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.

We give a Pieri-type formula for the sum of K-k-Schur functions μλ 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 g ˜ λ (k) . As an application of this, we also give a k-rectangle factorization formula g ˜ R t λ (k) =g ˜ R t (k) g ˜ λ (k) where R t =(t k+1-t ), analogous to that of k-Schur functions s R t λ (k) =s R t (k) s λ (k) .

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.45
Classification: 05E05, 20F55
Keywords: $K$-theoretic $k$-Schur functions, Pieri rule, Coxeter groups, affine symmetric groups
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
@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},
     pages = {447--480},
     publisher = {MathOA foundation},
     volume = {2},
     number = {4},
     year = {2019},
     doi = {10.5802/alco.45},
     zbl = {1421.05096},
     mrnumber = {3997506},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.45/}
}
TY  - JOUR
AU  - Takigiku, Motoki
TI  - A Pieri formula and a factorization formula for sums of $K$-theoretic $k$-Schur functions
JO  - Algebraic Combinatorics
PY  - 2019
SP  - 447
EP  - 480
VL  - 2
IS  - 4
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.45/
DO  - 10.5802/alco.45
LA  - en
ID  - ALCO_2019__2_4_447_0
ER  - 
%0 Journal Article
%A Takigiku, Motoki
%T A Pieri formula and a factorization formula for sums of $K$-theoretic $k$-Schur functions
%J Algebraic Combinatorics
%D 2019
%P 447-480
%V 2
%N 4
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.45/
%R 10.5802/alco.45
%G en
%F 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/articles/10.5802/alco.45/

[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

Cited by Sources: