Shifted combinatorial Hopf algebras from $K$-theory

Marberg, Eric

doi:10.5802/alco.362

Shifted combinatorial Hopf algebras from

K

-theory

Marberg, Eric ¹

¹ Department of Mathematics HKUST Clear Water Bay Hong Kong

Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 1123-1156.

Abstract

In prior joint work with Lewis, we developed a theory of enriched set-valued $P$ -partitions to construct a $K$ -theoretic generalization of the Hopf algebra of peak quasisymmetric functions. Here, we situate this object in a diagram of six Hopf algebras, providing a shifted version of the diagram of $K$ -theoretic combinatorial Hopf algebras studied by Lam and Pylyavskyy. This allows us to describe new $K$ -theoretic analogues of the classical peak algebra. We also study the Hopf algebras generated by Ikeda and Naruse’s $K$ -theoretic Schur $P$ - and $Q$ -functions, as well as their duals. Along the way, we derive several product, coproduct, and antipode formulas and outline a number of open problems and conjectures.

Received: 2023-05-18
Revised: 2024-01-26
Accepted: 2024-02-06
Published online: 2024-09-03

DOI: 10.5802/alco.362

Classification: 05E05, 16T30
Keywords: combinatorial Hopf algebras, Malvenuto-Reutenauer Hopf algebra, peak algebra, peak quasisymmetric functions, K-theoretic symmetric functions, shifted tableaux

Author's affiliations:

Marberg, Eric ¹

¹ Department of Mathematics HKUST Clear Water Bay Hong Kong

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{ALCO_2024__7_4_1123_0,
     author = {Marberg, Eric},
     title = {Shifted combinatorial {Hopf} algebras from $K$-theory},
     journal = {Algebraic Combinatorics},
     pages = {1123--1156},
     publisher = {The Combinatorics Consortium},
     volume = {7},
     number = {4},
     year = {2024},
     doi = {10.5802/alco.362},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.362/}
}

TY  - JOUR
AU  - Marberg, Eric
TI  - Shifted combinatorial Hopf algebras from $K$-theory
JO  - Algebraic Combinatorics
PY  - 2024
SP  - 1123
EP  - 1156
VL  - 7
IS  - 4
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.362/
DO  - 10.5802/alco.362
LA  - en
ID  - ALCO_2024__7_4_1123_0
ER  -

%0 Journal Article
%A Marberg, Eric
%T Shifted combinatorial Hopf algebras from $K$-theory
%J Algebraic Combinatorics
%D 2024
%P 1123-1156
%V 7
%N 4
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.362/
%R 10.5802/alco.362
%G en
%F ALCO_2024__7_4_1123_0

Marberg, Eric. Shifted combinatorial Hopf algebras from $K$-theory. Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 1123-1156. doi : 10.5802/alco.362. https://alco.centre-mersenne.org/articles/10.5802/alco.362/

References
Cited by

[1] Aguiar, Marcelo; Bergeron, Nantel; Nyman, Kathryn The peak algebra and the descent algebras of types $B$ and $D$ , Trans. Amer. Math. Soc., Volume 356 (2004) no. 7, pp. 2781-2824 | DOI | MR | Zbl

[2] Aguiar, Marcelo; Bergeron, Nantel; Sottile, Frank Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math., Volume 142 (2006) no. 1, pp. 1-30 | DOI | MR | Zbl

[3] Aguiar, Marcelo; Nyman, Kathryn; Orellana, Rosa New results on the peak algebra, J. Algebraic Combin., Volume 23 (2006) no. 2, pp. 149-188 | DOI | MR | Zbl

[4] Aguiar, Marcelo; Sottile, Frank Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, Adv. Math., Volume 191 (2005) no. 2, pp. 225-275 | DOI | MR | Zbl

[5] Bergeron, Nantel; Hivert, Florent; Thibon, Jean-Yves The peak algebra and the Hecke-Clifford algebras at $q = 0$ , J. Combin. Theory Ser. A, Volume 107 (2004) no. 1, pp. 1-19 | DOI | MR | Zbl

[6] Bergeron, Nantel; Mykytiuk, Stefan; Sottile, Frank; van Willigenburg, Stephanie Shifted quasi-symmetric functions and the Hopf algebra of peak functions, Discrete Math., Volume 246 (2002) no. 1-3, pp. 57-66 | DOI | MR | Zbl

[7] Billera, Louis J.; Hsiao, Samuel K.; van Willigenburg, Stephanie Peak quasisymmetric functions and Eulerian enumeration, Adv. Math., Volume 176 (2003) no. 2, pp. 248-276 | DOI | MR | Zbl

[8] Brewster Lewis, Joel; Marberg, Eric Enriched set-valued $P$ -partitions and shifted stable Grothendieck polynomials, Math. Z., Volume 299 (2021) no. 3-4, pp. 1929-1972 | DOI | MR | Zbl

[9] 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

[10] 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

[11] Buch, Anders Skovsted; Ravikumar, Vijay Pieri rules for the $K$ -theory of cominuscule Grassmannians, J. Reine Angew. Math., Volume 668 (2012), pp. 109-132 | DOI | MR | Zbl

[12] Chiu, Yu-Cheng; Marberg, Eric Expanding $K$ -theoretic Schur $Q$ -functions, Algebr. Comb., Volume 6 (2023) no. 6, pp. 1419-1445 | DOI | MR | Zbl

[13] Clifford, Edward; Thomas, Hugh; Yong, Alexander $K$ -theoretic Schubert calculus for $OG (n, 2 n + 1)$ and jeu de taquin for shifted increasing tableaux, J. Reine Angew. Math., Volume 690 (2014), pp. 51-63 | DOI | MR | Zbl

[14] Dieudonné, J. Introduction to the theory of formal groups, Pure and Applied Mathematics, 20, Marcel Dekker, Inc., New York, 1973, xii+265 pages | MR

[15] 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 (1994), pp. 183-189 | MR

[16] Grinberg, D.; Reiner, V. Hopf Algebras in Combinatorics, 2014 | arXiv

[17] Hamaker, Zachary; Keilthy, Adam; Patrias, Rebecca; Webster, Lillian; Zhang, Yinuo; Zhou, Shuqi Shifted Hecke insertion and $K$ -theory of $O G (n, 2 n + 1)$ , Sém. Lothar. Combin., Volume 78B (2017), Paper no. 14, 12 pages | MR | Zbl

[18] Hawkes, Graham; Scrimshaw, Travis Crystal structures for canonical Grothendieck functions, Algebr. Comb., Volume 3 (2020) no. 3, pp. 727-755 | DOI | Numdam | MR | Zbl

[19] Hsiao, Samuel K. Structure of the peak Hopf algebra of quasi-symmetric functions, preprint, 2002

[20] Hsiao, Samuel K.; Petersen, T. Kyle Colored posets and colored quasisymmetric functions, Ann. Comb., Volume 14 (2010) no. 2, pp. 251-289 | DOI | MR | Zbl

[21] Ikeda, Takeshi; Naruse, Hiroshi $K$ -theoretic analogues of factorial Schur $P$ - and $Q$ -functions, Adv. Math., Volume 243 (2013), pp. 22-66 | DOI | MR | Zbl

[22] Iwao, Shinsuke Neutral-fermionic presentation of the $K$ -theoretic $Q$ -function, J. Algebraic Combin., Volume 55 (2022) no. 2, pp. 629-662 | DOI | MR | Zbl

[23] Jing, Naihuan; Li, Yunnan A lift of Schur’s Q-functions to the peak algebra, J. Combin. Theory Ser. A, Volume 135 (2015), pp. 268-290 | DOI | MR | Zbl

[24] 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

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

[26] Lewis, Joel; Marberg, Eric Combinatorial formulas for shifted dual stable Grothendieck polynomials, Forum Math. Sigma, Volume 12 (2024), Paper no. e22, 45 pages | DOI | MR | Zbl

[27] Li, Yunnan Toward a polynomial basis of the algebra of peak quasisymmetric functions, J. Algebraic Combin., Volume 44 (2016) no. 4, pp. 931-946 | DOI | MR | Zbl

[28] Luoto, Kurt; Mykytiuk, Stefan; van Willigenburg, Stephanie An introduction to quasisymmetric Schur functions: Hopf algebras, quasisymmetric functions, and Young composition tableaux, SpringerBriefs in Mathematics, Springer, New York, 2013, xiv+89 pages | DOI | MR

[29] Malvenuto, Claudia Produits et coproduits des fonctions quasi-symétriques et de l’algèbre des descentes, Ph. D. Thesis, Université du Québec à Montréal, Montréal (1993)

[30] Marberg, E.; Scrimshaw, T. Key and Lascoux polynomials for symmetric orbit closures, 2023 | arXiv

[31] Marberg, Eric Linear compactness and combinatorial bialgebras, Electron. J. Combin., Volume 28 (2021) no. 3, Paper no. 3.9, 47 pages | DOI | MR | Zbl

[32] Marberg, Eric Shifted insertion algorithms for primed words, Comb. Theory, Volume 3 (2023) no. 3, Paper no. 14, 80 pages | MR | Zbl

[33] Marberg, Eric; Pawlowski, Brendan $K$ -theory formulas for orthogonal and symplectic orbit closures, Adv. Math., Volume 372 (2020), Paper no. 107299, 43 pages | DOI | MR | Zbl

[34] Marberg, Eric; Pawlowski, Brendan On some properties of symplectic Grothendieck polynomials, J. Pure Appl. Algebra, Volume 225 (2021) no. 1, Paper no. 106463, 22 pages | DOI | MR | Zbl

[35] Nakagawa, M.; Naruse, H. Universal factorial Schur $P, Q$ -functions and their duals, 2018 | arXiv

[36] Novelli, Jean-Christophe; Thibon, Jean-Yves Polynomial realizations of some trialgebras, 18th Formal Power Series and Algebraic Combinatorics (FPSAC’06), San Diego, USA (2006) no. 1, pp. 243-254 https://garsia.math.yorku.ca/fpsac06/papers/12.pdf

[37] Nyman, Kathryn L. The peak algebra of the symmetric group, J. Algebraic Combin., Volume 17 (2003) no. 3, pp. 309-322 | DOI | MR | Zbl

[38] Patrias, Rebecca Antipode formulas for some combinatorial Hopf algebras, Electron. J. Combin., Volume 23 (2016) no. 4, Paper no. 4.30, 32 pages | DOI | MR | Zbl

[39] Pechenik, Oliver; Yong, Alexander Genomic tableaux, J. Algebraic Combin., Volume 45 (2017) no. 3, pp. 649-685 | DOI | MR | Zbl

[40] Petersen, T. Kyle Enriched $P$ -partitions and peak algebras, Adv. Math., Volume 209 (2007) no. 2, pp. 561-610 | DOI | MR | Zbl

[41] Schocker, Manfred The peak algebra of the symmetric group revisited, Adv. Math., Volume 192 (2005) no. 2, pp. 259-309 | DOI | MR | Zbl

[42] Stembridge, John R. Enriched $P$ -partitions, Trans. Amer. Math. Soc., Volume 349 (1997) no. 2, pp. 763-788 | DOI | MR | Zbl

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

[44] Yeliussizov, Damir Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs, J. Combin. Theory Ser. A, Volume 161 (2019), pp. 453-485 | DOI | MR | Zbl

Cited by Sources: