Combinatorial relations on skew Schur and skew stable Grothendieck polynomials
Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 175-188.

We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in earlier joint work with López Martín and Teixidor i Bigas concerning Brill–Noether curves, and it generalizes a 2000 formula of Lenart and a recent result of Reiner–Tenner–Yong to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms of skew Grothendieck polynomials.

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Accepted:
Published online:
DOI: 10.5802/alco.144
Classification: 05E05, 05E14
Keywords: Schur functions, Grothendieck polynomials, insertion algorithms, set-valued tableaux, Brill–Noether theory.
Chan, Melody 1; Pflueger, Nathan 2

1 Brown University Department of Mathematics Box 1917 Providence RI 02912, USA
2 Amherst College Department of Mathematics and Statistics Amherst MA 01002, USA
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Chan, Melody; Pflueger, Nathan. Combinatorial relations on skew Schur and skew stable Grothendieck polynomials. Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 175-188. doi : 10.5802/alco.144. http://www.numdam.org/articles/10.5802/alco.144/

[1] Anderson, Dave; Chen, Linda; Tarasca, Nicola K-classes of Brill–Noether loci and a determinantal formula (2017) (https://arxiv.org/abs/1705.02992)

[2] Assaf, Sami H.; McNamara, Peter R. W. A Pieri rule for skew shapes, J. Combin. Theory Ser. A, Volume 118 (2011) no. 1, pp. 277-290 | DOI | MR | Zbl

[3] Bandlow, Jason; Morse, Jennifer Combinatorial expansions in K-theoretic bases, Electron. J. Combin., Volume 19 (2012) no. 4, 39, 27 pages | MR | Zbl

[4] Billey, Sara C.; Jockusch, William; Stanley, Richard P. Some combinatorial properties of Schubert polynomials, J. Algebraic Combin., Volume 2 (1993) no. 4, pp. 345-374 | DOI | MR | Zbl

[5] Buch, Anders Skovsted A Littlewood–Richardson rule for the K-theory of Grassmannians, Acta Math., Volume 189 (2002) no. 1, pp. 37-78 | DOI | MR | Zbl

[6] Buch, Anders Skovsted; Kresch, Andrew; Shimozono, Mark; Tamvakis, Harry; Yong, Alexander Stable Grothendieck polynomials and K-theoretic factor sequences, Math. Ann., Volume 340 (2008) no. 2, pp. 359-382 | DOI | MR | Zbl

[7] Chan, Melody; López Martín, Alberto; Pflueger, Nathan; Teixidor i Bigas, Montserrat Genera of Brill–Noether curves and staircase paths in Young tableaux, Trans. Amer. Math. Soc., Volume 370 (2018) no. 5, pp. 3405-3439 | DOI | MR | Zbl

[8] Chan, Melody; Pflueger, Nathan Euler characteristics of Brill–Noether varieties (to appear in Transactions of the AMS) | DOI

[9] Chan, Melody; Pflueger, Nathan Relative Richardson varieties (https://arxiv.org/abs/1909.12414)

[10] Fomin, Sergey; Greene, Curtis Noncommutative Schur functions and their applications, Discrete Math., Volume 193 (1998) no. 1-3, pp. 179-200 Selected papers in honor of Adriano Garsia (Taormina, 1994) | DOI | MR | Zbl

[11] Fomin, Sergey; Kirillov, Anatol N. Grothendieck polynomials and the Yang-Baxter equation, Formal power series and algebraic combinatorics/Séries formelles et combinatoire algébrique, DIMACS, Piscataway, NJ, sd, pp. 183-189 | MR

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

[13] Lascoux, Alain; Schützenberger, Marcel-Paul Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math., Volume 295 (1982) no. 11, pp. 629-633 | MR | Zbl

[14] Lenart, Cristian Combinatorial aspects of the K-theory of Grassmannians, Ann. Comb., Volume 4 (2000) no. 1, pp. 67-82 | DOI | MR | Zbl

[15] Reiner, Victor; Tenner, Bridget Eileen; Yong, Alexander Poset edge densities, nearly reduced words, and barely set-valued tableaux, J. Combin. Theory Ser. A, Volume 158 (2018), pp. 66-125 | DOI | MR | Zbl

[16] Sagan, Bruce E.; Stanley, Richard P. Robinson–Schensted algorithms for skew tableaux, J. Combin. Theory Ser. A, Volume 55 (1990) no. 2, pp. 161-193 | DOI | MR | Zbl

[17] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) | DOI | MR | Zbl

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