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 ¹

¹ Graduate School of Mathematical Sciences the University of Tokyo Japan

Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 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

MR Zbl

DOI: 10.5802/alco.45

Classification: 05E05, 20F55
Keywords: $K$-theoretic $k$-Schur functions, Pieri rule, Coxeter groups, affine symmetric groups

Author's affiliations:

Takigiku, Motoki ¹

¹ Graduate School of Mathematical Sciences the University of Tokyo Japan

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Takigiku, Motoki},
     title = {A {Pieri} formula and a factorization formula for sums of $K$-theoretic $k${-Schur} functions},
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     publisher = {MathOA foundation},
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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/articles/10.5802/alco.45/

References
Cited by

[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, Graduate Texts in Mathematics, 231, Springer, 2005, xiv+363 pages | MR | Zbl

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

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

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

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

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

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

[9] Ikeda, Takeshi; Iwao, Shinsuke; Maeno, Toshiaki Peterson isomorphism in $K$ -theory and relativistic Toda lattice, Int. Math. Res. Not. (2018), Paper no. rny051 | DOI

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

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

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

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

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

[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 | DOI | Zbl

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

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

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

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