A Combinatorial Description of the Syzygies of Certain Weyl Modules

Homotopies for the Generalized Bar Complex Associated to Certain 3-Rowed Weyl Modules

“…As it is said in Sano (2003), in addition to the intrinsic combinatorial and invariant theoretic interest of such basis, the author thinks of this basis esentially as a stepping stone for the construction of homotopies which lead to resolutions of Weyl modules. The present article closes this issue by constructing a splitting contracting homotopy assuming the condition q − p ≥ s − 1 (this condition implies at most one triple overlap in the case of skewshapes).…”

“…In Sano (2003), the author constructed and described a basis for the syzygies associated to the resolution of the aforementioned 3-rowed Weyl modules satisfying the additional condition q − p ≥ s − t 2 − 1 (Sano, 2004), where s is the number of overlaps between the second and third rows. As it is said in Sano (2003), in addition to the intrinsic combinatorial and invariant theoretic interest of such basis, the author thinks of this basis esentially as a stepping stone for the construction of homotopies which lead to resolutions of Weyl modules.…”

“…Let us remark that the completeness condition in Sano (2003) does not use the fact that the complex is a resolution; the essential elements M that satisfy d i M = d i N for nonessential elements N are found by hand computation.…”

“…The techniques used in the construction of basis for the syzygies in Sano (2003) is used in a fundamental way for guiding the construction of the homotopy; thus let us describe briefly the techniques of Sano (2003).…”

“…Then in Sano (2003) a basis for the syzygies of this resolution is constructed; assuming that the complex is in fact a resolution. The techniques used in the construction of basis for the syzygies in Sano (2003) is used in a fundamental way for guiding the construction of the homotopy; thus let us describe briefly the techniques of Sano (2003).…”