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#Gessel quasisymmetric functions

The Gessel quasisymmetric functions \(\gessel_\alpha(\xvec),\) 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 \[\gessel_\alpha(\xvec) = \sum_{\beta \leq \alpha} M_\beta(\xvec) = \sum_{\beta \leq \alpha} \sum_{i_1 \lt \dotsb \lt i_{\length(\beta)}} x_{i_1}^{\beta_1} \dotsm x_{i_{\length(\beta)}}^{\beta_{\length(\beta)}}\] where \(\leq\) is the 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 \(\gessel_{n,S}(\xvec)\) as \[\gessel_{n,S}(\xvec) = \sum_{\substack{1 \leq b_1 \leq \dotsb \leq b_n \\ i \in S \Rightarrow b_i \lt b_{i+1} }} x_{b_1} \dotsm x_{b_n}\]

To translate between the two definitions, note that \(\gessel_\alpha(\xvec) = \gessel_{n,S_\alpha}(\xvec),\) where, for \(\ell=\length(\alpha),\) \(S_\alpha = \{\alpha_1, \alpha_1+\alpha_2,\dotsc, \alpha_1+\dotsb + \alpha_{\ell-1}\}.\) Conversely, \(\gessel_{n,S}(\xvec) = \gessel_{comp(S)},\) 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 for 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 the following positive rule. 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 \(\tau\) after \(m\) 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 \(\binom{m+n}{m}\) shuffles in total. We then have \[\gessel_{m,S}(\xvec) \gessel_{n,T}(\xvec) = \sum_{\pi \text{ shuffle of } \sigma,\tau } \gessel_{m+n,\DES(\pi)}(\xvec).\]

#Specializations

For \(\length(S)\lt m,\) we have \[\gessel_{n,S}(1^m) = \binom{n+m-\length(S)-1}{n}\] and for \(\length(S)\lt m,\) \[\gessel_{n,S}(1,q,q^2,\dotsc,q^m) = q^{m \length(S)+m-|S|-1}\qbinom{n+m-1-\length(S)}{m}_q\] where \(|S|\) denotes the sum of the elements of \(S,\) and the final factor is a \(q\)-binomial coefficient.

#Egge–Loehr–Warrington and the slinky rule

In [ELW10], the following lifting from fundamental quasisymmetric functions to Schur functions was given. An alternative interpretation of this rule is proved in [GR18]. A short proof by I. 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 straightened into partitions. This can be done using the slinky rule: pictorially, the parts of the compositions fall 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.

          $ \;$         $ \;$                                     $ \; $       $ \; $                      

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.

          $ \; $         $ \; $                         $ \; $     $ \; $          

#Relationship with the power-sum basis

C. A. Athanasiadis [Ath15], also implicitly in [AR15], proves the following result. Suppose \(X(\xvec)\) is a symmetric function such that \[X(\xvec) = \sum_{S \subseteq [n-1]} a_S \gessel_{n,S}(\xvec).\] Then \[X(\xvec) = \sum_{\lambda \vdash n} z_{\lambda}^{-1} \powerSum_{\lambda}(\xvec) \sum_{S\in U_\lambda} (-1)^{|S\setminus S_\lambda|} a_S,\] 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 power-sum basis

In [BDHM+20], two versions of quasisymmetric power-sum bases are given. They give the fundamental quasisymmetric expansion of these two quasisymmetric power-sum bases.

In [AS19], we provide the quasisymmetric power-sum expansion of \(\gessel_\alpha(\xvec).\) We have \[\gessel_{n,S}(\xvec) = \sum_{\alpha} \frac{\qPsi_\alpha(\xvec)}{z_\alpha} (-1)^{|S\setminus S_\alpha|}\] where the sum is taken over all compositions \(\alpha\) such that the set \(S\) is \(\alpha\)-unimodal. This formula implies Athanasiadis’s formula above.

A. Roberts gives a criterion for promoting symmetric fundamental-positive expansions to Schur-positive expansions by grouping the fundamental terms into suitable equivalence classes [Rob16].

S. K. Mason and E. Niese introduce a super analogue of the fundamental quasisymmetric basis in their study of quasisymmetric hook Schur functions [MN18]. Their quasisymmetric hook Schur functions expand positively in this super Gessel fundamental basis.

#The 0-Hecke algebra

The 0-Hecke algebra gives a representation-theoretic source of the fundamental quasisymmetric functions. Under the quasisymmetric characteristic, simple \(0\)-Hecke modules map to the functions \(\gessel_\alpha.\) This parallels the way the usual Frobenius characteristic sends Specht modules for \(\symS_n\) to Schur functions.

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