ALGEBRAIC COMBINATORICS

no. 1
Top-degree components of Grothendieck and Lascoux polynomials
Pan, Jianping1; Yu, Tianyi2
1 Department of Mathematics, NC State University, Raleigh, NC 95616-8633, U.S.A.
2 Department of Mathematics, UC San Diego, La Jolla, CA 92093, U.S.A.
Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 109-135.

The Castelnuovo–Mumford polynomial 𝔊 ^ w with w∈S n is the highest homogeneous component of the Grothendieck polynomial 𝔊 w . Pechenik, Speyer and Weigandt define a statistic rajcode(·) on S n that gives the leading monomial of 𝔊 ^ w . We introduce a statistic rajcode(·) on any diagram D through a combinatorial construction “snow diagram” that augments and decorates D. When D is the Rothe diagram of a permutation w, rajcode(D) agrees with the aforementioned rajcode(w). When D is the key diagram of a weak composition α, rajcode(D) yields the leading monomial of 𝔏 ^ α , the highest homogeneous component of the Lascoux polynomials 𝔏 α . We use 𝔏 ^ α to construct a basis of V ^ n , the span of 𝔊 ^ w with w∈S n . Then we show V ^ n gives a natural algebraic interpretation of a classical q-analogue of Bell numbers.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.326
Classification: 05E05
Keywords: Grothendieck polynomials, Lascoux polynomials, Hilbert series, Castelnuovo–Mumford polynomials
Pan, Jianping 1; Yu, Tianyi 2

1 Department of Mathematics, NC State University, Raleigh, NC 95616-8633, U.S.A.
2 Department of Mathematics, UC San Diego, La Jolla, CA 92093, U.S.A.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Pan, Jianping; Yu, Tianyi. Top-degree components of Grothendieck and Lascoux polynomials. Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 109-135. doi : 10.5802/alco.326. https://alco.centre-mersenne.org/articles/10.5802/alco.326/

[1] 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

[2] Brion, Michel Positivity in the Grothendieck group of complex flag varieties, J. Algebra, Volume 258 (2002) no. 1, pp. 137-159 | DOI | MR | Zbl

[3] Butler, Fred; Can, Mahir; Haglund, Jim; Remmel, Jeffrey Rook Theory Notes, 2010 (book project, available at http://www.math.ucsd.edu/~remmel/files/Book.pdf)

[4] Claesson, Anders Generalized pattern avoidance, European J. Combin., Volume 22 (2001) no. 7, pp. 961-971 | DOI | MR | Zbl

[5] Demazure, Michel Une nouvelle formule des caractĂšres, Bull. Sci. Math. (2), Volume 98 (1974) no. 3, pp. 163-172 | MR | Zbl

[6] Dreyer, Matt; MĂ©szĂĄros, Karola; St. Dizier, Avery On the degree of Grothendieck polynomials, 2022 | arXiv

[7] 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

[8] Garsia, A. M.; Remmel, J. B. Q-counting rook configurations and a formula of Frobenius, J. Combin. Theory Ser. A, Volume 41 (1986) no. 2, pp. 246-275 | DOI | MR | Zbl

[9] Hafner, Elena S Vexillary Grothendieck Polynomials via Bumpless Pipe Dreams, 2022 | arXiv

[10] Knuth, Donald E. Permutations, matrices, and generalized Young tableaux, Pacific J. Math., Volume 34 (1970), pp. 709-727 http://projecteuclid.org/euclid.pjm/1102971948 | DOI | MR | Zbl

[11] Knutson, Allen; Miller, Ezra Gröbner geometry of Schubert polynomials, Ann. of Math. (2), Volume 161 (2005) no. 3, pp. 1245-1318 | DOI | MR | Zbl

[12] Lascoux, Alain Schubert & Grothendieck: un bilan bidécennal, Sém. Lothar. Combin., Volume 50 (2003/04), Paper no. B50i, 32 pages | MR | Zbl

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

[14] Lascoux, Alain; SchĂŒtzenberger, Marcel-Paul Tableaux and noncommutative Schubert polynomials, Funct. Anal. Appl., Volume 23 (1989) no. 3, pp. 223-225 | DOI

[15] Lenart, Cristian; Robinson, Shawn; Sottile, Frank Grothendieck polynomials via permutation patterns and chains in the Bruhat order, Amer. J. Math., Volume 128 (2006) no. 4, pp. 805-848 | DOI | MR | Zbl

[16] Milne, Stephen C. Restricted growth functions, rank row matchings of partition lattices, and q-Stirling numbers, Adv. in Math., Volume 43 (1982) no. 2, pp. 173-196 | DOI | MR | Zbl

[17] Pan, Jianping; Yu, Tianyi A bijection between K-Kohnert diagrams and reverse set-valued tableaux, Volume 30, 2023 no. 4, Paper no. 4.26, 38 pages | DOI | MR | Zbl

[18] Pechenik, Oliver; Speyer, David E; Weigandt, Anna Castelnuovo-Mumford regularity of matrix Schubert varieties, 2021 | arXiv

[19] Reiner, Victor; Shimozono, Mark Key polynomials and a flagged Littlewood-Richardson rule, J. Combin. Theory Ser. A, Volume 70 (1995) no. 1, pp. 107-143 | DOI | MR | Zbl

[20] Reiner, Victor; Yong, Alexander The “Grothendieck to Lascoux” conjecture, 2021 | arXiv

[21] Ross, Colleen; Yong, Alexander Combinatorial rules for three bases of polynomials, SĂ©m. Lothar. Combin., Volume 74 ([2015–2018]), Paper no. B74a, 11 pages | MR | Zbl

[22] Sagan, Bruce E. The symmetric group: representations, combinatorial algorithms, and symmetric functions, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001, xvi+238 pages | DOI | MR

[23] Schensted, C. Longest increasing and decreasing subsequences, Canadian J. Math., Volume 13 (1961), pp. 179-191 | DOI | MR | Zbl

[24] Shimozono, Mark; Yu, Tianyi Grothendieck-to-Lascoux expansions, Trans. Amer. Math. Soc., Volume 376 (2023) no. 7, pp. 5181-5220 | DOI | MR | Zbl

[25] Invariant theory and tableaux, The IMA Volumes in Mathematics and its Applications, 19 (1990), p. xii+298 | MR | Zbl

[26] Viennot, G. Une forme géométrique de la correspondance de Robinson-Schensted, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) (Lecture Notes in Math), Volume Vol. 579, Springer, Berlin, 1977, pp. pp 29-58 | MR | Zbl

[27] Wagner, Carl G. Partition statistics and q-Bell numbers (q=-1), J. Integer Seq., Volume 7 (2004) no. 1, Paper no. 04.1.1, 12 pages | MR | Zbl

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