The Stembridge equality for skew stable Grothendieck polynomials and skew dual stable Grothendieck polynomials
Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 187-208.

The Schur polynomials s λ are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For ρ=(n,n-1,,1) a staircase shape and μρ a subpartition, the Stembridge equality states that s ρ/μ =s ρ/μ T . This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials G λ , and the dual stable Grothendieck polynomials g λ , developed by Buch, Lam, and Pylyavskyy, are variants of the Schur polynomials and describe the K-theory of the Grassmannians. Using the Hopf algebra structure of the ring of symmetric functions and a generalized Littlewood–Richardson rule, we prove that G ρ/μ =G ρ/μ T and g ρ/μ =g ρ/μ T , the analogues of the Stembridge equality for the skew stable and skew dual stable Grothendieck polynomials.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.199
Classification: 05E05
Keywords: Stembridge equality, Grothendieck polynomial, Young tableau, Hopf algebra.
Abney-McPeek, Fiona 1; An, Serena 2; Ng, Jakin S. 2

1 Harvard University Cambridge MA USA.
2 Massachusetts Institute of Technology Cambridge MA USA.
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Abney-McPeek, Fiona; An, Serena; Ng, Jakin S. The Stembridge equality for skew stable Grothendieck polynomials and skew dual stable Grothendieck polynomials. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 187-208. doi : 10.5802/alco.199. https://alco.centre-mersenne.org/articles/10.5802/alco.199/

[1] Alwaise, Ethan; Chen, Shuli; Clifton, Alexander; Patrias, Rebecca; Prasad, Rohil; Shinners, Madeline; Zheng, Albert Coincidences among skew stable and dual stable Grothendieck polynomials, Involve, Volume 11 (2018) no. 1, pp. 143-167 | Article | MR: 3681354 | Zbl: 1368.05151

[2] Buch, Anders S. A Littlewood–Richardson rule for the K-theory of Grassmannians, Acta Math., Volume 189 (2002) no. 1, pp. 37-78 | Article | MR: 1946917 | Zbl: 1090.14015

[3] 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: 2307216

[4] Galashin, Pavel A Littlewood–Richardson rule for dual stable Grothendieck polynomials, J. Combin. Theory Ser. A, Volume 151 (2017), pp. 23-35 | Article | MR: 3663486 | Zbl: 1366.05116

[5] Grinberg, Darij; Reiner, Victor Hopf Algebras in Combinatorics (2020) (https://arxiv.org/abs/1409.8356)

[6] Lam, Thomas; Pylyavskyy, Pavlo Combinatorial Hopf algebras and K-homology of Grassmanians, Int. Math. Res. Not. (2007) no. 24, Paper no. rnm125, 48 pages | Article

[7] Reiner, Victor; Shaw, Kristin M.; van Willigenburg, Stephanie Coincidences among skew Schur functions, Adv. Math., Volume 216 (2007) no. 1, pp. 118-152 | Article | MR: 2353252 | Zbl: 1128.05051

[8] Stanley, Richard P. Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, 2012, xiv+626 pages (second edition) | MR: 2868112 | Zbl: 1247.05003

[9] Yeliussizov, Damir Duality and deformations of stable Grothendieck polynomials, J. Algebraic Combin., Volume 45 (2017) no. 1, pp. 295-344 | Article | MR: 3591379 | Zbl: 1355.05263

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