Alexander Miller, Victor Reiner
Published 2008-11-12Version 1
We conjecture a strong property for the up and down maps U and D in an r-differential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. In particular, this would determine the integral structure of the maps U, D, UD, DU, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice YF studied by Okada and its r-differential generalizations Z(r), as well as verifying many of its consequences for Young's lattice Y and the r-differential Cartesian products Y^r.
Comments: 29 pages, 9 figures
Path counting and rank gaps in differential posets
Recent results on $p$-ranks and Smith normal forms of some $2-(v,k,λ)$ designs
Smith Normal Forms of Incidence Matrices
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