The symmetric functions catalog

An overview of symmetric functions and related topics

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

  1. rows are weakly decreasing;
  2. the first column is strictly increasing with the row index;
  3. 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