#P-Grothendieck polynomials
The \(P\)-Grothendieck polynomials are symmetric, non-homogeneous symmetric functions attached to strict partitions. In [Haw23], G. Hawkes introduces weak \(P\)-Grothendieck polynomials, which play the same role for Schur \(P\) functions as dual stable Grothendieck polynomials play for Schur functions. The combinatorial model uses shifted multiset tableaux: shifted tableaux whose boxes contain nonempty multisets in the alphabet \(1'\lt 1\lt 2'\lt 2\lt \dotsb,\) with primed entries appearing with multiplicity at most one in each box and with shifted row and column semistandard conditions.
Following [Def. 4.5, Haw23], write \(a\lt_u z\) if \(a\lt z,\) or if \(a=z\) and both entries are unprimed. Similarly, write \(a\lt_p z\) if \(a\lt z,\) or if \(a=z\) and both entries are primed. If a box \(b'\) is directly to the right of a box \(b,\) then every entry \(a\in b\) must satisfy \(a\lt_u z\) for every \(z\in b'.\) If \(b'\) is directly below \(b,\) then for every \(z\in b'\) there must exist some \(a\in b\) with \(a\lt_p z.\) The set \(\mathrm{SMT}_0(\mu)\) consists of those shifted multiset tableaux for which the smallest entry in each row is unprimed.
Hawkes proves that these functions are Schur \(P\)-positive by homogeneous degree. This is analogous to the Schur-positivity of weak stable Grothendieck polynomials. The combinatorial definition is \[\grothendieckPDual_\mu(\xvec;\tvec) = \sum_{P\in \mathrm{SMT}_0(\mu)} \tvec^{\mathrm{dw}(P)}\xvec^{\mathrm{wt}(P)},\] where \(\mathrm{wt}(P)\) counts the values in \(P,\) ignoring primes, and \(\mathrm{dw}(P)\) records how many extra entries lie on each shifted diagonal; see [Thm. 4.17, Haw23].
Example (Shifted multiset tableaux for \(\mu=21\)).
Consider Hawkes’s \(t\)-deformation for \(\mu=21\) in two variables \(x_1,x_2.\) The first three homogeneous pieces are \[\begin{aligned} \left[\deg_{\xvec}=3\right]\grothendieckPDual_{21} &= x_1^2x_2+x_1x_2^2, \\ \left[\deg_{\xvec}=4\right]\grothendieckPDual_{21} &= (t_1+t_2) \left(x_1^3x_2+2x_1^2x_2^2+x_1x_2^3\right), \\ \left[\deg_{\xvec}=5\right]\grothendieckPDual_{21} &= (t_1^2+t_2^2) \left(x_1^4x_2+2x_1^3x_2^2+2x_1^2x_2^3+x_1x_2^4\right) \\ &\quad + t_1t_2 \left(x_1^4x_2+3x_1^3x_2^2+3x_1^2x_2^3+x_1x_2^4\right). \end{aligned}\] The degree-four part comes from the eight tableaux in \(\mathrm{SMT}_0(21)\) with exactly one extra entry beyond the three boxes of the shifted shape. Equivalently, this degree-four piece is \((t_1+t_2)\schurP_{31}(x_1,x_2)\); compare [Ex. 4.18, Haw23].
Suppose \(\mu\) is a strict partition with \(m\) parts. The weak symmetric \(P\)-Grothendieck polynomial \(\grothendieckPDual_\mu(\xvec)\) in \(n \geq m\) variables is defined as \[\grothendieckPDual_\mu(\xvec) = \prod_{i\lt j} (x_i-x_j)^{-1} \sum_{\sigma \in \symS_n / \symS_{n-m}} \sign(\sigma) \left( \prod_{i=1}^m \left( \frac{x_{\sigma_i}}{1-x_{\sigma_i}} \right)^{\mu_i} \right) \left( \prod_{i \lt j, i \leq m} x_{\sigma_i} + x_{\sigma_j} \right) \left( \prod_{m \lt i \lt j} x_{\sigma_i} - x_{\sigma_j} \right).\] Here, \(\symS_n / \symS_{n-m}\) is the set of permutations in \(\symS_n\) with no descent after position \(m.\)
Bibliography
- [Haw23]Graham Hawkes. P-schur positive P-Grothendieck polynomials. Australasian Journal of Combinatorics, 85(2):106–130, 2023.
.bib
@article{Hawkes2023, author = {Hawkes, Graham}, title = {{P}-Schur positive {P}-{G}rothendieck polynomials}, journal = {Australasian Journal of Combinatorics}, volume = {85}, number = {2}, pages = {106--130}, year = {2023}, issn = {2202-3518}, url = {https://ajc.maths.uq.edu.au/pdf/85/ajc_v85_p106.pdf} }