Resumen de Homology of Littlewood complexes
Steven V. Sam, Andrew Snowden, Jerzy Weyman
et VV be a symplectic vector space of dimension 2n2n . Given a partition λλ with at most nn parts, there is an associated irreducible representation S[λ](V)S[λ](V) of Sp(V)Sp(V) . This representation admits a resolution by a natural complex Lλ∙L∙λ , which we call the Littlewood complex, whose terms are restrictions of representations of GL(V)GL(V) . When λλ has more than nn parts, the representation S[λ](V)S[λ](V) is not defined, but the Littlewood complex Lλ∙L∙λ still makes sense. The purpose of this paper is to compute its homology. We find that either Lλ∙L∙λ is acyclic or it has a unique nonzero homology group, which forms an irreducible representation of Sp(V)Sp(V) . The nonzero homology group, if it exists, can be computed by a rule reminiscent of that occurring in the Borel–Weil–Bott theorem. This result can be interpreted as the computation of the “derived specialization” of irreducible representations of Sp(∞)Sp(∞) and as such categorifies earlier results of Koike–Terada on universal character rings. We prove analogous results for orthogonal and general linear groups. Along the way, we will see two topics from commutative algebra: the minimal free resolutions of determinantal ideals and Koszul homology.
