Peterson and Proctor obtained a formula which expresses the multivariate generating function for -partitions on a -complete poset as a product in terms of hooks in . In this paper, we give a skew generalization of Peterson–Proctor’s hook formula, i.e. a formula for the generating function of -partitions for a -complete poset and an order filter of , by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant -theory of Kac–Moody partial flag varieties. This generalization provides an alternate proof of Peterson–Proctor’s hook formula. As the equivariant cohomology version, we derive a skew generalization of a combinatorial reformulation of Nakada’s colored hook formula for roots.
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.54
Keywords: $d$-complete posets, hook formulas, $P$-partitions, Schubert calculus, equivariant $K$-theory
CC-BY 4.0
@article{ALCO_2019__2_4_541_0,
author = {Naruse, Hiroshi and Okada, Soichi},
title = {Skew hook formula for $d$-complete posets via equivariant $K$-theory},
journal = {Algebraic Combinatorics},
pages = {541--571},
publisher = {MathOA foundation},
volume = {2},
number = {4},
year = {2019},
doi = {10.5802/alco.54},
zbl = {1417.05011},
mrnumber = {3997510},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.54/}
}
TY - JOUR AU - Naruse, Hiroshi AU - Okada, Soichi TI - Skew hook formula for $d$-complete posets via equivariant $K$-theory JO - Algebraic Combinatorics PY - 2019 SP - 541 EP - 571 VL - 2 IS - 4 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.54/ DO - 10.5802/alco.54 LA - en ID - ALCO_2019__2_4_541_0 ER -
%0 Journal Article %A Naruse, Hiroshi %A Okada, Soichi %T Skew hook formula for $d$-complete posets via equivariant $K$-theory %J Algebraic Combinatorics %D 2019 %P 541-571 %V 2 %N 4 %I MathOA foundation %U https://alco.centre-mersenne.org/articles/10.5802/alco.54/ %R 10.5802/alco.54 %G en %F ALCO_2019__2_4_541_0
Naruse, Hiroshi; Okada, Soichi. Skew hook formula for $d$-complete posets via equivariant $K$-theory. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 541-571. doi : 10.5802/alco.54. https://alco.centre-mersenne.org/articles/10.5802/alco.54/
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