#Eulerian quasisymmetric functions
The Eulerian quasisymmetric functions are in fact 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 defined as the word obtained from \(\sigma\) by replacing \(\sigma_i\) with \(\overline{\sigma}_i\) whenever \(\sigma_i \gt i.\) Let \[\DEX(\sigma) = \{ i \in [n-1] : \overline{\sigma}(i+1)\gt{}\overline{\sigma}(i) \}\] 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
For example, \[\begin{aligned} \eulerianQ_{33,2}(\xvec) &= \gessel_{\emptyset,6} + \gessel_{\{2\},6} + \gessel_{\{3\},6} + \gessel_{\{4\},6} + \gessel_{\{1,3\},6} \\ &+\gessel_{\{1,4\},6} + \gessel_{\{1,5\},6} + \gessel_{\{2,4\},6} + \gessel_{\{2,5\},6} + \gessel_{\{3,5\},6}. \end{aligned}\]
#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 Frobenius 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.
Conjecture (Alexandersson (2018)).
It seems like all \(\eulerianQ_{\lambda,j}\) are \(\powerSum_\mu\)-positive.
The stronger statement, positivity in the complete homogeneous 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 \[\left(\symS_{\lambda,j}, C_n, \sum_{\sigma \in \symS_{\lambda,j}} q^{\maj(\sigma) - \exc(\sigma)} \right)\] exhibits the cyclic sieving phenomenon, where \(C_n\) acts by conjugation.
Conjecture (Alexandersson (2018)).
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 phenomenon.
#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 allows 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 [Lin13], the author shows that these are symmetric functions. These are believed to satisfy a cyclic sieving phenomenon as above. V.-D. Moustakas studies specializations of colored quasisymmetric functions and obtains Euler–Mahonian identities for colored permutations [Mou21]. The paper uses colored fundamental quasisymmetric functions as the main basis.
Question
Can the above be generalized to wreath product symmetric functions?
Bibliography
- [HW12]Anthony Henderson and Michelle L. Wachs. Unimodality of Eulerian quasisymmetric functions. Journal of Combinatorial Theory, Series A, 119(1):135–145, January 2012.
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@article{HendersonWachs2012, doi = {10.1016/j.jcta.2011.07.004}, url2 = {https://doi.org/10.1016/j.jcta.2011.07.004}, year = {2012}, month = jan, publisher = {Elsevier {BV}}, volume = {119}, number = {1}, pages = {135--145}, author = {Anthony Henderson and Michelle L. Wachs}, title = {Unimodality of {E}ulerian quasisymmetric functions}, journal = {Journal of Combinatorial Theory, Series A} } - [Hya12]Matthew Hyatt. Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups. Advances in Applied Mathematics, 48(3):465–505, March 2012.
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@article{Hyatt2012, doi = {10.1016/j.aam.2011.11.005}, url2 = {https://doi.org/10.1016/j.aam.2011.11.005}, year = {2012}, month = mar, publisher = {Elsevier {BV}}, volume = {48}, number = {3}, pages = {465--505}, author = {Matthew Hyatt}, title = {Eulerian quasisymmetric functions for the type {B} {C}oxeter group and other wreath product groups}, journal = {Advances in Applied Mathematics} } - [Lin13]Zhicong Lin. On some colored Eulerian quasisymmetric functions. arXiv:1309.6368, 2013.
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@article{Lin2013, Author = {Zhicong Lin}, Title = {On some colored {E}ulerian quasisymmetric functions}, journal = {arXiv e-prints}, Year = {2013}, Eprint = {1309.6368}, url={https://arxiv.org/abs/1309.6368} } - [Mou21]Vassilis-Dionyssis Moustakas. Specializations of colored quasisymmetric functions and Euler–Mahonian identities. Advances in Applied Mathematics, 127:102177, 2021.
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@article{Moustakas2021, author = {Moustakas, Vassilis-Dionyssis}, title = {Specializations of colored quasisymmetric functions and {E}uler--{M}ahonian identities}, year = {2021}, journal = {Advances in Applied Mathematics}, volume = {127}, pages = {102177}, publisher = {Elsevier BV}, doi = {10.1016/j.aam.2021.102177}, url = {http://dx.doi.org/10.1016/j.aam.2021.102177}, issn = {0196-8858}, eprint = {2003.07879} }