Lattice Paths, Lefschetz Properties, and Almkvist’s Conjecture in Two Variables. arXiv:2404.05098
Preprint, arXiv:2404.05098 [math.AC] (2024).
Summary: We study a certain two-parameter family of non-standard graded complete intersections \(A(m,n)\). In case \(n=2\), we show that \(A(m,2)\) has the strong Lefschetz property and the complex Hodge-Riemann property if and only if \(m\) is even. This supports a strengthening of a conjecture of Almkvist on the unimodality of the Hilbert function of \(A(m,n)\).
MSC:
| 13E10 | Commutative Artinian rings and modules, finite-dimensional algebras |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
| 11B83 | Special sequences and polynomials |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 14F45 | Topological properties in algebraic geometry |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
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