2022-12-28
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 $\beta.$
Taken from [AS19b].
\[ 112|341|21|34|2 \quad\text{corresponds to}\quad \alpha = 11234121342, \quad \beta = 48372. \]The most prevalent quasisymmetric function is perhaps the Gessel quasisymmetric functions.
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 [BDHMN20], 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 [BDHMN20] 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)
Powersum quasisymmetric functions (p)
In [AWW21], the authors introduce a third quasisymmetric refinement of the power-sum basis, denoted $\qRho_\alpha.$ They refer to this basis as the combinatorial quasisymmetric power-sum basis.
Let $\alpha$ be a composition and let $\lambda$ be the partition obtained from $\alpha$ by rearranging the parts in decreasing order. The monomial expansion of $\qRho_\alpha$ is given by
\[ \qRho_\alpha(\xvec) = \sum_{\beta} R_{\alpha \beta} \qmonom_\beta(\xvec) \]where $R_{\alpha \beta}$ is the number of ordered set-partitions (see A000670)
\[ \gamma_1 | \gamma_2 | \dotsb | \gamma_k \]with the following two properties. The word
\[ \lambda_{\gamma_{11}}, \lambda_{\gamma_{12}},\dotsc,\; \lambda_{\gamma_{21}}, \lambda_{\gamma_{22}},\dotsc,\; \dotsc, \lambda_{\gamma_{k1}}, \lambda_{\gamma_{k2}},\dotsc, \]is equal to $\alpha,$ and $ \lambda_{\gamma_{i1}}+ \lambda_{\gamma_{i2}}+\dotsb = \beta_i $ for all $i.$ For example, if $\lambda = 322211,$ then the set-partition $235|16|4$ contributes to $R_{\alpha \beta}$ where
\[ \alpha = (2,2,1,3,1,2), \qquad \beta = (5,4,2). \]The original article [AWW21] uses the enumeration of certain matrices to define the $R_{\alpha \beta},$ but the above definition is equivalent to theirs.
The following table shows the monomial expansion of $\qRho_\alpha.$
$\alpha$ | $\qRho_\alpha$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$4$ | $\qmonom_4$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$13$ | $\qmonom_{13}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$22$ | $\qmonom_4+2 \qmonom_{22}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$31$ | $\qmonom_4+\qmonom_{31}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$112$ | $\qmonom_{22}+2 \qmonom_{112}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$121$ | $2 \qmonom_{13}+2 \qmonom_{121}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$211$ | $\qmonom_4+\qmonom_{22}+2 \qmonom_{31}+2 \qmonom_{211}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
$1111$ | $\qmonom_4+4 \qmonom_{13}+6 \qmonom_{22}+4 \qmonom_{31}+12 \qmonom_{112}+12 \qmonom_{121}+12 \qmonom_{211}+24 \qmonom_{1111}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The authors also give formulas for computing product and coproduct.
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 [BDHMN20], it is shown that $ \omega\left( \qPsi_\alpha \right) = (-1)^{|\alpha|-\length(\alpha)}\qPsi_{\alpha^r}, $ where $\alpha^r$ denotes the reverse of $\alpha.$ The same property holds for $\qRho_\alpha,$ see [Thm. 5.7, AWW21].
References
- [AS19b] Per Alexandersson and Robin Sulzgruber. P-partitions and P-positivity. International Mathematics Research Notices, July 2019.
- [AWW21] Farid Aliniaeifard, Victor Wang and Stephanie van Willigenburg. $p$-partition power sums. arXiv e-prints, 2021.
- [BDHMN20] Cristina Ballantine, Zajj Daugherty, Angela Hicks, Sarah Mason and Elizabeth Niese. On quasisymmetric power sums. Journal of Combinatorial Theory, Series A, 175:105273, October 2020.
- [Ges84] Ira M. Gessel. Multipartite $P$-partitions and inner products of skew Schur functions. In Combinatorics and algebra (Boulder, Colo., 1983). Amer. Math. Soc., Providence, RI, 1984.
- [LMvW13] Kurt W. Luoto, Stefan Mykytiuk and Stephanie van Willigenburg. An introduction to quasisymmetric Schur functions: Hopf algebras, quasisymmetric functions, and Young composition tableaux (springerbriefs in mathematics). Springer, 2013.