On the ring of [formula omitted] generic matrices

Abstract

Let F be a field of characteristic 0. We investigate the rationality of the center C n 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 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 n is stably isomorphic to F ( A * − ) S n , the invariants of F ( A * − ) under the action of S n . Here A * − is A * ⊗ Z − and Z − is the sign representation of S n . Lebruyn and Bessenrodt proved the cases n = 5 and 7 by showing that C n is stably isomorphic to F ( A * ) S n . We show that for n = 13 there exists a Z S n -lattice M, which is stably permutation when restricted to the alternating group, such that F ( A * − ⊕ M ) S n 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.