We study a certain two-parameter family of non-standard graded complete intersection algebras $A(m,n)$. In case $n=2$, we show that if $m$ is even then $A(m,2)$ has the strong Lefschetz property and satisfies the complex Hodge–Riemann relations, while if $m$ is odd then $A(m,2)$ satisfies these properties only up to a certain degree. This supports a strengthening of a conjecture of Almkvist on the unimodality of the Hilbert function of $A(m,n)$.
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Keywords: Hodge–Riemann property, strong Lefschetz property, higher Hessian, NE lattice paths, binomial determinants, pseudo-reflection group
Abdallah, Nancy 1; McDaniel, Chris 2
CC-BY 4.0
@article{ALCO_2025__8_2_295_0,
author = {Abdallah, Nancy and McDaniel, Chris},
title = {Lattice {Paths,} {Lefschetz} {Properties,} and {Almkvist{\textquoteright}s} {Conjecture} in {Two} {Variables}},
journal = {Algebraic Combinatorics},
pages = {295--317},
publisher = {The Combinatorics Consortium},
volume = {8},
number = {2},
year = {2025},
doi = {10.5802/alco.414},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.414/}
}
TY - JOUR AU - Abdallah, Nancy AU - McDaniel, Chris TI - Lattice Paths, Lefschetz Properties, and Almkvist’s Conjecture in Two Variables JO - Algebraic Combinatorics PY - 2025 SP - 295 EP - 317 VL - 8 IS - 2 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.414/ DO - 10.5802/alco.414 LA - en ID - ALCO_2025__8_2_295_0 ER -
%0 Journal Article %A Abdallah, Nancy %A McDaniel, Chris %T Lattice Paths, Lefschetz Properties, and Almkvist’s Conjecture in Two Variables %J Algebraic Combinatorics %D 2025 %P 295-317 %V 8 %N 2 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.414/ %R 10.5802/alco.414 %G en %F ALCO_2025__8_2_295_0
Abdallah, Nancy; McDaniel, Chris. Lattice Paths, Lefschetz Properties, and Almkvist’s Conjecture in Two Variables. Algebraic Combinatorics, Volume 8 (2025) no. 2, pp. 295-317. doi : 10.5802/alco.414. https://alco.centre-mersenne.org/articles/10.5802/alco.414/
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