Homology of Littlewood complexes
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 ar...
| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Springer-Verlag
2014
|
| Online Access: | http://hdl.handle.net/1721.1/85656 |
| _version_ | 1826203530922295296 |
|---|---|
| author | Sam, Steven V Weyman, Jerzy Snowden, Andrew WIlson |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Sam, Steven V Weyman, Jerzy Snowden, Andrew WIlson |
| author_sort | Sam, Steven V |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T12:38:29Z |
| format | Article |
| id | mit-1721.1/85656 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:38:29Z |
| publishDate | 2014 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | mit-1721.1/856562022-09-28T09:09:56Z Homology of Littlewood complexes Sam, Steven V Weyman, Jerzy Snowden, Andrew WIlson Massachusetts Institute of Technology. Department of Mathematics Snowden, Andrew Snowden, Andrew WIlson 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. National Science Foundation (U.S.) (Fellowship DMS-0902661) 2014-03-14T20:03:19Z 2014-03-14T20:03:19Z 2013-02 Article http://purl.org/eprint/type/JournalArticle 1022-1824 1420-9020 http://hdl.handle.net/1721.1/85656 Sam, Steven V, Andrew Snowden, and Jerzy Weyman. “Homology of Littlewood Complexes.” Sel. Math. New Ser. 19, no. 3 (August 2013): 655–698. en_US http://dx.doi.org/10.1007/s00029-013-0119-5 Selecta Mathematica Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Springer-Verlag Snowden |
| spellingShingle | Sam, Steven V Weyman, Jerzy Snowden, Andrew WIlson Homology of Littlewood complexes |
| title | Homology of Littlewood complexes |
| title_full | Homology of Littlewood complexes |
| title_fullStr | Homology of Littlewood complexes |
| title_full_unstemmed | Homology of Littlewood complexes |
| title_short | Homology of Littlewood complexes |
| title_sort | homology of littlewood complexes |
| url | http://hdl.handle.net/1721.1/85656 |
| work_keys_str_mv | AT samstevenv homologyoflittlewoodcomplexes AT weymanjerzy homologyoflittlewoodcomplexes AT snowdenandrewwilson homologyoflittlewoodcomplexes |