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#Lascoux polynomials

The Lascoux polynomials are a family of non-symmetric, non-homogeneous polynomials, first introduced by A. Lascoux in [Las01]. The set \(\{ \lascoux_\alpha(x_1,\dotsc,x_n) \}_\alpha\) where \(\alpha\) ranges over all compositions of length \(n,\) is a basis for \(\setC[x_1,\dotsc,x_n].\)

The Lascoux polynomials are the \(K\)-theoretic analogue of the key polynomials, and they generalize the Grothendieck polynomials. Similarly, there are Lascoux-atom polynomials, which are \(K\)-theoretic analogues of the Demazure atom polynomials.

Note: There seem to be other types of polynomials referred to as Lascoux polynomials, e.g. [BCMV+21].

#Definition

We use the same notation as in the operator definition of key polynomials. Define the \(\beta\)-divided difference operators \(\partial_i^{(\beta)}\) and \(\pi_i^{(\beta)}:\) \[\partial_i^{(\beta)}(f) \coloneqq \partial_i(f + \beta x_{i+1} f) \qquad \pi_i^{(\beta)}(f) \coloneqq \partial_i^{(\beta)}(x_i f).\] The Lascoux polynomial is then defined as \[\lascoux^{(\beta)}_{\alpha}(\xvec) \coloneqq \begin{cases} \xvec^{\alpha} & \text{ if $\alpha$ is a partition} \\ \pi_i^{(\beta)} \lascoux^{(\beta)}_{s_i \alpha}(\xvec) & \text{ if $\alpha_i \lt \alpha_{i+1}$}. \end{cases}\] Note that \(\key_{\alpha}(\xvec) = \lascoux^{(0)}_{\alpha}(\xvec),\) that is, at \(\beta=0\) we recover a key polynomial.

The first combinatorial model for Lascoux polynomials was conjectured by C. Monical in [Mon17], where these are given as a sum over certain set-valued tableaux. They also provide a combinatorial formula involving set-valued skyline fillings (terminology from non-symmetric Macdonald polynomials). A proof of this conjecture was later given in [Thm. 4.1, BSW20].

Another set-valued tableau formula is proved in [Thm. 1.1, RY21]. A set-valued tableau formula is also given in [Thm. 3.16, Yu23]. Here, the tableaux have composition shape, and the author argues that his formula is simpler, as it does not use the Lusztig involution.

In [MPS18], the question is raised whether there is some type of \(K\)-theoretic crystal structure for Lascoux polynomials. Yu’s set-valued-tableau operators give an answer in the context of abstract Kashiwara crystals: they behave as a \(K\)-theoretic analogue of Demazure crystals and satisfy the corresponding generation properties [Sec. 5, Yu23].

The Lascoux polynomials also have a diagram formula using \(K\)-Kohnert diagrams. This was conjectured by C. Ross and A. Yong, and proved by J. Pan and T. Yu via a weight-preserving bijection with reverse set-valued tableaux [Thm. 2, PY23]. The same \(K\)-Kohnert model leads to snow diagrams for determining the top degree of a Lascoux polynomial [Thms. 1.1 and 1.2, PY24]. K. Hanser and N. Mayers study the corresponding maximal-ghost problem for more general diagrams and prove the snow-diagram formula for several additional diagram classes [HM26].

Example (A small \(K\)-Kohnert expansion).

Let \(\alpha=(0,2,1).\) Pan and Yu list the eleven \(K\)-Kohnert diagrams of \(\alpha\) explicitly in [Ex. 2.13, PY24]. Translating their diagram weights gives \[\begin{split} \lascoux^{(\beta)}_{(0,2,1)} &=x_2^2x_3+x_1x_2x_3+x_1^2x_3+x_1x_2^2+x_1^2x_2\\ &\phantom{=} +\beta\left(2x_1x_2^2x_3+2x_1^2x_2x_3+x_1^2x_2^2\right) +\beta^2x_1^2x_2^2x_3 . \end{split}\] The first line is the key-polynomial part, and the powers of \(\beta\) record how many ghost cells were left by the \(K\)-Kohnert moves. For the key diagram \(D(\alpha),\) the snow-diagram algorithm places dark clouds at \((3,1)\) and \((2,2),\) and snowflakes at \((1,1)\) and \((1,2).\) Thus \(\operatorname{rajcode}(\alpha)=(2,2,1)\) and \(\operatorname{raj}(\alpha)=5,\) so the top monomial is \(x_1^2x_2^2x_3,\) matching the final term above [Thm. 1.2, PY24].

Example (A small Pan–Yu bijection example).

Pan and Yu’s bijection is stated for reverse set-valued tableaux, represented as diagram pairs, rather than for the skyline-filling convention above. The following source-backed example uses their row convention, where row \(1\) is at the bottom [Exs. 3.2 and 3.5, PY23].

Write \(\bullet\) for ordinary cells and \(\times\) for ghost cells. The \(K\)-Kohnert diagram pair \[K=\{(1,1),(1,2),(2,1)\},\qquad G=\{(1,3),(2,2)\}\] is drawn as \[\begin{array}{c|cc} 3 & \times & \\ 2 & \bullet & \times \\ 1 & \bullet & \bullet \\ \hline & 1 & 2 \end{array}.\] It has excess \(2\) and weight \((2,2,1),\) so it contributes \[\beta^2 x_1^2x_2^2x_3.\] Under Pan–Yu’s map \(\Psi_\alpha,\) for \(\alpha=(0,2,1),\) this diagram pair is sent to the reverse set-valued tableau diagram pair \[L=\{(1,2),(1,3),(2,2)\},\qquad E=\{(1,1),(2,1)\}.\] Equivalently, the corresponding reverse set-valued tableau is \[\begin{array}{cc} \{3\} & \{2,1\}\\ \{2,1\} & \end{array}.\] The image has the same excess and weight, so it contributes the same monomial \(\beta^2x_1^2x_2^2x_3\) on the tableau side.

#Properties of Lascoux polynomials

The Schubert polynomials expand positively into key polynomials. In the same manner, in [Thm. 1.9, SY23], M. Shimozono and T. Yu give a formula for the expansion of Grothendieck polynomials into Lascoux polynomials, thus proving an earlier conjecture by V. Reiner and A. Yong [RY21].

G. Orelowitz and T. Yu prove a product rule for Lascoux polynomials and stable Grothendieck polynomials. Their tableau formula expands \(\lascoux_{\alpha}^{(\beta)} \grothendieckStable_w^{(\beta)}\) as a graded nonnegative sum of Lascoux polynomials [Thm. 1.3, OY23]. The proof uses the row analogue of Hecke column insertion introduced by D. Huang, M. Shimozono, and T. Yu [HSY24].

There are also strong connections between Lascoux polynomials and Schubert polynomials at top degree. The top homogeneous components of Lascoux polynomials form a basis for a graded algebra, and T. Yu proves that they are related to Schubert polynomials by a reverse-complement operator [Thm. 3.6 and Cor. 3.8, Yu25]. This transfers the Schubert structure constants to the top-Lascoux structure constants [Thm. 4.2, Yu25].

L. Setiabrata and A. S. Dizier give double orthodontia formulas for double Grothendieck polynomials. As an application, they prove a Lascoux-positivity theorem for a reversed specialization of double Schubert polynomials indexed by vexillary permutations [Cor. 1.3, SD24]; they conjecture that the same positivity holds without the vexillary hypothesis.

E. Presnova and E. Smirnov give a Gelfand–Zetlin polytope model for Lascoux polynomials. They construct a cellular decomposition of a Gelfand–Zetlin polytope and express a Lascoux polynomial as a weighted sum over cells lying in the relevant union of dual Kogan faces [Thms. 4.3.1–4.3.3, PS24]. This extends the Gelfand–Zetlin model for key polynomials.

E. Marberg and T. Scrimshaw introduce shifted \(P\)- and \(Q\)-analogues of key polynomials and their K-theoretic analogues. Their shifted \(P\)-Lascoux polynomials and atoms are \(\setN[\beta]\)-linear combinations of ordinary Lascoux polynomials and Lascoux atoms [Prop. 3.21, MS26], while their stable shifted Grothendieck functions have positive finite expansions in stable Grothendieck functions [Thms. 3.28 and 3.41, MS26].

C. Monical, O. Pechenik, and D. Searles introduce the quasiLascoux basis and kaon basis in combinatorial \(K\)-theory [MPS21]. The quasiLascoux basis is a \(K\)-theoretic deformation of the quasikey basis and a polynomial lift of quasiGrothendieck polynomials; it expands positively into both glide and Lascoux atom bases, while the Lascoux basis expands positively into it. Kaons are \(K\)-theoretic deformations of fundamental particles, and both glide polynomials and Lascoux atoms expand positively into the kaon basis. L. Pierson proves a conjecture of C. Monical, O. Pechenik, and D. Searles on cancellation in two K-theoretic polynomial expansions [Pie25]. The result says that certain alternating sums obtained by specializing the Lascoux-atom–to–kaon and quasiLascoux–to–glide transition coefficients at \(\beta=-1\) are always \(0\) or \(1,\) and the proof is by a sign-reversing involution.

#Lascoux atom

#A vertex model for Lascoux atoms

In [BSW20], the authors construct a 5-vertex model whose partition function is a Lascoux atom. This is the first combinatorial model for the Lascoux atoms.

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