2022-12-09

## Permuted basement Macdonald E polynomials

The permuted-basement Macdonald polynomials
(also called *relative Macdonald polynomials*, [GR21a]) generalize the
non-symmetric Macdonald polynomials, by introducing an additional parameter $\sigma \in \symS_n,$
the *basement*.
They were introduced in [Fer11b] by J. Ferreira, as eigenpolynomials of certain operators.
Later in [Ale19a], a combinatorial model was introduced.
For some properties of these formulas, see [AS17aAS19a].

A good overview and introduction to this topic, is [GR21aGR21b].

### Definitions

The following section is mainly from [Ale19a].

Let $\sigma = (\sigma_1,\dots,\sigma_n)$ be a list of $n$ different positive integers and let $\alpha=(\alpha_1,\dots,\alpha_n)$ be a weak integer composition. An augmented filling of shape $\alpha$ and basement $\sigma$ is a filling of a Young diagram of shape $(\alpha_1,\dotsc,\alpha_n)$ with positive integers, augmented with a zeroth column filled from top to bottom with $\sigma_1,\dotsc,\sigma_n.$

Note that we use *English notation* rather than the
skyline fillings used in [HHL08Mas09].
For example, the following figure illustrates the difference,
where the English notation is used in the left diagram, while the skyline convention is
used in the right diagram.

$6$ | $5$ | $5$ | |

$5$ | |||

$4$ | |||

$3$ | $3$ | $4$ | $2$ |

$2$ | $2$ | ||

$1$ | $1$ | $6$ |

$2$ | |||||

$6$ | $4$ | $5$ | |||

$1$ | $2$ | $3$ | $5$ | ||

$1$ | $2$ | $3$ | $4$ | $5$ | $6$ |

In the skyline convention, the basement appears in the bottom of the diagram, thus explaining the peculiar choice of terminology.

**Definition.**

Let $F$ be an augmented filling. Two boxes $a,$ $b,$ are attacking if $F(a)=F(b)$ and the boxes are either in the same column, or they are in adjacent columns, with the rightmost box in a row strictly below the other box.

$a$ |

$\vdots$ |

$b$ |

$a$ | |

$\vdots$ | |

$b$ |

A filling is non-attacking if there are no attacking pairs of boxes.

**Definition.**

A triple of type $A$ is an arrangement of boxes, $a,$ $b,$ $c,$
located such that $a$ is immediately to the left of $b,$ and $c$ is somewhere below $b,$
and the row containing $a$ and $b$ is at least as long as the row containing $c.$
Similarly, a triple of type $B$ is an arrangement of boxes, $a,$ $b,$ $c,$
located such that $a$ is immediately to the left of $b,$ and $c$ is somewhere above $a,$
and the row containing $a$ and $b$ is *strictly* longer than the row containing $c.$

A type $A$ triple is an inversion triple if the entries ordered increasingly
form a *counter-clockwise* orientation. Similarly, a type $B$ triple is an inversion triple
if the entries ordered increasingly form a *clockwise* orientation.
If two entries are equal, the one with largest subscript in the figures below
is considered largest.

$a_3$ | $b_1$ |

$\vdots$ | |

$c_2$ |

$c_2$ | |

$\vdots$ | |

$a_3$ | $b_1$ |

If $u = (i,j)$ let $d(u)$ denote $(i,j-1).$ A descent in $F$ is a non-basement box $u$ such that $F(d(u)) \lt F(u).$ The set of descents in $F$ is denoted $\Des(F).$

**Example.**

Here is a non-attacking filling of shape $(4,1,3,0,1)$ and basement $(4,5,3,2,1).$ The bold entries are descents and the underlined entries form a type $A$ inversion triple. There are in total $7$ inversion triples (of type $A$ and $B$).

$\underline{4}$ | $\underline{2}$ | $1$ | $\textbf{2}$ | $4$ |

$5$ | $5$ | |||

$3$ | $3$ | $\textbf{4}$ | $3$ | |

$2$ | ||||

$1$ | $\underline{1}$ |

The leg, $\leg(u),$ of a box $u$ in a diagram is the number of boxes to the right of $u$ in the diagram. The arm, denoted $\arm(u),$ of a box $u = (r,c)$ in a diagram $\alpha$ is defined as the cardinality of the sets

\begin{align*} \{ (r', c) \in \alpha : r < r' \text{ and } \alpha_{r'} \leq \alpha_r \} \text{ and } \\ \{ (r', c-1) \in \alpha : r' < r \text{ and } \alpha_{r'} < \alpha_r \}. \end{align*}The major index of an augmented filling $F$ is defined as

\begin{align*} \maj(F) = \sum_{ u \in \Des(F) } \leg(u)+1. \end{align*}The number of inversions, $\inv(F)$ of a filling is the number of inversion triples of either type.
The number of coinversions, $\coinv(F),$ is the number of type $A$ and type $B$ triples which are *not*
inversion triples.

Let $\mathrm{NAF}_\sigma(\alpha)$ denote all non-attacking fillings of shape $\alpha,$ augmented with the basement $\sigma \in \symS_n,$ and all entries in the fillings are from $[n].$

**Definition.**

Let $\sigma \in \symS_n$ and let $\alpha$ be a weak composition with $n$ parts. The non-symmetric permuted basement Macdonald polynomial $\macdonaldE^\sigma_\alpha(\xvec;q,t)$ is defined as

\begin{equation} \macdonaldE^\sigma_\alpha(\xvec; q,t) = \sum_{ F \in \mathrm{NAF}_\sigma(\alpha)} \xvec^F q^{\maj(F)} t^{\coinv(F)} \prod_{ \substack{ u \in F \\ u \text{ is in the basement or} \\ F(d(u))\neq F(u) }} \frac{1-t}{1-q^{1+\leg(u)} t^{1+\arm(u)}}. \end{equation}The product is over all boxes $u$ in $F$ such that either $u$ is in the basement or $F(d(u))\neq F(u).$

When $\sigma = \omega_0,$ we recover
the non-symmetric Macdonald polynomials
defined in [HHL08], $\macdonaldE_\alpha(\xvec;q,t).$
*There is a slight difference in notation, the index $\alpha$ is reversed compared to [HHL08].*

A formula for the $\macdonaldE^\sigma_\alpha$ using set-valued tableaux can be found in [Thm. 2.2, DR22]. The proof relies on a bijection with alcove walks.

### Properties

**Proposition (See [CMW18]).**

Let $\lambda$ be a partition, of length at most $n.$ Then

\[ \macdonaldP_\lambda(\xvec;q,t) = \sum_{\mu\in \symS_n(\lambda)} \macdonaldE^{\sigma}_{inc(\mu)}(\xvec;q,t), \]where $\sigma$ is the longest permutation which sorts the entries of $\mu$ in increasing order, i.e.,

\[ \mu_{\sigma(1)} \leq \mu_{\sigma(2)} \leq \dotsb \leq \mu_{\sigma(n)}. \]**Conjecture (Olya Mandelshtam, 2019 personal communication).**

We can find coefficients $R_\mu(q,t) \in \setQ(q,t),$
such that for any *composition* composition $\mu,$ we have

Combining [Prop 1.1, GR21b] with [Eq 1.10, GR21b], one can more or less find the expansion

\[ \macdonaldP_\mu(\xvec;q,t) = \sum_{\sigma \in \symS_n} R'_{\mu,\alpha}(q,t) \macdonaldE^{\sigma}_\mu(\xvec;q,t), \]see also [Eq. 5.7.8, Mac96a].

## Quasisymmetric Macdonald E polynomials

A quasisymmetric version of the non-symmetric Macdonald polynomials were introduced in [CHMMW19a].

They specialize to the quasisymmetric Schur polynomials at $q=t=0.$

**Conjecture (Alexandersson 2020).**

Let $\alpha$ be a composition. Then the coefficients $K_{\alpha\gamma}(q)$ in the expansion

\[ \macdonaldEQuasi_\alpha(\xvec;q,0) = \sum_{\gamma} K_{\alpha\gamma}(q) \schurQS_\gamma(\xvec) \]are in $\setN[q].$ Note that this resemblence the fact that $\macdonaldE_\alpha(\xvec;q,0)$ are key-positive, with versions of Kostka–Foulkes polynomials as coefficients.

## References

- [Ale19a] Per Alexandersson. Non-symmetric Macdonald polynomials and Demazure–Lusztig operators. Séminaire Lotharingien de Combinatoire, 76, 2019.
- [AS17a] Per Alexandersson and Mehtaab Sawhney. A major-index preserving map on fillings. Electronic Journal of Combinatorics, 24(4):1–30, 2017.
- [AS19a] Per Alexandersson and Mehtaab Sawhney. Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$. Annals of Combinatorics, 23(2):219–239, May 2019.
- [CHMMW19a] Sylvie Corteel, James Haglund, Olya Mandelshtam, Sarah Mason and Lauren Williams. Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials. arXiv e-prints, 2019.
- [CMW18] Sylvie Corteel, Olya Mandelshtam and Lauren Williams. From multiline queues to Macdonald polynomials via the exclusion process. arXiv e-prints, 2018.
- [DR22] Zajj Daugherty and Arun Ram. Set-valued tableaux for Macdonald polynomials. , 2022.
- [Fer11b] Jeffrey Paul Ferreira. Row-strict quasisymmetric Schur functions, characterizations of Demazure atoms, and permuted basement nonsymmetric Macdonald polynomials. University of California Davis. 2011.
- [GR21a] Weiying Guo and Arun Ram. Comparing formulas for type $gl_n$ Macdonald polynomials. arXiv e-prints, 2021.
- [GR21b] Weiying Guo and Arun Ram. Comparing formulas for type $gl_n$ Macdonald polynomials: supplement. arXiv e-prints, 2021.
- [HHL08] James Haglund, Mark D. Haiman and Nicholas A. Loehr. A combinatorial formula for nonsymmetric Macdonald polynomials. American Journal of Mathematics, 130(2):359–383, 2008.
- [Mac96a] Ian G. Macdonald. Affine Hecke algebras and orthogonal polynomials. In Séminaire Bourbaki. Société Mathématique de France, 1996.
- [Mas09] Sarah K. Mason. An explicit construction of type A Demazure atoms. Journal of Algebraic Combinatorics, 29(3):295–313, 2009.