The Center of the Division Ring of pxp Generic Matrices

  • Beneish, Esther (PI)

Grant Details

Description

The main objective of the proposed work is to investigate the following problem. Let p be a prime. Is the center, Cp, of the division ring of p x p generic matrices stably rational over the base field? This is an old problem which has been studied extensively over several decades, in particular for its connections to important problems in other fields, such as geometric invariant theory and Brauer groups. If G is a finite group, and M is a ZG-lattice, then F(M) denotes the quotient field of the group algebra of the abelian group M written multiplicatively. Procesi and Formanek have shown that Cn is isomorphic to the fixed field under the action of Sn of F(M) for a specific ZSn-lattice M. The findings of the investigator can be briefly described as two reduction steps. Let p be a prime. First, the investigator has shown that under certain conditions induction-restriction from Sp to the normalizer N, of a p-sylow subgroup H of Sp, does not affect the stable type of the field.. Second, the investigator has proved that if we twist the Sp-action by a 1-cocycle a in Ext1SP(J(A), L*), then Cp is stably isomorphic to La(J(A))Sp, where A is the root lattice, L=F(ZG/H) and, J represents induction-restriction from Sp to H. Note that now we are inducing from H which is a cyclic group of order p. The cocycle a corresponds to an element of the relative Brauer group of L over LH. The importance of this result lies in the connection, described by Saltman., between fixed fields under twisted H-actions and corresponding fixed fields under twisted Sp-actions. The above results naturally lead to the study of a-twisted actions on fields, and this is also an area the investigator will study; more precisely actions induced from subgroups. The investigator will be also working on Noether's problem which is naturally related to the above question. In particular the investigator will study its connections with monomial and a-twisted group actions and on fields.

The focus of this project is to investigate the following problem. Let p be a prime number. Is the center of the division ring of p x p generic matrices stably rational over the base field? Generic matrices over a field F are n x n matrices whose entries are independent variables. This problem has been studied extensively over several decades. Its importance lies in part, in its connections to problems in many other fields such as, geometric invariant theory, representation theory of algebras, and Brauer groups. Invariant theory and representation theory are classical mathematical fields whose results are also used in applied mathematics and physics. Brauer groups are a fundamental structure in the study of central simple algebras. The importance of this problem also lies in the fact that matrices, which are rectangular arrays of elements, are an essential tool for storing data. Little is known about this problem. Positive results have been proved for the primes 2, 3,5 and 7, the first of which was found in 1883 by Sylvester.

StatusFinished
Effective start/end date09/1/0204/30/04

Funding

  • National Science Foundation: $8,396.00

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