSym}_n is the R-span of all quasisymmetric functions that are homogeneous of degree n. Two natural bases for \mathrm{QSym}_n are the monomial basis \{M_{\alpha} \} and the fundamental basis \{F_{\alpha} \} indexed by compositions \alpha = (\alpha_1, \alpha_2, \ldots , \alpha_k) of n, denoted \alpha \vDash n. The monomial basis consists of M_0=1 and all formal power series

M_{\alpha} = \sum_{i_1 < i_2 < \cdots < i_k} x_{i_1}^{\alpha_1} x_{i_2}^{\alpha_2} \cdots x_{i_k}^{\alpha_k}. \,

The fundamental basis consists F_0=1 and all formal power series

F_\alpha = \sum_{\alpha \succeq \beta} M_\beta, \,

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

\mathrm{QSym}_n = \mathrm{span}_{\mathbb{Q}} \{ M_\alpha | \alpha \vDash n \} = \mathrm{span}_{\mathbb{Q}} \{ F_{\alpha} | \alpha \vDash n \}. \,

Then one can define the algebra of symmetric functions \Lambda = \Lambda _0 \oplus \Lambda _1 \oplus \cdots as the subalgebra of QSym spanned by the monomial symmetric functions m_0=1 and all formal power series m_{\lambda} = \sum M_{\alpha}, where the sum is over all compositions \alpha which rearrange to the partition \lambda. Moreover, we have \Lambda_n = \Lambda \cap \mathrm{QSym}_n. For example, F_{(1,2)}=M_{(1,2)}+M_{(1,1,1)} and m_{(2,1)}=M_{(2,1)}+M_{(1,2)}.

Other important bases for quasisymmetric functions include the basis of quasisymmetric Schur functions,[2] and bases related to enumeration in matroids.[3][4]

Applications

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,[5][6] permutations,[7][8][9][10] tableaux,[11] chains of posets,[11][12] reduced decompositions in finite Coxeter groups (via Stanley symmetric functions),[11] and parking functions.[13] In symmetric function theory and representation theory, applications include the study of Schubert polynomials,[14][15] Macdonald polynomials,[16] Hecke algebras,[17] and Kazhdan-Lusztig polynomials.[18] Often quasisymmetric functions provide a powerful bridge between combinatorial structures and symmetric functions.

Related algebras

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.[19] Hence any such Hopf algebra has a morphism to the ring of quasisymmetric functions.

One example of this is the peak algebra.[20]

Other Related Algebras: The Malvenuto-Reutenauer algebra[21] 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 .[22]

One can also construct the ring of quasisymmetric functions in noncommuting variables.[23][24]

External links

References

  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. ↑ Haglund, J.; Luoto, K.; Mason, S.; van Willigenburg, S. (2011), "Quasisymmetric Schur functions", J. Combin. Theory Ser. A 118 (2): 463–490, doi:10.1016/j.jcta.2009.11.002
  3. ↑ Luoto, K. (2008), "A matroid-friendly basis for the quasisymmetric functions", J. Combin. Theory Ser. A 115 (5): 777–798, arXiv:0704.0836, Bibcode:2007arXiv0704.0836L, doi:10.1016/j.jcta.2007.10.003
  4. ↑ Billera, L.; Jia, N.; Reiner, V. (2009), "A quasisymmetric function for matroids", European J. Combin. 30 (8): 1727–1757, arXiv:math/0606646, Bibcode:2006math......6646B, doi:10.1016/j.ejc.2008.12.007
  5. ↑ Stanley, Richard P. Ordered structures and partitions, Memoirs of the American Mathematical Society, No. 119, American Mathematical Society, 1972.
  6. ↑ 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.
  7. ↑ Gessel, Ira; Reutenauer, Christophe (1993), "Counting permutations with given cycle structure and descent set", J. Combin. Theory Ser. A 64 (2): 189–215, doi:10.1016/0097-3165(93)90095-P
  8. ↑ Shareshian, John; Wachs, Michelle L. (2007), "q-Eulerian polynomials: excedance number and major index", Electron. Res. Announc. Amer. Math. Soc. 13 (4): 33–45, doi:10.1090/S1079-6762-07-00172-2
  9. ↑ Shareshian, John; Wachs, Michelle L. (2010), "Eulerian quasisymmetric functions", Advances in Mathematics 225 (6): 2921–2966, doi:10.1016/j.aim.2010.05.009
  10. ↑ Hyatt, Matthew (2010), Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups 1007, p. 459, arXiv:1007.0459, Bibcode:2010arXiv1007.0459H
  11. ↑ 11.0 11.1 11.2 Stanley, Richard P. (1984), "On the number of reduced decompositions of elements of Coxeter groups", European J. Combin. 5 (4): 359–372, doi:10.1016/s0195-6698(84)80039-6
  12. ↑ Ehrenborg, Richard (1996), "On posets and Hopf algebras", Adv. Math. 119 (1): 1–25, doi:10.1006/aima.1996.0026
  13. ↑ 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
  14. ↑ Billey, Sara C.; Jockusch, William; Stanley, Richard P. (1993), "Some combinatorial properties of Schubert polynomials", Journal of Algebraic Combinatorics 2 (4): 345–374, doi:10.1023/A:1022419800503
  15. ↑ 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
  16. ↑ Assaf, Sami, Dual Equivalence Graphs I: A combinatorial proof of LLT and Macdonald positivity, arXiv:1005.3759, Bibcode:2010arXiv1005.3759A
  17. ↑ Duchamp, Gérard; Krob, Daniel; Leclerc, Bernard; Thibon, Jean-Yves (1996), "Fonctions quasi-symétriques, fonctions symétriques non commutatives et algèbres de Hecke à q=0", C. R. Acad. Sci. Paris, Sér. I Math. 322 (2): 107–112
  18. ↑ Billera, Louis J.; Brenti, Francesco (2007), Quasisymmetric functions and Kazhdan-Lusztig polynomials 0710, p. 3965, arXiv:0710.3965, Bibcode:2007arXiv0710.3965B
  19. ↑ 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
  20. ↑ Stembridge, John R. (1997), "Enriched P-partitions", Trans. Amer. Math. Soc. 349 (2): 763–788, doi:10.1090/S0002-9947-97-01804-7
  21. ↑ 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
  22. ↑ Aguiar, Marcelo; Mahajan, Swapneel Monoidal Functors, Species and Hopf Algebras CRM Monograph Series, no. 29. American Mathematical Society, Providence, RI, 2010.
  23. ↑ Hivert, Florent, Ph.D. Thesis, Marne-la-Vallée
  24. ↑ Bergeron, Nantel; Zabrocki, Mike (2009), "The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free", J. Algebra Appl. 8 (4): 581–600, doi:10.1142/S0219498809003485