Top

Published in:

2016 | OriginalPaper | Chapter

Schur–Weyl Dualities Old and New

Author : Anne Henke

Published in: From Arithmetic to Zeta-Functions

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

We give an overview of Schur–Weyl dualities involved in the representation theory of orthogonal and symplectic groups and of the properties of a new class of algebras occurring in this context, the Brauer Schur algebras.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Sum of the Lerch Zeta-Function over Nontrivial Zeros of the Dirichlet -Function

next chapter Arithmetic Functions: A Pivotal Topic in the Scientific Work of Wolfgang Schwarz

A.M. Adamovich, G.L. Rybnikov, Tilting modules for classical groups and Howe duality in positive characteristic. Transform. Groups 1 (1–2), 1–34 (1996)MathSciNetCrossRefMATH

R. Brauer, On algebras which are connected with the semisimple continuous groups. Ann. Math. 38, 854–872 (1937)MathSciNetCrossRefMATH

W.P. Brown, The semisimplicity of ω _f ⁿ. Ann. Math. (2) 63, 324–335 (1956)

R. Carter, G. Lusztig, On the modular representations of the general linear and symmetric groups. Math. Z. 136, 193–242 (1974)MathSciNetCrossRefMATH

E. Cline, B. Parshall, L. Scott, Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)

E. Cline, B. Parshall, L. Scott, Algebraic stratification in representation categories. J. Algebra 117, 504–521 (1988)MathSciNetCrossRefMATH

C. De Concini, C. Procesi, A characteristic free approach to invariant theory. Adv. Math. 21 (3), 330–354 (1976)MathSciNetCrossRefMATH

R. Dipper, S. Doty, J. Hu, Brauer’s centralizer algebras, symplectic Schur algebras, and Schur-Weyl duality. Trans. Am. Math. Soc. 360, 189–213 (2008)MathSciNetCrossRefMATH

V. Dlab, C.M. Ringel, Quasi-hereditary algebras. Illinois J. Math. 33, 280–291 (1989)MathSciNetMATH

10.

V. Dlab, C.M. Ringel, A construction for quasi-hereditary algebras. Compos. Math. 70, 155–175 (1989)MathSciNetMATH

11.

V. Dlab, C.M. Ringel, The module theoretical approach to quasi-hereditary algebras, in Representations of Algebras and Related Topics (Kyoto, 1990). London Mathematical Society Lecture Note Series, vol. 168 (Cambridge University Press, Cambridge, 1992), pp. 200–224

12.

S. Donkin, On Schur algebras and related algebras. I. J. Algebra 104 (2), 310–328 (1986); On Schur algebras and related algebras. II. J. Algebra 111 (2), 354–364 (1987)

13.

S. Donkin, On tilting modules for algebraic groups. Math. Z. 212 (1), 39–60 (1993)MathSciNetCrossRefMATH

14.

S. Donkin, On tilting modules and invariants for algebraic groups, in Finite-Dimensional Algebras and Related Topics (Ottawa, ON, 1992). NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 424 (Kluwer Academic Publishers, Dordrecht, 1994), pp. 59–77

15.

S. Donkin, The q-Schur Algebra. London Mathematical Society Lecture Note Series, vol. 253 (Cambridge University Press, Cambridge, 1998), x+179 pp.

16.

S. Donkin, R. Tange, The Brauer algebra and the symplectic Schur algebra. Math. Z. 265, 187–219 (2010)MathSciNetCrossRefMATH

17.

S. Doty, J. Hu, Schur-Weyl duality for orthogonal groups. Proc. Lond. Math. Soc. (3) 98, 679–713 (2009)

18.

K. Erdmann, A. Henke, On Ringel duality for Schur algebras. Math. Proc. Camb. Philos. Soc. 132, 97–116 (2002)MathSciNetCrossRefMATH

19.

M. Fang, S. Koenig, Schur functors and dominant dimension. Trans. Am. Math. Soc. 363 (3), 1555–1576 (2011)MathSciNetCrossRefMATH

20.

R. Goodman, N. Wallach, Representations and invariants of the classical groups, in Encyclopedia of Mathematics and Its Applications, vol. 68 (Cambridge University Press, Cambridge, 1998), xvi+685 pp.

21.

J. Graham, G. Lehrer, Cellular algebras. Invent. Math. 123, 1–34 (1996)MathSciNetCrossRefMATH

22.

J.A. Green, Polynomial Representations of GL_n. Lecture Notes in Mathematics, vol. 830 (Springer, Berlin/New York, 1980)

23.

P. Hanlon, D. Wales, On the decomposition of Brauer’s centralizer algebras. J. Algebra 121, 409–445 (1989)MathSciNetCrossRefMATH

24.

R. Hartmann, R. Paget, Young modules and filtration multiplicities for Brauer algebras. Math. Z. 254, 333–357 (2006)MathSciNetCrossRefMATH

25.

R. Hartmann, A. Henke, S. Koenig, R. Paget, Cohomological stratification of diagram algebras. Math. Ann. 347, 765–804 (2010)MathSciNetCrossRefMATH

26.

D. Hemmer, D. Nakano, Specht filtrations for Hecke algebras of type A. J. Lond. Math. Soc. (2) 69, 623–638 (2004)

27.

A. Henke, S. Koenig, Schur algebras of Brauer algebras, I. Math. Z. 272 (3–4), 729–759 (2012)MathSciNetCrossRefMATH

28.

A. Henke, S. Koenig, Schur algebras of Brauer algebras, II. Math. Z. 276 (3), 1077–1099 (2014)MathSciNetCrossRefMATH

29.

A. Henke, R. Paget, Brauer algebras with parameter n = 2 acting on tensor space. Algebr. Represent. Theory 11 (6), 545–575 (2008)

30.

S. Koenig, C.C. Xi, A characteristic free approach to Brauer algebras. Trans. Am. Math. Soc. 353, 1489–1505 (2001)MathSciNetCrossRefMATH

31.

S. Koenig, I.H. Slungard, C.C. Xi, Double centralizer properties, dominant dimension, and tilting modules. J. Algebra 240, 393–412 (2001)MathSciNetCrossRef

32.

G.I. Lehrer, R.B. Zhang, The Brauer category and invariant theory. J. Eur. Math. Soc. (JEMS) 17 (9), 2311–2351 (2015)

33.

Q. Liu, Schur algebras of classical groups. J. Algebra 301, 867–887 (2006)MathSciNetCrossRefMATH

34.

Q. Liu, Schur algebras of classical groups II. Commun. Algebra 38, 2656–2676 (2010)MathSciNetCrossRefMATH

35.

S. Oehms, Centralizer coalgebras, FRT-construction, and symplectic monoids. J. Algebra 244, 19–44 (2001)MathSciNetCrossRefMATH

36.

C.M. Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences. Math. Z. 208, 209–223 (1991)MathSciNetCrossRefMATH

37.

R. Rouquier, q-Schur algebras and complex reflection groups. Moscow Math. J. 8, 119–158 (2008)

38.

I. Schur, Über die rationalen Darstellungen der allgemeinen linearen Gruppe. Sitzungsberichte Akad. Berlin, 58–75 (1927) (Reprinted as: I. Schur, Gesammelte Abhandlungen III (Springer, Berlin, 1973), pp. 68–85)

39.

L.L. Scott, Simulating algebraic geometry with algebra I: the algebraic theory of derived categories. Proc. Symp. Pure Math. 47, 271–281 (1987)MathSciNetCrossRef

40.

R. Tange, The symplectic ideal and a double centraliser theorem. J. Lond. Math. Soc. (2) 77 (3), 687–699 (2008)

41.

H. Wenzl, On the structure of Brauer’s centralizer algebras. Ann. Math (2) 128 (1), 173–193 (1988)

Title: Schur–Weyl Dualities Old and New
Author: Anne Henke
Publisher: Springer International Publishing
Book: From Arithmetic to Zeta-Functions
Print ISBN: 978-3-319-28202-2

Electronic ISBN: 978-3-319-28203-9

Copyright Year: 2016
DOI: https://doi.org/10.1007/978-3-319-28203-9_11

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner