Spanning two papers from 1989 and 2018, Weyman unearthed a fascinating connection between commutative algebra and representation theory in his study of generic free resolutions of length three. This thesis is devoted to analyzing this connection further. In the first half, we show that certain Kazhdan-Lusztig varieties provide generic examples of ideals in the linkage class of a complete intersection. For those of embedding codimension three, we also compute the free resolutions of their coordinate rings. We later show that these specialize to resolutions of all grade three licci ideals.

In the second half, we develop the machinery of higher structure maps originating from Weyman's generic ring. Using the free resolutions constructed previously, we disprove Hochster's conjecture on finite generation of generic rings. The two perspectives converge in the final chapter of the thesis, in which we develop an ADE correspondence to completely classify grade three perfect ideals with small type and deviation.