Lattice invariants and the center of the generic division ring

Author:
Esther Beneish

Journal:
Trans. Amer. Math. Soc. **356** (2004), 1609-1622

MSC (2000):
Primary 20C10, 16R30, 13A50, 16K20

DOI:
https://doi.org/10.1090/S0002-9947-03-03331-2

Published electronically:
October 21, 2003

MathSciNet review:
2034321

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a finite group, let be a -lattice, and let be a field of characteristic zero containing primitive roots of 1. Let be the quotient field of the group algebra of the abelian group . It is well known that if is quasi-permutation and -faithful, then is stably equivalent to . Let be the center of the division ring of generic matrices over . Let be the symmetric group on symbols. Let be a prime. We show that there exist a split group extension of by a -elementary group, a -faithful quasi-permutation -lattice , and a one-cocycle in such that is stably isomorphic to . This represents a reduction of the problem since we have a quasi-permutation action; however, the twist introduces a new level of complexity. The second result, which is a consequence of the first, is that, if is algebraically closed, there is a group extension of by an abelian -group such that is stably equivalent to the invariants of the Noether setting .

**[B1]**E. Beneish,*Induction theorems on the center of the ring of generic matrices*, Transactions of the AMS,**350**(1998), no. 9, 3571-3585. MR**98k:16034****[B2]**E. Beneish,*Monomial actions of the symmetric group*, J. of Algebra**265**(2003), 405-419.**[BJ]**J. Barge,*Cohomologie des groupes et corps d'invariants multiplicatifs tordus*, Comment. Math. Helv.**72**(1997), 1-15. MR**98g:12006****[BK]**K. Brown,*Cohomology of Groups*, Springer-Verlag, 1982. MR**83k:20002****[BL]**C. Bessenrodt and L. Lebruyn,*Stable rationality of certain -quotients*, Inventiones Mathematica**104**(1991), 179-199. MR**92m:14060****[CR]**C. Curtis and I. Reiner,*Methods of Representation Theory*, Vol. 1, Springer-Verlag, 1982.**[CTS]**J.-L. Colliot-Thelene et J.-P. Sansuc,*La -equivalence sur les Tores*, Ann. Sci. Ecole Normale Sup. (4)**10**(1977), 175-230. MR**56:8576****[EM]**S. Endo and T. Miyata,*On the classification of the function fields of algebraic tori*, Nagoya Math. J.**56**(1974), 85-104. MR**51:458****[F1]**E. Formanek,*The center of the ring of generic matrices*, Linear Multilinear Algebra**7**, (1979), 203-212. MR**80h:16019****[F2]**E. Formanek,*The center of the ring of generic matrices*, J. of Algebra**62**(1980), 304-319. MR**81g:15032****[L]**H. W. Lenstra,*Rational functions invariant under a finite abelian group*, Inventiones Mathematica**25**(1974), 299-325. MR**50:289****[S]**J. Sylvester,*On the involution of two matrices of second order*, Southport: British Assoc. Report (1883), 430-432.**[SD1]**D. Saltman,*Retract rational fields and cyclic Galois extensions*, Israel J. of Math.**47**(1984), 165-215. MR**85j:13008****[SD2]**D. Saltman,*Multiplicative field invariants*, Journal of Algebra**105**, (1987), 221-238. MR**88f:12007****[SD3]**D. Saltman,*The Schur index and Moody's theorem*, -Theory, Art No. 168, PIPS. No. 35888 (1993). MR**94k:16049****[SR]**R. Swan,*Noether's problem in Galois theory*. In: Srinivasan, B. Sally, J. (eds.) Emmy Noether in Bryn Mawr. Berlin, Heidelberg, New York, Springer, 1983, pp. 21-40. MR**84k:12013****[W]**D. J. Winter,*The structure of fields*, Springer-Verlag, 1974. MR**52:10703**

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Additional Information

**Esther Beneish**

Affiliation:
Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan 48859

Email:
benei1e@cmich.edu

DOI:
https://doi.org/10.1090/S0002-9947-03-03331-2

Received by editor(s):
May 13, 2002

Received by editor(s) in revised form:
March 7, 2003

Published electronically:
October 21, 2003

Additional Notes:
This work was partially supported by NSF grant #DMS-0070665

Article copyright:
© Copyright 2003
American Mathematical Society