Schur functors and dominant dimension

Fang, Ming; Koenig, Steffen

doi:10.1090/S0002-9947-2010-05177-3

Schur functors and dominant dimension
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by Ming Fang and Steffen Koenig PDF

Trans. Amer. Math. Soc. 363 (2011), 1555-1576 Request permission

Abstract:

The dominant dimension of an algebra $A$ provides information about the connection between $A\textrm {-mod}$ and $B\textrm {-mod}$ for $B=eAe$, a certain centralizer subalgebra of $A$. Well-known examples of such a situation are the connection (given by Schur-Weyl duality) between Schur algebras and group algebras of symmetric groups, and the connection (given by Soergel’s ’Struktursatz’) between blocks of the category $\mathcal O$ of a complex semisimple Lie algebra and the coinvariant algebra. We study cohomological aspects of such connections, in the framework of highest weight categories. In this setup we characterize the dominant dimension of $A$ by the vanishing of certain extension groups over $A$, we determine the range of degrees, for which certain cohomology groups over $A$ and over $eAe$ get identified, we show that Ringel duality does not change dominant dimensions and we determine the dominant dimension of Schur algebras.

References

Kaan Akin, Extensions of symmetric tensors by alternating tensors, J. Algebra 121 (1989), no. 2, 358–363. MR 992770, DOI 10.1016/0021-8693(89)90071-9
Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422, DOI 10.1017/CBO9780511623608
M. Auslander and Ø. Solberg, Relative homology and representation theory. III. Cotilting modules and Wedderburn correspondence, Comm. Algebra 21 (1993), no. 9, 3081–3097. MR 1228753, DOI 10.1080/00927879308824719
D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR 1110581
Jonathan Brundan, Richard Dipper, and Alexander Kleshchev, Quantum linear groups and representations of $\textrm {GL}_n(\textbf {F}_q)$, Mem. Amer. Math. Soc. 149 (2001), no. 706, viii+112. MR 1804485, DOI 10.1090/memo/0706
E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99. MR 961165
Vlastimil Dlab and Claus Michael Ringel, Quasi-hereditary algebras, Illinois J. Math. 33 (1989), no. 2, 280–291. MR 987824
Vlastimil Dlab and Claus Michael Ringel, The module theoretical approach to quasi-hereditary algebras, Representations of algebras and related topics (Kyoto, 1990) London Math. Soc. Lecture Note Ser., vol. 168, Cambridge Univ. Press, Cambridge, 1992, pp. 200–224. MR 1211481
Stephen Donkin, On tilting modules for algebraic groups, Math. Z. 212 (1993), no. 1, 39–60. MR 1200163, DOI 10.1007/BF02571640
S. Donkin, The $q$-Schur algebra, London Mathematical Society Lecture Note Series, vol. 253, Cambridge University Press, Cambridge, 1998. MR 1707336, DOI 10.1017/CBO9780511600708
Stephen R. Doty, Karin Erdmann, and Daniel K. Nakano, Extensions of modules over Schur algebras, symmetric groups and Hecke algebras, Algebr. Represent. Theory 7 (2004), no. 1, 67–100. MR 2046956, DOI 10.1023/B:ALGE.0000019454.27331.59
Stephen R. Doty and Daniel K. Nakano, Relating the cohomology of general linear groups and symmetric groups, Modular representation theory of finite groups (Charlottesville, VA, 1998) de Gruyter, Berlin, 2001, pp. 175–187. MR 1889344
Ming Fang, Schur functors on QF-3 standardly stratified algebras, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 2, 311–318. MR 2383358, DOI 10.1007/s10114-007-0984-y
Vincent Franjou, Eric M. Friedlander, Alexander Scorichenko, and Andrei Suslin, General linear and functor cohomology over finite fields, Ann. of Math. (2) 150 (1999), no. 2, 663–728. MR 1726705, DOI 10.2307/121092
James A. Green, Polynomial representations of $\textrm {GL}_{n}$, Lecture Notes in Mathematics, vol. 830, Springer-Verlag, Berlin-New York, 1980. MR 606556, DOI 10.1007/BFb0092296
David J. Hemmer and Daniel K. Nakano, Specht filtrations for Hecke algebras of type A, J. London Math. Soc. (2) 69 (2004), no. 3, 623–638. MR 2050037, DOI 10.1112/S0024610704005186
Gordon D. James, The decomposition of tensors over fields of prime characteristic, Math. Z. 172 (1980), no. 2, 161–178. MR 580858, DOI 10.1007/BF01182401
Alexander S. Kleshchev and Daniel K. Nakano, On comparing the cohomology of general linear and symmetric groups, Pacific J. Math. 201 (2001), no. 2, 339–355. MR 1875898, DOI 10.2140/pjm.2001.201.339
Steffen König, Inger Heidi Slungård, and Changchang Xi, Double centralizer properties, dominant dimension, and tilting modules, J. Algebra 240 (2001), no. 1, 393–412. MR 1830559, DOI 10.1006/jabr.2000.8726
Volodymyr Mazorchuk and Serge Ovsienko, Finitistic dimension of properly stratified algebras, Adv. Math. 186 (2004), no. 1, 251–265. MR 2065514, DOI 10.1016/j.aim.2003.08.001
Yoichi Miyashita, Tilting modules of finite projective dimension, Math. Z. 193 (1986), no. 1, 113–146. MR 852914, DOI 10.1007/BF01163359
Kiiti Morita, Duality in $\textrm {QF}-3$ rings, Math. Z. 108 (1969), 237–252. MR 241470, DOI 10.1007/BF01112025
Bruno J. Müller, The classification of algebras by dominant dimension, Canadian J. Math. 20 (1968), 398–409. MR 224656, DOI 10.4153/CJM-1968-037-9
Daniel K. Nakano, Some recent developments in the representation theory of general linear and symmetric groups, Representations and quantizations (Shanghai, 1998) China High. Educ. Press, Beijing, 2000, pp. 357–373. MR 1802183
Claus Michael Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), no. 2, 209–223. MR 1128706, DOI 10.1007/BF02571521
Wolfgang Soergel, Kategorie $\scr O$, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421–445 (German, with English summary). MR 1029692, DOI 10.1090/S0894-0347-1990-1029692-5
Yasutaka Suzuki, Dominant dimension of double centralizers, Math. Z. 122 (1971), 53–56. MR 289582, DOI 10.1007/BF01113563
Hiroyuki Tachikawa, On dominant dimensions of $\textrm {QF}$-3 algebras, Trans. Amer. Math. Soc. 112 (1964), 249–266. MR 161888, DOI 10.1090/S0002-9947-1964-0161888-8
Hiroyuki Tachikawa, Double centralizers and dominant dimensions, Math. Z. 116 (1970), 79–88. MR 265407, DOI 10.1007/BF01110189
Hiroyuki Tachikawa, Quasi-Frobenius rings and generalizations. $\textrm {QF}-3$ and $\textrm {QF}-1$ rings, Lecture Notes in Mathematics, Vol. 351, Springer-Verlag, Berlin-New York, 1973. Notes by Claus Michael Ringel. MR 0349740, DOI 10.1007/BFb0059997
Burt Totaro, Projective resolutions of representations of $\textrm {GL}(n)$, J. Reine Angew. Math. 482 (1997), 1–13. MR 1427655, DOI 10.1515/crll.1997.482.1
Kunio Yamagata, Frobenius algebras, Handbook of algebra, Vol. 1, Handb. Algebr., vol. 1, Elsevier/North-Holland, Amsterdam, 1996, pp. 841–887. MR 1421820, DOI 10.1016/S1570-7954(96)80028-3