Boij-S\"oderberg theory has had a dramatic impact on commutative algebra. We determine explicit Boij-S\"oderberg coefficients for ideals with linear resolutions and illustrate how these arise from the usual Eliahou-Kervaire computations for Borel ideals. In addition, we explore a new numerical decomposition for resolutions based on a row-by-row approach; here, the coefficients of the Betti diagrams are not necessarily positive. Finally, we demonstrate how the Boij-S\"oderberg decomposition of an arbitrary homogeneous ideal with a pure resolution changes when multiplying the ideal by a homogeneous polynomial.
"Boij-Söderberg and Veronese decompositions." J. Commut. Algebra 9 (3) 367 - 386, 2017. https://doi.org/10.1216/JCA-2017-9-3-367