Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions. (English) Zbl 1344.05148

Galashin, Pavel; Grinberg, Darij; Liu, Gaku

Electron. J. Comb. 23, No. 3, Research Paper P3.14, 28 p. (2016).

Summary: The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the \(K\)-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2.

MSC:

05E05	Symmetric functions and generalizations
05E10	Combinatorial aspects of representation theory
14M15	Grassmannians, Schubert varieties, flag manifolds

Keywords:

dual stable Grothendieck polynomials; symmetric functions; Schur functions; plane partitions; Young tableaux

References:

[1]	Anders Buch, A Littlewood-Richardson rule for the K-theory of Grassmannians, Acta Math. 189 (2002), 37-78. · Zbl 1090.14015
[2]	Sergey Fomin, Curtis Greene, Noncommutative Schur functions and their applications, Discrete Mathematics 306 (2006) 1080-1096. doi:10.1016/S0012 365X(98)00140-X. · Zbl 1096.05051
[3]	Sergey Fomin, Anatol N. Kirillov, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math. 153 (1996), 1-3, 123-143. 1 This follows from Lemma 23 (c) (applied to the skew shape gλ/µ and k = 1). Here we are using the fact that if we apply Lemma 23 (a) to gλ/µ instead of λ/µ, then we get r = 1 (because if r > 2, then λ/µg #ν= #ν⊆[a= b 2− a 2 in contradiction to the irreducibility of λ/µ). ⊆[a 2,b 2)2,b 2) the electronic journal of combinatorics 23(3) (2016), #P3.14 27
[4]	William Fulton, Young Tableaux, London Mathematical Society Student Texts 35, Cambridge University Press 1997. · Zbl 0878.14034
[5]	Pavel Galashin, Darij Grinberg, Gaku Liu, Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions (extended abstract), FPSAC 2016, to appear in: Discrete Mathematics and Theoretical Computer Science, http://www.lix.polytechnique.fr/ pilaud/FPSAC16/final_138. · Zbl 1344.05148
[6]	Darij Grinberg, Victor Reiner, Hopf algebras in Combinatorics, August 25, 2015,arXiv:1409.8356v3. See also http://web.mit.edu/ darij/www/algebra/ HopfComb.pdf for a version which is more frequently updated. · Zbl 1458.16036
[7]	Thomas Lam, Pavlo Pylyavskyy, Combinatorial Hopf algebras and K homology of Grassmanians,arXiv:0705.2189v1. An updated version was later pub lished in: International Mathematics Research Notices, Vol. 2007, Article ID rnm125, 48 pages. doi:10.1093/imrn/rnm125. · Zbl 1134.16017
[8]	Alain Lascoux, Marcel-Paul Sch¨utzenberger, Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´et´e de drapeaux, C. R. Acad. Sci. Paris Sr. I Math 295 (1982), 11, 629-633. · Zbl 0542.14030
[9]	Ian G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press 1995. · Zbl 0824.05059
[10]	Richard Stanley, Enumerative Combinatorics, volume 2, Cambridge University Press, 1999. the electronic journal of combinatorics 23(3) (2016), #P3.14 28 · Zbl 0928.05001