Homology of Littlewood complexes
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Snowden_Homology of.pdf
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Author(s)
Sam, Steven V
Weyman, Jerzy
Snowden, Andrew WIlson
Date Issued
February 2013
Journal
Selecta Mathematica
Publisher
Springer-Verlag
Citation
Sam, Steven V, Andrew Snowden, and Jerzy Weyman. “Homology of Littlewood Complexes.” Sel. Math. New Ser. 19, no. 3 (August 2013): 655–698.
Version
Author's final manuscript
Abstract
Let V be a symplectic vector space of dimension 2n. Given a partition λ with at most n parts, there is an associated irreducible representation S[subscript [λ]](V) of Sp(V). This representation admits a resolution by a natural complex L[λ over ∙], which we call the Littlewood complex, whose terms are restrictions of representations of GL(V). When λ has more than n parts, the representation S[subscript [λ]](V) is not defined, but the Littlewood complex L[λ over ∙] still makes sense. The purpose of this paper is to compute its homology. We find that either L[λ over ∙] is acyclic or it has a unique nonzero homology group, which forms an irreducible representation of 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(∞) 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00029-013-0119-5
