The symmetric functions catalog

An overview of symmetric functions and related topics

2022-12-19

Gessel quasisymmetric functions

The Gessel quasisymmetric functions $\gessel_\alpha(x),$ also called the fundamental quasisymmetric functions, were introduced by I. Gessel in [Ges84]. They are indexed by integer compositions and constitute a basis for the space of quasisymmetric functions.

Definition (compositions)

For $\alpha \vDash n,$ we define

\begin{equation*} \gessel_\alpha(x) = \sum_{\beta \leq \alpha} M_\alpha(x) = \sum_{\beta \leq \alpha} \sum_{i_1 \lt \dotsb \lt i_n } x_{i_1}^{\alpha_1} \dotsm x_{i_n}^{\alpha_n} \end{equation*}

where $\leq$ is refinement partial order. There is a second way to index the Gessel quasisymmetric functions, by using subsets.

Definition (subsets)

Let $S \subseteq [n-1].$ Then we define $\gessel_{n,S}(x)$ as

\begin{equation*} \gessel_{n,S}(x) = \sum_{\substack{1 \leq b_1 \leq \dotsb \leq b_n \\ i \in S \Rightarrow b_i \lt b_{i+1} }} x_{b_1} \dotsm x_{b_n} \end{equation*}

To translate between the two definitions, note that $\gessel_\alpha(x) = \gessel_{n,S_\alpha}(x)$ where $S_\alpha = \{\alpha_1, \alpha_1+\alpha_2,\dotsc, \alpha_1+\dotsb + \alpha_{n-1}\}.$ Conversely, $\gessel_{n,S}(x) = \gessel_{comp(S)},$ where $comp(S) = (s_1,s_2-s_1,s_3-s_2,\dotsc,s_{\ell} - s_{\ell-1},n-s_{\ell}).$

In terms of the monomial quasisymmetric functions, we have $\gessel_{\alpha}(\xvec) = \sum_{\beta \preceq \alpha} \qmonom_\beta(\xvec),$ where $\preceq$ denotes composition refinement.

Properties

See the page on quasisymmetric functions on how the standard involution on symmetric functions that sends $\schurS_\lambda$ to $\schurS_{\lambda'}$ can be extended to quasisymmetric functions.

Product rule

The product $\gessel_{m,S} \gessel_{n,T}$ has a positive rule as follows. Choose $\sigma \in \symS_m$ and $\tau \in \symS_n$ such that the descent sets of the permutations satisfy $S = \DES(\sigma)$ and $T = \DES(\tau).$ A shuffle of $\sigma$ and $\tau$ is a permutation in $\symS_{m+n}$ such that entries $1,\dotsc,m$ appear in the same relative order as in $\sigma,$ and the entries $m+1,\dotsc,m+n$ appear in the same relative order as in $\tau,$ when $m$ is subtracted from these entries. For example, the shuffles of the permutations $21$ and $12$ are given by

\[ 2134, 2314, 3214, 2341, 3241, 3421. \]

There are of course $\binom{m+n}{m}$ shuffles in total. We then have that

\[ \gessel_{m,S}(\xvec) \gessel_{n,T}(\xvec) = \sum_{\pi \text{ shuffle of } \sigma,\tau } F_{m+n,\DES(\pi)}(\xvec). \]

Specializations

We have that for $\length(S)\lt m,$

\begin{equation*} \gessel_{n,S}(1^m) = \binom{n+m-\length(S)-1}{n} \end{equation*}

and for $\length(S)\lt m,$

\begin{equation*} \gessel_{n,S}(1,q,q^2,\dotsc,q^m) = q^{m \length(S)+m-|S|-1}\qbinom{n+m-1-\length(S)}{m}_q \end{equation*}

where $|S|$ denotes the sum of the elements.

Egge–Loehr–Warrington and the slinky rule

In [ELW10], the following lifting of the fundamental quasisymmetric functions to the Schur functions was given. An alternative interpretation of this rule is proved in [GR18]. A short proof by Ira Gessel is given in [Ges18].

Suppose $X(\xvec)$ is a symmetric function, with fundamental quasisymmetric expansion

\[ X(\xvec) = \sum_\alpha c_\alpha \gessel_\alpha(\xvec) \]

Then

\[ X(\xvec) = \sum_\alpha c_\alpha \schurS_\alpha(\xvec). \]

The composition indices in $\schurS_\alpha(\xvec)$ should be computed via the Jacobi–Trudi identity using the complete homogeneous symmetric functions. By using determinant formulas, the compositions can be "straighened" into partitions, which can be done using the slinky rule as it can be described pictorially as the parts of the compositions falling down when $\alpha$ is illustrated in French notation, see [ELW10].

Example (The slinky rule).

We use the slinky rule on the composition $(4,3,8).$

        
        
        
       
       
       
      
      
      

The resulting shape is $(6,5,4)$ and the sign is $(-1)^2$ as the height of the ribbon of size $8$ ends two levels below its starting row. That is, the height of a ribbon is computed in the same manner as in the Murnaghan–Nakayama rule. Hence, $\schurS_{438}=\schurS_{654}.$

As a second example, $\schurS_{25115} = -\schurS_{43322}$ according to the slinky rule.

     
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Note that the sign is $(-1)^{2}(-1)^{1} = -1.$

Finally, $\schurS_{12} = 0,$ since there are not enough boxes to perform a slinky move. Similarly, $\schurS_{2115} = 0,$ since the slinky rule does not give a partition.

     
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Relationship with the powersum basis

Athanasiadis [Ath15], (also implicitly in [AR15]) prove the following: Suppose $X(x)$ is a symmetric function, such that

\begin{equation*} X(\xvec) = \sum_{S \subseteq [n-1]} a_S \gessel_{n,S}(\xvec). \end{equation*}

Then

\begin{equation*} X(\xvec) = \sum_{\lambda \vdash n} z_{\lambda}^{-1} \powerSum_{\lambda}(\xvec) \sum_{S\in U_\lambda} (-1)^{|S\setminus S_\lambda|} a_S, \end{equation*}

where $U_\lambda$ is the set of $\lambda$-unimodal subsets of $[n-1]$ and $S_\lambda \coloneqq \{ s_1,s_2,\dotsc,s_{\ell-1} \}.$

Here, $s_i$ is defined as $\lambda_1+\dotsb + \lambda_{i}$ and a set $A \subseteq [n-1]$ is $\lambda$-unimodal if for $0\leq i \lt \ell$ the intersection of $A$ with the intervals $\{s_i+1,s_i+2,\dotsc,s_{i+1}-1\}$ is a prefix of the latter.

Relationship with a quasisymmetric powersum basis

In [BDHMN20], two versions of quasisymmetric powersum bases are given. They give the fundamental quasisymmetric expansion of these two quasisymmetric powersum bases.

In [AS19b], we provide the quasisymmetric powersum expansion of $\gessel_\alpha(\xvec).$ We have

\begin{equation*} \gessel_{n,S}(\xvec) = \sum_{\alpha} \frac{\qPsi_\alpha(\xvec)}{z_\alpha} (-1)^{|S\setminus S_\alpha|} \end{equation*}

where the sum is taken over all compositions $\alpha$ such that the set $S$ is $\alpha$-unimodal. This formula implies Athanasiadis formula above.

The 0-Hecke algebra

The 0-Hecke algebra $H_n(0)$ in type $A$ is the $\setC$-algebra generated by $T_i,$ $i=1,\dotsc,n-1,$ subject to the relations

\[ T_i^2 = T_i \qquad T_iT_{i+1}T_i = T_{i+1}T_iT_{i+1}\quad T_{j}T_i = T_{i}T_j \text{ whenever } |i-j|\geq 2. \]

Note that $T_{s_1}\dotsc T_{s_{\ell}}$ is independent for any choice of reduced word $w = s_1\dotsm s_{\ell}$ with $w \in \symS_n.$ It follows that the dimension of $H_n(0)$ is $n!.$

As an example of such operators, the divided difference operators generating the key polynomials are such operators. This also highlights the connection with the non-symmetric Macdonald polynomials.

Similar to how $\symS_n$-modules decompose into irreducible modules indexed by partitions, we can completely characterize indecomposable $H(0)$-modules, see [KT97]. There are $2^{n-1}$ irreducible representations $F^\alpha,$ indexed by integer compositions of $n.$

The map $\frobChar$ on a $H_n(0)$-module $M,$ sending $F^\alpha \to \gessel_\alpha$ is the quasisymmetric characteristic of $M.$ This is an analogue of the Frobenius characteristic.

For example, in [TvW15], the authors show that the quasisymmetric Schur functions are given as the quasisymmetric characteristic of certain $H_n(0)$-modules.

See also [Hua13] for some examples of graded vector spaces.

References