The symmetric functions catalog

An overview of symmetric functions and related topics

2019-05-20

Eulerian quasisymmetric functions

The Eulerain quasisymmetric functions are actually symmetric. They were introduced in [SW10].

Definition

Let $\bar{1} \lt \bar{2} \lt \dotsb \lt \bar{n} \lt 1 \lt 2 \lt \dotsb \lt n.$ For $\sigma \in \symS_n,$ let $\overline{\sigma}$ be define as the word obtained from $\sigma$ by replacing $\sigma_i$ with $\overline{\sigma}_i$ whenever $\sigma_i \gt i.$ Let

$\DEX(\sigma) = \{ i \in [n-1] : \overline{\sigma}(i+1)>\overline{\sigma}(i) \}$

that is, the set of descents in the modified alphabet.

The Eulerian quasisymmetric function indexed by $n$ and $j$ is then described using the Gessel quasisymmetric functions:

$\eulerianQ_{\lambda,j}(\xvec) \coloneqq \sum_{\substack{\sigma \in \symS_n \\ \type(\sigma) = \lambda}} \gessel_{n,\DEX(\sigma)}(\xvec).$
Example.

We have that

\begin{align*} \eulerianQ_{33,2}(\xvec) &= \gessel_{\emptyset,6} + \gessel_{\{2\},6} + \gessel_{\{3\},6} + \gessel_{\{4\},6} + \gessel_{\{1,3\},6} \\ &+\gessel_{\{1,4\},6} + \gessel_{\{1,5\},6} + \gessel_{\{2,4\},6} + \gessel_{\{2,5\},6} + \gessel_{\{3,5\},6}. \end{align*}

Plethystic formula

Theorem (See [SW10]).

Let $\lambda$ have $m_i$ parts of size $i.$ Then

$\sum_{j=0}^{|\lambda|-1} \eulerianQ_{\lambda,j}(\xvec)t^j = \prod_{i \geq 1} \completeH_{m_i}\left[ \sum_{j=0}^{i-1} \eulerianQ_{(i),j} t^j \right].$

Schur positivity

In [HW12] it is proved that $\eulerianQ_{\lambda,j} = \frobChar V_{\lambda,j},$ that is, it is the Frobenous characteristic of a certain vector space $V_{\lambda,j}$ spanned by forests of marked trees. This implies Schur positivity.

Power-sum positivity

In [SSW11], it is proved that $\eulerianQ_{(n),j}$ expand positively in the power-sum basis.

It seems like all $\eulerianQ_{\lambda,j}$ are $\powerSum_\mu$-positive.

The stronger statement, being positive in the complete homogenous basis, is not true. For example,

$\eulerianQ_{(6),3} = \completeH_{321} - \completeH_{411}+2\completeH_{42} + \completeH_{51}.$

Cyclic sieving phenomena

Let $\symS_{\lambda,j}$ be the subset of permutations in $\symS_n$ with cycle type $\lambda$ and exactly $j$ excedances. In [SSW11], it is shown that

$\left(\symS_{\lambda,j}, C_n, \sum_{\sigma \in \symS_{\lambda,j}} q^{\maj(\sigma) - \exc(\sigma)} \right)$

exhibit the cyclic sieving phenomena, where $C_n$ act by conjugation.

Let $\lambda$ be a partition of $n.$ Then

$\eulerianQ_{\lambda,j}(1,q,q^2,\dotsc,q^{n-1})$

evaluates to non-negative integers at $q = e^{2 \pi i k/n},$ and there should be some action $C_n$ that completes this to a cyclic sieving phenomena.

Colored Eulerian quasisymmetric functions

In [Hya12], the notion of Eulerian quasisymmetric functions is generalized to the wreath product of a cyclic group and the symmetric group, also known as the group of colored permutations. The notion of descents, fixed points and so on generalizes naturally to this setting, and thus allow for a definition of colored Eulerian quasisymmetric functions.

A special case would be to choose the cyclic group to be $\setZ_2,$ which can then be called a type $B$ Eulerian quasisymmetric function.

In this preprint they are shown to be symmetric. These are believed to be symmetric and satsify a cyclic sieving phenomena as above.