Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Induction theorems on the stable rationality
of the center of the ring of generic matrices


Author: Esther Beneish
Journal: Trans. Amer. Math. Soc. 350 (1998), 3571-3585
MSC (1991): Primary 13A50, 20C10
DOI: https://doi.org/10.1090/S0002-9947-98-02202-8
MathSciNet review: 1475675
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Following Procesi and Formanek, the center of the division ring of $n\times n$ generic matrices over the complex numbers $\mathbf C$ is stably equivalent to the fixed field under the action of $S_n$, of the function field of the group algebra of a $ZS_n$-lattice, denoted by $G_n$. We study the question of the stable rationality of the center $C_n$ over the complex numbers when $n$ is a prime, in this module theoretic setting. Let $N$ be the normalizer of an $n$-sylow subgroup of $S_n$. Let $M$ be a $ZS_n$-lattice. We show that under certain conditions on $M$, induction-restriction from $N$ to $S_n$ does not affect the stable type of the corresponding field. In particular, $\mathbf C (G_n)$ and $\mathbf C(ZG\otimes _{ZN}G_n)$ are stably isomorphic and the isomorphism preserves the $S_n$-action. We further reduce the problem to the study of the localization of $G_n$ at the prime $n$; all other primes behave well. We also present new simple proofs for the stable rationality of $C_n$ over $\mathbf C$, in the cases $n=5$ and $n=7$.


References [Enhancements On Off] (What's this?)

  • [B] E. Beneish, Invertible Modules, J. Algebra 128 (1990), 101-125. MR 91b:20006
  • [B1] -, The Grothendieck ring of invertible modules over nilpotent groups, J. Algebra 159 (1993), 400-418. MR 94i:20009
  • [BK] K. Brown, Cohomology of groups, Springer-Verlag, New York, 1982. MR 83k:20002
  • [BL] C. Bessenrodt and L. Le Bruyn, Stable rationality of certain $PGL_n$-quotients, Invent. Math. 104 (1991), 179-199. MR 92m:14060
  • [CT] J.-L. Colliot-Thelene et J.-J. Sansuc, La $R$-equivalence sur les tores, Ann. Sci. Ecole Norm. Sup. (4) 10 (1977), 175-230. MR 56:8576
  • [CR1] C. W. Curtis and I. Reiner, Methods of Representation Theory, vol. 1, Wiley, New York, 1981. MR 90k:20001
  • [CR2] -, Methods of Representation Theory, vol. 2, Wiley, New York, 1987. MR 88f:20002
  • [EM] S. Endo and T. Miyata, On the projective class group of finite groups, Osaka J. Math. 13 (1976), 109-122. MR 53:13315
  • [F1] E. Formanek, The center of the ring of $3\times 3$ generic matrices, Linear and Multilinear Algebra 7 (1979), 203-212. MR 80h:16019
  • [F2] -, The center of the ring of $4\times 4$ generic matrices, J. Algebra 62 (1980), 304-319. MR 81g:15032
  • [L] H. W. Lenstra, Rational functions invariant under a finite abelian group, Invent. Math. 25 (1974), 299-325. MR 50:289
  • [LL] L. Le Bruyn, Centers of generic division algebras, the rationality problem 1965-1990, Israel Journal of Math. 76 (1991), 97-111. MR 93f:16024
  • [P] C. Procesi, Non-commutative affine rings, Atti Accad. Naz. Lincei 8 (1967), 237-255. MR 37:256
  • [SD] D. Saltman, Retract rational fields and cyclic Galois extensions, Israel J. Math. 47 (1984), 165-215. MR 85j:13008
  • [SR] R. Swan, Invariant rational functions and a problem of Steenrod, Invent. Math. 7 (1969), 148-158. MR 39:5532
  • [S] J. Sylvester, On the involution of two matrices of second order, Southport: British Assoc. Report (1883), 430-432.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 13A50, 20C10

Retrieve articles in all journals with MSC (1991): 13A50, 20C10


Additional Information

Esther Beneish
Affiliation: Department of Mathematics, University of Wisconsin-Parkside, Kenosha, Wisconsin 53141-2000

DOI: https://doi.org/10.1090/S0002-9947-98-02202-8
Received by editor(s): September 22, 1996
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society