2022-11-21

## Quasisymmetric Schur functions

The quasisymmetric Schur functions, $\{\qSchur_\alpha \},$ were introduced by J. Haglund, K. Luoto, S. Mason and S. van Willigenburg [HLMvW11a]. These refine the classical Schur functions in the sense that

\[ \schurS_\lambda(\xvec) = \sum_{\alpha \sim \lambda} \qSchur_\alpha(\xvec), \]where the sum is taken over all compositions $\alpha$ whose parts rearrange to the parts of $\lambda.$ The $\{\qSchur_\alpha \}$ constitute a basis for the space of quasisymmetric functions.

A skew version is introduced in [BLW11], and they show that the skew quasisymmetric Schur functions expand positively into quasisymmetric Schur functions.

The quasisymmetric Schur functions can be described as certain characters of the 0-Hecke algebra, see [TvW15].

The original definition of the quasisymmetric Schur functions is in terms of Demazure atoms:

**Definition.**

Let $\alpha$ be a composition. Then

\[ \qSchur_\alpha(\xvec) \coloneqq \sum_{\gamma : \mathrm{comp}(\gamma) = \alpha} \atom_{\gamma}(\xvec) \]where the sum ranges over all *weak compositions* $\gamma$ which can be obtained by
inserting $0$s between parts of $\alpha.$

We can compute the expansion of $\qSchur_\alpha$ in terms of fundamental quasisymmetric functions by using standard reverse composition tableaux. A semistandard reverse composition tableaux, (SSRCT), of shape $\alpha \vDash n$ is a filling of the diagram $\alpha$ (English notation) with positive integers such that

- rows are weakly decreasing;
- the first column is strictly increasing with the row index;
- for every triple $a,$ $b$ $c$ arranged as
$b$ $c$ $ \vdots$ $a$ we have that $a \leq b$ implies that the box $c$ is present and $a\lt c.$

A standard reverse composition tableaux (SRCT) uses each integer $1,2,\dotsc,n$ exactly once in the filling. The descent set $\des(T)$ of a SRCT is defined as

\[ \des(T) := \{ i : i+1 \text{ appears weakly to the right of } i \}. \]**Example (Example of a SRCT and its descent set).**

(From [TvW15]) The tableau below is an element in $SRCT(3,4,3,2)$ and has descent set $\{1,2,5,8,9,11\}.$

$5$ | $4$ | $2$ | |

$8$ | $7$ | $6$ | $3$ |

$11$ | $10$ | $1$ | |

$12$ | $9$ |

**Theorem.**

We have that the fundamental quasisymmetric expansion of $\qSchur_\alpha$ is given by

\[ \qSchur_\alpha(\xvec) = \sum_{T \in \mathrm{SRTC}(\alpha)} \gessel_{\DES(T)}(\xvec). \]Skew quasisymmetric Schur functions have a similar expansion.

**Example (Fundamental expansion of $\qSchur_{(2,1,3)}$).**

(From [Ex. 2.7, TvW15]) We have that $\qSchur_{(2,1,3)} = \gessel_{(2,1,3)} + \gessel_{(2,2,2)}+\gessel_{(1,2,1,2)},$ corresponding to the three SRTCs

$\mathbf{2}$ | $1$ | |

$\mathbf{3}$ | ||

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

$\mathbf{2}$ | $1$ | |

$\mathbf{4}$ | ||

$6$ | $5$ | $3$ |

$\mathbf{3}$ | $\mathbf{1}$ | |

$\mathbf{4}$ | ||

$6$ | $5$ | $2$ |

where the descents are the bold entries.

## Young quasisymmetric Schur functions

The Young quasisymmetric Schur functions, introduced in [LMW13], are obtained from the quasisymmetric Schur functions via the relation $\yqSchur_\alpha = \rho(\qSchur_{\rev(\alpha)})$ where $\rho$ is a certain involution on quasisymmetric functions.

## Row-strict quasisymmetric Schur functions

The row-strict quasisymmetric Schur functions were first studied by S. Mason and J. Remmel [MR13] in 2013. These quasisymmetric functions are indexed by compositions, and refine the Schur functions as $\schurS_\lambda(\xvec) = \sum_{\alpha \sim \lambda'} \rsqSchur_\alpha(\xvec).$

The relation with the quasisymmetric Schur functions is straightforward; we have that $qSchur_\alpha = \omega(\rsqSchur_\alpha),$ where $\omega$ is a certain involution on quasisymmetric functions.

A product rule for multiplying a row-strict quasisymmetric Schur function with a regular Schur function, and expanding into row-strict Schur functions, is given in [Thm 13, Fer11a]. See also [Thm. 10, Thm. 11, MN15].

## Row-strict Young quasisymmetric Schur functions

Row-strict Young quasisymmetric Schur functions and their skew versions are introduced in [MN15]. They can be defined via the relation with Young quasisymmetric Schur functions: $\rsyqSchur_\alpha = \omega(\yqSchur_\alpha),$ see [Thm. 12, MN15].

A 0-Hecke algebra module whose charateristic is given by $\rsyqSchur_\alpha$ is provided in \cref{https://arxiv.org/pdf/2012.12568.pdf}.

## Dual immaculate Schur functions

The dual immaculate Schur functions (and their skew version),
$\{\diSchur_\alpha\},$ were introduced in [BBSSZ14].
They constitute a basis for the space of quasisymmetric functions.
The original definition is that they are dual to the *immaculate Schur functions*,
which are non-commutative.

The definition is as follows:

\[ \diSchur_\alpha = \sum_{T \in IT(\alpha)} \xvec^T \]where we sum over all *immaculate tableaux* of shape $\alpha.$
Fillings of the diagram $\alpha,$ such that the rows are weakly increasing,
and the first column is *strictly increasing* with respect to the row indexing.
The weight $\xvec^T$ is computed in the same manner as for semistandard Young tableaux.

We have [Prop. 3.37, BBSSZ14] that

\[ \diSchur_\alpha = \sum_{\beta \leq_\ell \alpha} L_{\alpha,\beta} \gessel_\beta \]where $L_{\alpha,\beta}$ are certain non-negative coefficients, and $\leq_\ell$ denotes lexicographic ordering.

*Note:* The Schur functions are not positive in the dual immaculate basis!

**Theorem.**

From [BBSSZ14]. The number of standard immaculate tableaux of shape $\alpha = (\alpha_1,\dotsc,\alpha_\ell),$ with $|\alpha|=n,$ is equal to

\[ \frac{ (n-1)! }{ \prod_{k=1}^{\ell-1}(n-\alpha_1-\alpha_2-\dotsb - \alpha_k) \prod_{k=1}^\ell (\alpha_k - 1)! }. \]Alexandersson 2022: Can one find a q-analog of this and some nice instance of cyclic sieving?

## Row-strict dual immaculate Schur functions

The row-strict dual immaculate Schur functions $\{\rsdiSchur_\alpha\},$ were introduced in [NSWVW22], and are defined in a very similar manner to the dual immaculate Schur functions. They constitute a basis for the space of quasisymmetric functions. For an introduction, see Sheila Sundaram's AlCoVE 2022 video.

The definition is as follows:

\[ \rsdiSchur_\alpha = \sum_{T \in IT(\alpha)} \xvec^T \]where we sum over all *row-strict immaculate tableaux* of shape $\alpha.$
These are fillings of the diagram $\alpha,$ such that the rows are strictly increasing,
and the first column is *weakly increasing* with respect to the row indexing.

One can show that the $\rsdiSchur_\alpha$ are positive in the fundamental quasisymmetric basis.

## Extended Schur functions

The extended Schur functions were introduced by S. Assaf and D. Searles in [AS22b].
They are obtained by taking the stable limit of lock polynomials.
Alternatively, the extended Schur functions are dual
to the *shin polynomials* (see [CFLSX14])
which constitute a basis for the ring of non-commutative symmetric functions.

For a connection with 0-Hecke algebra, see [Sea20], where these are shown to be characters of certain 0-Hecke modules.

## References

- [AS22b] Sami Assaf and Dominic Searles. Kohnert polynomials. Experimental Mathematics, 31(1):93–119, April 2022.
- [BBSSZ14] Chris Berg, Nantel Bergeron, Franco Saliola, Luis Serrano and Mike Zabrocki. A lift of the Schur and Hall–Littlewood bases to non-commutative symmetric functions. Canadian Journal of Mathematics, 66(3):525–565, June 2014.
- [BLW11] C. Bessenrodt, K. Luoto and S. van Willigenburg. Skew quasisymmetric Schur functions and noncommutative Schur functions. Advances in Mathematics, 226(5):4492–4532, March 2011.
- [CFLSX14] John Maxwell Campbell, Karen Feldman, Jennifer Light, Pavel Shuldiner and Yan Xu. A Schur-like basis of NSYM defined by a Pieri rule. The Electronic Journal of Combinatorics, 21(3), September 2014.
- [Fer11a] Jeffrey Ferreira. A Littlewood–Richardson type rule for row-strict quasisymmetric Schur functions. 23th International Conference on Formal Power Series and Algebraic Combinatorics. Discrete Mathematics and Theoretical Computer Science, 2011.
- [HLMvW11a] James Haglund, Kurt W. Luoto, Sarah K. Mason and Stephanie van Willigenburg. Quasisymmetric Schur functions. Journal of Combinatorial Theory, Series A, 118(2):463–490, 2011.
- [LMW13] Kurt Luoto, Stefan Mykytiuk and Stephanie van Willigenburg. An introduction to quasisymmetric Schur functions. Springer New York, 2013.
- [MN15] Sarah K. Mason and Elizabeth Niese. Skew row-strict quasisymmetric Schur functions. Journal of Algebraic Combinatorics, 42(3):763–791, May 2015.
- [MR13] Sarah Mason and Jeffrey Remmel. Row-strict quasisymmetric Schur functions. Annals of Combinatorics, 18(1):127–148, November 2013.
- [NSWVW22] Elizabeth Niese, Sheila Sundaram, Stephanie van Willigenburg, Julianne Vega and Shiyun Wang. 0-hecke modules for row-strict dual immaculate functions. arXiv e-prints, 2022.
- [Sea20] Dominic Searles. Indecomposable 0-Hecke modules for extended Schur functions. Proceedings of the American Mathematical Society, 148(5):1933–1943, February 2020.
- [TvW15] Vasu V. Tewari and Stephanie van Willigenburg. Modules of the 0-Hecke algebra and quasisymmetric Schur functions. Advances in Mathematics, 285:1025–1065, November 2015.