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**The Brauer algebra and the symplectic Schur algebra.** / Donkin, Stephen; Tange, Rudolf.

Research output: Contribution to journal › Article › peer-review

Donkin, S & Tange, R 2010, 'The Brauer algebra and the symplectic Schur algebra', *Mathematische Zeitschrift*, vol. 265, no. 1, pp. 187-219. https://doi.org/10.1007/s00209-009-0510-2

Donkin, S., & Tange, R. (2010). The Brauer algebra and the symplectic Schur algebra. *Mathematische Zeitschrift*, *265*(1), 187-219. https://doi.org/10.1007/s00209-009-0510-2

Donkin S, Tange R. The Brauer algebra and the symplectic Schur algebra. Mathematische Zeitschrift. 2010 May 1;265(1):187-219. https://doi.org/10.1007/s00209-009-0510-2

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title = "The Brauer algebra and the symplectic Schur algebra",

abstract = "Let k be an algebraically closed field of characteristic p > 0, let m, r be integers with m a parts per thousand yen 1, r a parts per thousand yen 0 and m a parts per thousand yen r and let S (0)(2m, r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogonal Schur algebra under similar assumptions. We also show that, in general, the block relations of the Brauer algebra and the symplectic and orthogonal Schur algebra are the same. Finally, we deduce from the previous results a new proof of the description of the blocks of the Brauer algebra in characteristic 0 as obtained by Cox, De Visscher and Martin.",

keywords = "Brauer algebra, Symplectic Schur algebra, Jantzen sum formula, Young modules, Blocks, CENTRALIZER ALGEBRAS, SEMISIMPLICITY, REPRESENTATIONS, MODULES, MONOIDS",

author = "Stephen Donkin and Rudolf Tange",

year = "2010",

month = may,

day = "1",

doi = "10.1007/s00209-009-0510-2",

language = "English",

volume = "265",

pages = "187--219",

journal = "Mathematische Zeitschrift",

issn = "0025-5874",

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T1 - The Brauer algebra and the symplectic Schur algebra

AU - Donkin, Stephen

AU - Tange, Rudolf

PY - 2010/5/1

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N2 - Let k be an algebraically closed field of characteristic p > 0, let m, r be integers with m a parts per thousand yen 1, r a parts per thousand yen 0 and m a parts per thousand yen r and let S (0)(2m, r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogonal Schur algebra under similar assumptions. We also show that, in general, the block relations of the Brauer algebra and the symplectic and orthogonal Schur algebra are the same. Finally, we deduce from the previous results a new proof of the description of the blocks of the Brauer algebra in characteristic 0 as obtained by Cox, De Visscher and Martin.

AB - Let k be an algebraically closed field of characteristic p > 0, let m, r be integers with m a parts per thousand yen 1, r a parts per thousand yen 0 and m a parts per thousand yen r and let S (0)(2m, r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogonal Schur algebra under similar assumptions. We also show that, in general, the block relations of the Brauer algebra and the symplectic and orthogonal Schur algebra are the same. Finally, we deduce from the previous results a new proof of the description of the blocks of the Brauer algebra in characteristic 0 as obtained by Cox, De Visscher and Martin.

KW - Brauer algebra

KW - Symplectic Schur algebra

KW - Jantzen sum formula

KW - Young modules

KW - Blocks

KW - CENTRALIZER ALGEBRAS

KW - SEMISIMPLICITY

KW - REPRESENTATIONS

KW - MODULES

KW - MONOIDS

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U2 - 10.1007/s00209-009-0510-2

DO - 10.1007/s00209-009-0510-2

M3 - Article

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SP - 187

EP - 219

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 1

ER -