The symmetric functions catalog

An overview of symmetric functions and related topics

2020-04-17

Quasisymmetric functions

Quasisymmetric functions were formally introduced by I. Gessel in [Ges84]. Earlier work on P-partitions anticipated this development. For an introduction to quasisymmetric functions, see [LMvW13].

A function $f$ is quasisymmetric if for every composition $\alpha$ of length $\ell,$ the coefficient of $x_1^{\alpha_1} \dotsm x_\ell^{\alpha_\ell}$ is the same as the coefficient of $x_{i_1}^{\alpha_1} \dotsm x_{i_\ell}^{\alpha_\ell},$ for any $0 \lt i_1 \lt i_2 \lt \dotsb \lt i_{\ell}.$ The set of quasisymmetric functions form a graded ring, $\spaceQSym.$

Quasisymmetric functions of degree $n$ are usually indexed by either integer compositions of $n,$ or subsets of $[n-1].$ Given a composition $\alpha \vDash n$ with $\ell$ parts, define

\[ S_\alpha \coloneqq \{\alpha_1, \alpha_1+\alpha_2,\dotsc, \alpha_1+\alpha_2+\dotsb+\alpha_{\ell-1}\}. \]

The bijection $\alpha \mapsto S_\alpha$ maps compositions of $n$ to subsets of $[n-1].$ The partial order $\alpha \leq \beta$ denotes refinement. That is, $\beta$ can be obtained from $\alpha$ by adding consecutive parts of $\alpha.$ When $\alpha \leq \beta,$ we illustrate this relationship with bars between parts of $\alpha,$ such that parts between bars add to parts of $\beta.$

Example (Refinement and bars).

Taken from [AS19b].

\[ 112|341|21|34|2 \quad\text{corresponds to}\quad \alpha = 11234121342, \quad \beta = 48372. \]

Monomial quasisymmetric functions

Given a composition $\alpha$ with $\ell$ parts, we define the monomial quasisymmetric functions as

\[ \qmonom_\alpha(\xvec) \coloneqq \sum_{i_1 < i_2 < \dotsb < i_\ell} x_{i_1}^{\alpha_1} x_{i_2}^{\alpha_2} \dotsm x_{i_\ell}^{\alpha_\ell}. \]

The functions $\qmonom_\alpha$ constitute a basis for the space of homogeneous quasisymmetric functions of degree $n$ as $\alpha$ ranges over all compositions of $n.$

The monomial quasisymmetric functions refine the monomial symmetric functions,

\[ \monomial_{\lambda}(\xvec) = \sum_{\alpha \sim \lambda}\qmonom_{\alpha}(\xvec) \]

where we sum over all compositions $\alpha$ that are a permutation of $\lambda.$

Powersum quasisymmetric functions (Psi)

There are two quasisymmetric refinements of the power-sum symmetric functions introduced in [BDHMN17], denoted $\qPsi_\alpha$ and $\qPhi_\alpha.$

Given a pair of compositions of $n,$ $\alpha \leq \beta,$ related by

\begin{equation*} \alpha_{11} \alpha_{12} \dotsc \alpha_{1,i_1}| \alpha_{21} \alpha_{22} \dotsc \alpha_{2,i_2} | \dotsb | \alpha_{k1} \alpha_{k2} \dotsc \alpha_{k,i_k} \end{equation*}

let

\begin{equation*} \pi(\alpha,\beta) \coloneqq \prod_{j=1}^k (\alpha_{j1})(\alpha_{j1}+\alpha_{j2})\dotsb (\alpha_{j1}+\alpha_{j2}+\dotsb +\alpha_{j,i_j}). \end{equation*}

The quasisymmetric power sum $\qPsi_\alpha$ is defined as

\begin{equation*} \qPsi_\alpha(\xvec) \coloneqq z_\alpha \sum_{\beta \geq \alpha } \frac{1}{\pi(\alpha,\beta)} \qmonom_\beta(\xvec). \end{equation*}

For example, $\Psi_{231} = \frac{1}{10}\qmonom_6 + \frac{1}{4}\qmonom_{24}+ \frac{3}{5}\qmonom_{51}+\qmonom_{231}.$ It was shown in [Thm. 3.11, BDHMN17] that quasisymmetric power sums refine the usual power sums as

\[ \powerSum_{\lambda}(\xvec) = \sum_{\alpha \sim \lambda}\qPsi_{\alpha}(\xvec). \]

The $\qPsi_\alpha$ have a nice relationship with Gessel quasisymmetric functions.

Powersum quasisymmetric functions (Phi)

Involution on symmetric functions

The standard involution on symmetric functions that sends $\schurS_\lambda$ to $\schurS_{\lambda'}$ can be extended to quasisymmetric functions in two ways, here defined via the Gessel quasisymmetric functions.

\[ \omega \gessel_{n,S}(x) = \gessel_{n,[n-1]\setminus (n-S)}(x) \qquad \text{ or } \qquad \psi \gessel_{n,S}(x) = \gessel_{n, [n-1]\setminus S}(x). \]

Here, $n-S$ is the set $\{n-s : s \in S\}.$

In [BDHMN17], it is shown that $ \omega\left( \qPsi_\alpha \right) = (-1)^{|\alpha|-\length(\alpha)}\qPsi_{\alpha^r}, $ where $\alpha^r$ denotes the reverse of $\alpha.$

References