On the ring of 13 x 13 generic matrices

[摘要] Let F be a field of characteristic 0. We investigate the rationality of the center C-n of the division algebra of two n x n generic matrices over F for n = 13. If n is a prime power, then the only cases for which C-n is known to be stably rational over F are n = 2, 3, 4, 5, and 7 with rationality proven for 2, 3, and 4. There is a certain Z S-n-lattice, denoted A*, which has played an essential role in the proofs of these results. Formanek proved the case n = 4 by showing that C, is stably isomorphic to F(A*(-))(Sn), the invariants of F(A*(-)) under the action of S-n. Here A*(-) is A* circle times Z(-) and Z(-) is the sign representation of S, Lebruyn and Bessenrodt proved the cases n = 5 and 7 by showing that C-n is stably isomorphic to F(A*)(Sn). We show that for n = 13 there exists a ZSn-lattice M, which is stably permutation when restricted to the alternating group, such that F(A*(-) circle plusM)(Sn) and C-n are stably isomorphic. We then show that a field extension of degree 2 of C-n is stably isomorphic to a field extension of degree 2 of a rational extension of F. (C) 2004 Elsevier Inc. All rights reserved.

[发布日期] 2004-12-15

[关键词] rationality;generic matrices;flasque classes