On the ring of 13×13 generic matrices

Let F be a field of characteristic 0. We investigate the rationality of the center Cn of the division algebra of two n×n generic matrices over F for n=13. If n is a prime power, then the only cases for which Cn 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 ZSn-lattice, denoted A*, which has played an essential role in the proofs of these results. Formanek proved the case n=4 by showing that Cn is stably isomorphic to F(A*−)Sn, the invariants of F(A*−) under the action of Sn. Here A*− is A* Z− and Z− is the sign representation of Sn. Lebruyn and Bessenrodt proved the cases n=5 and 7 by showing that Cn 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*− M)Sn and Cn are stably isomorphic. We then show that a field extension of degree 2 of Cn is stably isomorphic to a field extension of degree 2 of a rational extension of F.