nLab quasi-symmetric function

Quasi-symmetric functions


Quasi symmetric functions are a generalisation of symmetric functions and are closely related to noncommutative symmetric functions.



Let XX be a totally ordered set of indeterminants. Let RR be a ring. A polynomial in R[X]R[X] or a power series in R[[X]]R[ [X] ] is said to be quasi-symmetric if whenever X 1<X 2<<X nX_1 \lt X_2 \lt \dots \lt X_n and Y 1<Y 2<<Y nY_1 \lt Y_2 \lt \dots \lt Y_n are finite sets of indeterminants then the coefficients of X 1 i 1X 2 i 2X n i nX_1^{i_1} X_2^{i_2} \cdots X_n^{i_n} and Y 1 i 1Y 2 i 2Y n i nY_1^{i_1} Y_2^{i_2} \cdots Y_n^{i_n} are the same.


The ring QSymm ^\QSymm^{\hat{}} is defined as the ring of quasi-symmetric power series over \mathbb{Z} in countably many variables. Its subring QSymm\QSymm is defined as the ring of quasi-symmetric polynomials (meaning, power series of bounded degree).


(Copied from noncommutative symmetric function as the two concepts are often studied together.)

Research articles

  • G. Duchamp, F. Hivert, J.-Y. Thibon, Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras, Internat. J. Alg. Comput. 12 (2002), 671–717.
  • I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. in Math. 112 (1995), 218–348, hep-th/9407124
  • Jean-Christophe Novelli, Jean-Yves Thibon, Noncommutative symmetric functions and Lagrange inversion, math.CO/0512570; Noncommutative symmetric functions and an amazing matrix arxiv/1109.1184
  • Lenny Tevlin, Noncommutative Monomial Symmetric Functions, Formal Power Series and Algebraic Combinatorics Nankai University, Tianjin, China, 2007, proceedings pdf
  • D. Krob, J.-Y. Thibon, Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at q=0q = 0, pdf
  • Christos A. Athanasiadis, Power sum expansion of chromatic quasisymmetric functions, arxiv/1409.2595

Long surveys and lecture notes

  • Michael Hazewinkel, Symmetric functions, noncommutative symmetric functions and quasisymmetric functions, pdf
  • V. Retakh and R. Wilson, Advanced Course on Quasideterminants and Universal Localization: pdf (see the part Factorization of Noncommutative Polynomials

    and Noncommutative Symmetric Functions_)

Expositions/short summaries

  • Mike Zabrocki, Non-commutative symmetric functions II: Combinatorics and coinvariants, slides from a talk pdf, III: A representation theoretical approach pdf
  • Lenny Tevlin, Introduction to quasisymmetric and noncommutative symmetric functions, slides, Fields Institute 2010 pdf
category: combinatorics

Last revised on August 23, 2015 at 02:46:50. See the history of this page for a list of all contributions to it.