A Pieri formula and a factorization formula for sums of $K$-theoretic $k$-Schur functions

Takigiku, Motoki

doi:10.5802/alco.45

A Pieri formula and a factorization formula for sums of

K

-theoretic

k

-Schur functions

Takigiku, Motoki

Algebraic Combinatorics, Volume 2 (2019) no. 4, p. 447-480

Abstract

We give a Pieri-type formula for the sum of $K$ - $k$ -Schur functions $\sum_{μ \leq λ} g_{μ}^{(k)}$ over a principal order ideal of the poset of $k$ -bounded partitions under the strong Bruhat order, whose sum we denote by ${\tilde{g}}_{λ}^{(k)}$ . As an application of this, we also give a $k$ -rectangle factorization formula ${\tilde{g}}_{R_{t} \cup λ}^{(k)} = {\tilde{g}}_{R_{t}}^{(k)} {\tilde{g}}_{λ}^{(k)}$ where $R_{t} = (t^{k + 1 - t})$ , analogous to that of $k$ -Schur functions $s_{R_{t} \cup λ}^{(k)} = s_{R_{t}}^{(k)} s_{λ}^{(k)}$ .

Received : 2018-05-06
Revised : 2018-10-19
Accepted : 2018-10-28
Published online : 2019-08-01
DOI : https://doi.org/10.5802/alco.45
Classification: 05E05, 20F55
Keywords:

K

-theoretic

k

-Schur functions, Pieri rule, Coxeter groups, affine symmetric groups

@article{ALCO_2019__2_4_447_0,
     author = {Takigiku, Motoki},
     title = {A Pieri formula and a factorization formula for sums of $K$-theoretic $k$-Schur functions},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {2},
     number = {4},
     year = {2019},
     pages = {447-480},
     doi = {10.5802/alco.45},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_4_447_0}
}

Takigiku, Motoki. A Pieri formula and a factorization formula for sums of $K$-theoretic $k$-Schur functions. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 447-480. doi : 10.5802/alco.45. https://alco.centre-mersenne.org/item/ALCO_2019__2_4_447_0/

References

[1] Anderson, David; Chen, Linda; Tseng, Hsian-Hua On the quantum $K$ -ring of the flag manifold (2017) (https://arxiv.org/abs/1711.08414 )

[2] Björner, Anders; Brenti, Francesco Combinatorics of Coxeter groups, Springer, Graduate Texts in Mathematics, Volume 231 (2005), xiv+363 pages | MR 2133266 | Zbl 1110.05001

[3] Björner, Anders; Wachs, Michelle Bruhat order of Coxeter groups and shellability, Adv. Math., Volume 43 (1982) no. 1, pp. 87-100 | Article | MR 644668 | Zbl 0481.06002

[4] Björner, Anders; Wachs, Michelle Generalized quotients in Coxeter groups, Trans. Am. Math. Soc., Volume 308 (1988) no. 1, pp. 1-37 | Article | MR 946427 | Zbl 0659.05007

[5] Buch, Anders S.; Mihalcea, Leonardo C. Curve neighborhoods of Schubert varieties, J. Differ. Geom., Volume 99 (2015) no. 2, pp. 255-283 | Article | MR 3302040 | Zbl 06423472

[6] Denton, Tom Canonical decompositions of affine permutations, affine codes, and split $k$ -Schur functions, Electron. J. Comb., Volume 19 (2012) no. 4, 19, 41 pages | MR 3001656 | Zbl 1267.05290

[7] Deodhar, Vinay V. A splitting criterion for the Bruhat orderings on Coxeter groups, Commun. Algebra, Volume 15 (1987) no. 9, pp. 1889-1894 | Article | MR 898299 | Zbl 0631.20030

[8] Humphreys, James E. Reflection groups and Coxeter groups, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Volume 29 (1990), xii+204 pages | Article | MR 1066460 | Zbl 0725.20028

[9] Ikeda, Takeshi; Iwao, Shinsuke; Maeno, Toshiaki Peterson isomorphism in $K$ -theory and relativistic Toda lattice, Int. Math. Res. Not. (2018), rny051 | Article

[10] Kato, Syu Loop structure on equivariant $K$ -theory of semi-infinite flag manifolds (2018) (https://arxiv.org/abs/1805.01718 ) | Zbl 1398.14053

[11] Knutson, Allen; Miller, Ezra Subword complexes in Coxeter groups, Adv. Math., Volume 184 (2004) no. 1, pp. 161-176 | Article | MR 2047852 | Zbl 1069.20026

[12] Lam, Thomas Schubert polynomials for the affine Grassmannian, J. Am. Math. Soc., Volume 21 (2008) no. 1, pp. 259-281 | MR 2350056 | Zbl 1149.05045

[13] Lam, Thomas; Lapointe, Luc; Morse, Jennifer; Schilling, Anne; Shimozono, Mark; Zabrocki, Mike $k$ -Schur functions and affine Schubert calculus, The Fields Institute for Research in the Mathematical Sciences, Fields Institute Monographs, Volume 33 (2014), viii+219 pages | MR 3379711 | Zbl 1360.14004

[14] Lam, Thomas; Li, Changzheng; Mihalcea, Leonardo C.; Shimozono, Mark A conjectural Peterson isomorphism in $K$ -theory, J. Algebra, Volume 513 (2018), pp. 326-343 | Article | MR 3849889 | Zbl 06942901

[15] Lam, Thomas; Schilling, Anne; Shimozono, Mark $K$ -theory Schubert calculus of the affine Grassmannian, Compos. Math., Volume 146 (2010) no. 4, pp. 811-852 | MR 2660675 | Zbl 1256.14056

[16] Lam, Thomas; Shimozono, Mark Quantum cohomology of $G / P$ and homology of affine Grassmannian, Acta Math., Volume 204 (2010) no. 1, pp. 49-90 | Article | MR 2600433 | Zbl 1216.14052

[17] Lam, Thomas; Shimozono, Mark From quantum Schubert polynomials to $k$ -Schur functions via the Toda lattice, Math. Res. Lett., Volume 19 (2012) no. 1, pp. 81-93 | MR 2923177 | Zbl 1269.05113

[18] Lapointe, Luc; Lascoux, Alain; Morse, Jennifer Tableau atoms and a new Macdonald positivity conjecture, Duke Math. J., Volume 116 (2003) no. 1, pp. 103-146 | MR 1950481 | Zbl 1020.05069

[19] Lapointe, Luc; Morse, Jennifer Order ideals in weak subposets of Young’s lattice and associated unimodality conjectures, Ann. Comb., Volume 8 (2004) no. 2, pp. 197-219 | Article | MR 2079931 | Zbl 1063.06001

[20] Lapointe, Luc; Morse, Jennifer Tableaux on $k + 1$ -cores, reduced words for affine permutations, and $k$ -Schur expansions, J. Comb. Theory, Ser. A, Volume 112 (2005) no. 1, pp. 44-81 | Article | MR 2167475 | Zbl 1120.05093

[21] Lapointe, Luc; Morse, Jennifer A $k$ -tableau characterization of $k$ -Schur functions, Adv. Math., Volume 213 (2007) no. 1, pp. 183-204 | Article | MR 2331242 | Zbl 1118.05096

[22] Macdonald, Ian G. Symmetric functions and Hall polynomials, Clarendon Press, Oxford Science Publications (1995), x+475 pages (With contributions by A. Zelevinsky) | Zbl 0824.05059

[23] Morse, Jennifer Combinatorics of the $K$ -theory of affine Grassmannians, Adv. Math., Volume 229 (2012) no. 5, pp. 2950-2984 | Article | MR 2889152 | Zbl 1238.05276

[24] Shimozono, Mark (private communication, 2018)

[25] Stembridge, John R. A short derivation of the Möbius function for the Bruhat order, J. Algebr. Comb., Volume 25 (2007) no. 2, pp. 141-148 | Article | Zbl 1150.20028

[26] Takigiku, Motoki Factorization formulas of $K$ - $k$ -Schur functions I (2017) (https://arxiv.org/abs/1704.08643 ) | MR 3678641 | Zbl 1397.05201

[27] Takigiku, Motoki Factorization formulas of $K$ - $k$ -Schur functions II (2017) (https://arxiv.org/abs/1704.08660 ) | Zbl 1397.05201

[28] Waugh, Debra J. Upper bounds in affine Weyl groups under the weak order, Order, Volume 16 (1999) no. 1, pp. 77-87 | Article | MR 1740744 | Zbl 0952.20031

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