Book Review
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Book Information:
Author:
Stuart Martin
Title:
Schur algebras and representation theory
Additional book information:
Cambridge Tracts in Math., vol. 112, Cambridge University Press,
Cambridge,
1993,
xv + 232 pp.,
ISBN 0-521-41591-8,
$44.95$
Michael Artin, William Schelter, and John Tate, Quantum deformations of $\textrm {GL}_n$, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 879–895. MR 1127037, DOI 10.1002/cpa.3160440804
Roger W. Carter and George Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242. MR 354887, DOI 10.1007/BF01214125
E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99. MR 961165
Richard Dipper, Polynomial representations of finite general linear groups in nondescribing characteristic, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 343–370. MR 1112168
Richard Dipper and Stephen Donkin, Quantum $\textrm {GL}_n$, Proc. London Math. Soc. (3) 63 (1991), no. 1, 165–211. MR 1105721, DOI 10.1112/plms/s3-63.1.165
Richard Dipper and Gordon James, The $q$-Schur algebra, Proc. London Math. Soc. (3) 59 (1989), no. 1, 23–50. MR 997250, DOI 10.1112/plms/s3-59.1.23
[Do1] S. Donkin, The blocks of a semisimple group, J. Algebra 67 (1980), 36--53.
S. Donkin, On Schur algebras and related algebras. I, J. Algebra 104 (1986), no. 2, 310–328. MR 866778, DOI 10.1016/0021-8693(86)90218-8
Stephen Donkin, On Schur algebras and related algebras. IV. The blocks of the Schur algebras, J. Algebra 168 (1994), no. 2, 400–429. MR 1292772, DOI 10.1006/jabr.1994.1236
Jie Du, Brian Parshall, and Jian Pan Wang, Two-parameter quantum linear groups and the hyperbolic invariance of $q$-Schur algebras, J. London Math. Soc. (2) 44 (1991), no. 3, 420–436. MR 1149005, DOI 10.1112/jlms/s2-44.3.420
Ferdinand Georg Frobenius, Gesammelte Abhandlungen. Bände I, II, III, Springer-Verlag, Berlin-New York, 1968 (German). Herausgegeben von J.-P. Serre. MR 0235974
James A. Green, Polynomial representations of $\textrm {GL}_{n}$, Lecture Notes in Mathematics, vol. 830, Springer-Verlag, Berlin-New York, 1980. MR 606556
G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
Michio Jimbo, A $q$-analogue of $U({\mathfrak {g}}{\mathfrak {l}}(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247–252. MR 841713, DOI 10.1007/BF00400222
I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 553598
Brian J. Parshall, Finite-dimensional algebras and algebraic groups, Classical groups and related topics (Beijing, 1987) Contemp. Math., vol. 82, Amer. Math. Soc., Providence, RI, 1989, pp. 97–114. MR 982281, DOI 10.1090/conm/082/982281
[PS1] B. Parshall and L. Scott, Derived categories, quasi-hereditary algebras, and algebraic groups, Carlton Univ. Math. Notes 3 (1989), 1--105.
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
Brian Parshall and Jian Pan Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89 (1991), no. 439, vi+157. MR 1048073, DOI 10.1090/memo/0439
Issai Schur, Gesammelte Abhandlungen. Band I, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. MR 0462891
Issai Schur, Gesammelte Abhandlungen. Band I, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. MR 0462891
Leonard L. Scott, Simulating algebraic geometry with algebra. I. The algebraic theory of derived categories, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 271–281. MR 933417
Robert Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33–56. MR 155937
[W] H. Weyl, The classical groups, Princeton: Princeton U. Press, 1939.
- [AST]
- M. Artin, W. Schelter, J. Tate, Quantum deformations of
, Comm. Pure and Applied Math. 44 (1991), 879--895. MR 1127037
- [CL]
- R. Carter and G. Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193--242. MR 0354887
- [CPS]
- E. Cline, B. Parshall, and L. Scott, Finite dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85--99. MR 0961165
- [D]
- R. Dipper, Polynomial representations of finite general linear groups in non-describing characteristic, Progress in Mathematics, vol. 95, Birkhäuser, Basel, 1991, pp. 343--370. MR 1112168
- [DD]
- R. Dipper and S. Donkin, Quantum
, Proc. London Math. Soc. 63 (1991), 165--211. MR 1105721
- [DJ]
- R. Dipper and G. James, The
-Schur algebra, J. London Math. Soc. 59 (1989), 23--50. MR 0997250
- [Do1]
- S. Donkin, The blocks of a semisimple group, J. Algebra 67 (1980), 36--53.
- [Do2]
- ------, On Schur algebras and related algebras II, J. Algebra 111 (1987), 354--364. MR 0866778
- [Do3]
- ------, On Schur algebras and related algebras IV: The blocks of the Schur algebras, J. Algebra 168 (1994), 400--429. MR 1292772
- [DPW]
- J. Du, B. Parshall, and J.-p. Wang, Two-parameter quantum linear groups and the hyperbolic invariance of
-Schur algebras, J. London Math. Soc. 44 (1991), 420--436. MR 1149005
- [F]
- G. Frobenius, Über die Charaktere der symmetrischen Groupe, G. Frobenius: Gesammelte Abhandlungen 3 (1968), Heidelberg: Springer-Verlag, pp. 167--179. MR 0235974
- [G]
- J.A. Green, Polynomial representations of
, Springer Lecture Notes in Math., vol. 830, Springer-Verlag, Berlin and New York, 1980. MR 0606556
- [J]
- G. James, The representation theory of the symmetric group, Springer Lecture Notes in Math., vol. 682, Springer, Berlin, 1978. MR 0513828
- [Ji]
- M. Jimbo, A
-analogue of 
, Hecke algebras, and the Yang-Baxter equation, Letters Math. Physics 11 (1986), 247--252. MR 0841713
- [M]
- I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford: Clarendon Press, 1979. MR 0553598
- [P]
- B. Parshall, Finite dimensional algebras and algebraic groups, Contemp. Math., vol. 82, Amer. Math. Soc., Providence, RI, 1989, pp. 97--114. MR 0982281
- [PS1]
- B. Parshall and L. Scott, Derived categories, quasi-hereditary algebras, and algebraic groups, Carlton Univ. Math. Notes 3 (1989), 1--105.
- [PS2]
- ------, Koszul algebras and the Frobenius morphism, Quart. Jour. Math. 46 (1995), 345--384. MR 1:348 822
- [PW]
- B. Parshall and J. -p. Wang, Quantum Linear Groups, Amer. Math. Soc. ``Memoirs'', vol. 89, no. 439 (1991). MR 1048073
- [S1]
- I. Schur, Über eine Klasse von Matrizen, die sich einer gegeben Matrix zuordnen lassen, I. Schur: Gesammelte Abhandlungen 1 (1973), Berlin: Springer-Verlag, pp. 1--71. MR 0462891
- [S2]
- ------, Über die rationalen Darstellungen der allgemeinen linearen Gruppe, I. Schur: Gesammelte Abhandlungen 3 (1973), Berlin: Springer-Verlag, pp. 68--85. MR 0462891
- [S]
- L. Scott, Simulating algebraic geometry with algebra I: The algebraic theory of derived categories, Proc. Symposia of Pure Math., vol. 47, Part 2, Amer. Math. Soc., Providence, RI, 1987, pp. 271--281. MR 0933417
- [St]
- R. Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33--56. MR 0155937
- [W]
- H. Weyl, The classical groups, Princeton: Princeton U. Press, 1939.
Review Information:
Reviewer:
Brian Parshall
Affiliation:
University of Virginia
Email:
bjp8w@virginia.edu
Journal:
Bull. Amer. Math. Soc.
33 (1996), 371-375
DOI:
https://doi.org/10.1090/S0273-0979-96-00663-5
Review copyright:
© Copyright 1996
American Mathematical Society
