Browse issues
no. 1
Jacobi–Trudi formulas for flagged refined dual stable Grothendieck polynomials
Kim, Jang Soo1
1 Department of Mathematics Sungkyunkwan University (SKKU) Suwon Gyeonggi-do 16419 South Korea
Algebraic Combinatorics, Volume 5 (2022) no. 1, pp. 121-148.

Recently Galashin, Grinberg, and Liu introduced the refined dual stable Grothendieck polynomials, which are symmetric functions in x=(x 1 ,x 2 ,...) with additional parameters t=(t 1 ,t 2 ,...). The refined dual stable Grothendieck polynomials are defined as a generating function for reverse plane partitions of a given shape. They interpolate between Schur functions and dual stable Grothendieck polynomials introduced by Lam and Pylyavskyy in 2007. Flagged refined dual stable Grothendieck polynomials are a more refined version of refined dual stable Grothendieck polynomials, where lower and upper bounds are given for the entries of each row or column. In this paper Jacobi–Trudi-type formulas for flagged refined dual stable Grothendieck polynomials are proved using plethystic substitution. This resolves a conjecture of Grinberg and generalizes a result by Iwao and Amanov–Yeliussizov.

Received:
Accepted:
Revised after acceptance:
Published online:
DOI: 10.5802/alco.203
Classification: 05E05, 05A15, 05E10
Keywords: Jacobi–Trudi formula, Grothendieck polynomial, symmetric function

Kim, Jang Soo 1

1 Department of Mathematics Sungkyunkwan University (SKKU) Suwon Gyeonggi-do 16419 South Korea
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2022__5_1_121_0,
     author = {Kim, Jang Soo},
     title = {Jacobi{\textendash}Trudi formulas for flagged refined dual stable {Grothendieck} polynomials},
     journal = {Algebraic Combinatorics},
     pages = {121--148},
     publisher = {MathOA foundation},
     volume = {5},
     number = {1},
     year = {2022},
     doi = {10.5802/alco.203},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.203/}
}
TY  - JOUR
AU  - Kim, Jang Soo
TI  - Jacobi–Trudi formulas for flagged refined dual stable Grothendieck polynomials
JO  - Algebraic Combinatorics
PY  - 2022
SP  - 121
EP  - 148
VL  - 5
IS  - 1
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.203/
DO  - 10.5802/alco.203
LA  - en
ID  - ALCO_2022__5_1_121_0
ER  - 
%0 Journal Article
%A Kim, Jang Soo
%T Jacobi–Trudi formulas for flagged refined dual stable Grothendieck polynomials
%J Algebraic Combinatorics
%D 2022
%P 121-148
%V 5
%N 1
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.203/
%R 10.5802/alco.203
%G en
%F ALCO_2022__5_1_121_0
Kim, Jang Soo. Jacobi–Trudi formulas for flagged refined dual stable Grothendieck polynomials. Algebraic Combinatorics, Volume 5 (2022) no. 1, pp. 121-148. doi : 10.5802/alco.203. https://alco.centre-mersenne.org/articles/10.5802/alco.203/

[1] Amanov, Alimzhan; Yeliussizov, Damir Determinantal formulas for dual Grothendieck polynomials (Preprint, https://arxiv.org/abs/2003.03907v1)

[2] Chen, William Y. C.; Li, Bingqing; Louck, J. D. The flagged double Schur function, J. Algebraic Combin., Volume 15 (2002) no. 1, pp. 7-26 | DOI | MR | Zbl

[3] Galashin, Pavel; Grinberg, Darij; Liu, Gaku Refined dual stable Grothendieck polynomials and generalized Bender–Knuth involutions, Electron. J. Combin., Volume 23 (2016) no. 3, Paper no. 3.14, 28 pages | DOI | MR | Zbl

[4] Gessel, Ira M. Determinants and plane partitions, Unpublished manuscript

[5] Grinberg, Darij Refined dual stable Grothendieck polynomials, http://www.cip.ifi.lmu.de/~grinberg/algebra/chicago2015.pdf

[6] Grinberg, Darij; Reiner, Victor Hopf algebras in combinatorics (Preprint, https://arxiv.org/abs/1409.8356v7)

[7] Iwao, Shinsuke Free-fermions and adjoint actions on stable β-Grothendieck polynomials (2020) (Preprint, https://arxiv.org/abs/2004.09499)

[8] Kim, Jang Soo Jacobi–Trudi formula for refined dual stable Grothendieck polynomials, J. Combin. Theory Ser. A, Volume 180 (2021), Paper no. 105415, 33 pages | DOI | MR | Zbl

[9] Lam, Thomas; Pylyavskyy, Pavlo Combinatorial Hopf algebras and K-homology of Grassmannians, Int. Math. Res. Not. IMRN (2007) no. 24, Paper no. rnm125, 48 pages | DOI | MR | Zbl

[10] Lascoux, Alain; Schützenberger, Marcel-Paul Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math., Volume 294 (1982) no. 13, pp. 447-450 | MR | Zbl

[11] Loehr, Nicholas A.; Remmel, Jeffrey B. A computational and combinatorial exposé of plethystic calculus, J. Algebraic Combin., Volume 33 (2011) no. 2, pp. 163-198 | DOI | MR | Zbl

[12] Merzon, Grigory; Smirnov, Evgeny Determinantal identities for flagged Schur and Schubert polynomials, Eur. J. Math., Volume 2 (2016) no. 1, pp. 227-245 | DOI | MR | Zbl

[13] Motegi, Kohei; Scrimshaw, Travis Refined dual Grothendieck polynomials, integrability, and the Schur measure (Preprint, https://arxiv.org/abs/2012.15011v1)

[14] Wachs, Michelle L. Flagged Schur functions, Schubert polynomials, and symmetrizing operators, J. Combin. Theory Ser. A, Volume 40 (1985) no. 2, pp. 276-289 | DOI | MR | Zbl

Cited by Sources: