#Monomial slide polynomials
The monomial slide polynomials were introduced by S. Assaf and D. Searles in [AS17]. The slide polynomials form a basis for the space of polynomials, and can be seen as a lift of the monomial quasisymmetric functions. Together with the fundamental slide polynomials, they give finite-variable lifts of the two basic bases of quasisymmetric functions. This page also records nearby nonsymmetric families such as glide, forest, and lock polynomials.
Definition
For a weak composition \(\alpha,\) set \[\slideM_\alpha(\xvec) \coloneqq \sum_{\substack{b \trianglerighteq \alpha \\ \mathrm{flat}(b)=\mathrm{flat}(\alpha)}} \xvec^b\] where \(\mathrm{flat}(\beta)\) is the flat composition obtained by removing all 0s from \(\beta,\) and \(\trianglerighteq\) denotes dominance order.
As an example (from [Eq. 3.7, AS17]), \[\slideM_{(0,2,0,3)}(\xvec) = x_1^2x_2^3 + x_1^2x_3^3 + x_1^2 x_4^3 + x_2^2 x_3^3 + x_2^2 x_4^3.\]
The monomial slide polynomials have the monomial quasisymmetric functions as stable limit: \[\lim_{m \to \infty} \slideM_{0^m\times \alpha}(\xvec) = \qmonom_{\mathrm{flat}(\alpha)}(\xvec).\]
#Fundamental slide polynomials
The fundamental slide polynomials (or sometimes just slide polynomials) were introduced by S. Assaf and D. Searles in [AS17]. The slide polynomials form a basis for the space of polynomials, and can be seen as a lift of the Gessel quasisymmetric functions. O. Pechenik and D. Searles give a survey of asymmetric function theory [PS20]. It is a useful entry point for slide, glide, key, Demazure atom, and related nonsymmetric or quasisymmetric families.
The \(K\)-theoretic analogue of fundamental slide polynomials is given by the glide polynomials [PS17]. They refine stable Grothendieck polynomials in the same way that fundamental slide polynomials refine Schubert polynomials.
The fundamental slide polynomials expand positively in the monomial slide polynomials. Moreover, the Schubert polynomials expand positively in the fundamental slide polynomials [Thm. 3.13, AS17]. The main motivation for introducing the fundamental slide polynomials, is that products of Schubert polynomials can be expanded, with a combinatorial formula, into fundamental slide polynomials.
Definition
For a weak composition \(\alpha,\) the fundamental slide polynomial \(\slideF_\alpha\) is defined as \[\slideF_\alpha(\xvec) \coloneqq \sum_{\substack{b \trianglerighteq \alpha \\ \mathrm{flat}(b) \text{ refines } \mathrm{flat}(\alpha)}} \xvec^b\] where refinement is the composition refinement order, and \(\trianglerighteq\) denotes dominance order.
The fundamental slide polynomials have the Gessel quasisymmetric functions as stable limit: \[\lim_{m \to \infty} \slideF_{0^m\times \alpha}(\xvec) = \gessel_{\mathrm{flat}(\alpha)}(\xvec).\]
Example
For \(\alpha=(1,0,2),\) the monomial slide polynomial is \[\slideM_{(1,0,2)}(\xvec) = x_1x_3^2+x_1x_2^2.\] The support can be seen directly from the weak-composition diagrams. We record the same information by listing the rows from top to bottom: row \(3,\) row \(2,\) then row \(1.\) \[\begin{aligned} b=(1,0,2)&:\quad (\bullet\bullet,\ \emptyset,\ \bullet) &&\longmapsto x_1x_3^2,\\ b=(1,1,1)&:\quad (\bullet,\ \bullet,\ \bullet) &&\longmapsto x_1x_2x_3,\\ b=(1,2,0)&:\quad (\emptyset,\ \bullet\bullet,\ \bullet) &&\longmapsto x_1x_2^2. \end{aligned}\] The first and last diagrams have flat composition \((1,2),\) so they contribute to \(\slideM_{(1,0,2)}.\) The middle diagram has flat composition \((1,1,1),\) which refines \((1,2)\) but is not equal to it. The fundamental slide polynomial permits the refinement \((1,1,1)\) of \(\mathrm{flat}(\alpha)=(1,2),\) so \[\slideF_{(1,0,2)}(\xvec) = x_1x_3^2+x_1x_2x_3+x_1x_2^2 = \slideM_{(1,0,2)}(\xvec)+\slideM_{(1,1,1)}(\xvec).\]
#Fundamental particle basis
D. Searles introduced the fundamental particle basis [Sea19]. A. Hicks and E. Niese identify this basis with the fundamental Demazure atoms obtained by replacing the usual Demazure atom operators with Hivert’s quasisymmetric divided difference operators [HN24]. In this form the basis has an explicit monomial-positive formula, and the fundamental slide polynomials expand positively in the fundamental particle basis.
#Slide positive families
Key polynomials expand positively in the fundamental slide basis [Thm. 2.13, AS18]. In [CW22], the authors determine for which \(\alpha,\) the key polynomials \(\key_\alpha\) expanded into slide polynomials are multiplicity free.
The flagged \((P,w)\)-partition generating functions of S. Assaf and N. Bergeron are slide-positive [AB19]. In [TWZ22], it is shown that certain polynomials similar to chromatic symmetric functions are slide-positive.
See [ST21] for the notion of slide complexes.
The Kohnert polynomial page gives a diagrammatic model which contains keys, Schubert polynomials, locks, and other Assaf–Searles families as special cases. For \(K\)-theoretic diagram models related to Lascoux and glide phenomena, see the Lascoux polynomial page.
#Forest polynomials
P. Nadeau and V. Tewari [NT23] introduce forest polynomials, indexed by forests with labeled vertices. These polynomials can be defined as generating functions for labelings of forest posets with weak and strict inequalities along the two types of edges. This puts them close in spirit to flagged P-partitions.
Forest polynomials form a basis of the polynomial ring, and they expand positively in the fundamental slide basis. Conversely, Schubert polynomials expand positively in the forest polynomial basis.
P. Nadeau, H. Spink, and V. Tewari [NST24] develop a quasisymmetric divided-difference formalism in which forest polynomials play the role of Schubert polynomials. Their trimming operators are indexed by forests and governed by the Thompson monoid. In this setting forest polynomials are characterized recursively by the action of the trimming operators, and products of forest polynomials are forest-positive. A. Guo and D. Woodruff characterize when a Schubert polynomial is itself a forest polynomial [GW26]. Their criterion is that the indexing permutation avoids a specific set of six patterns, giving another pattern-avoidance description of a special Schubert class. M. J. Samuel gives a Littlewood–Richardson rule for products of forest polynomials using forest RC-graphs and a Schubert bialgebra [Sam26]. The same bialgebra framework yields product rules for dual Schubert, key, and slide polynomials.
#Lock polynomials
The lock polynomials were introduced by S. Assaf and D. Searles in [AS22]. The lock polynomials form a basis for the polynomial ring, and are indexed by weak compositions. The combinatorial formula for these is \[\lock_\alpha(\xvec) \coloneqq \sum_{T \in LT(\alpha)} \xvec^T\] where the sum is over all lock tableaux.
As an example (from [Wan20]), we have \[\lock_{(0,2,3)}(\xvec) = x_2^2 x_3^2 + x_1x_2x_3^3 + x_1^2x_3^3 + x_1x_2^2x_3^2 + x_1^2x_2x_3^2 + x_1^2x_2^2x_3 + x_1^2 x_2^3\]
Whenever the nonzero parts of \(\alpha\) are weakly decreasing, the lock polynomial coincides with a key polynomial; \(\lock_\alpha(\xvec)=\key_\alpha(\xvec),\) see [Thm. 6.12, AS22].
There is a crystal structure on lock polynomials, explored by G. Wang in [Wan20]. This crystal structure embeds naturally into Demazure crystals.
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