Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Remote Access
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Standard polynomials in matrix algebras


Author: Louis H. Rowen
Journal: Trans. Amer. Math. Soc. 190 (1974), 253-284
MSC: Primary 15A30
MathSciNet review: 0349715
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {M_n}(F)$ be an $ n \times n$ matrix ring with entries in the field F, and let $ {S_k}({X_1}, \ldots ,{X_k})$ be the standard polynomial in k variables. Amitsur-Levitzki have shown that $ {S_{2n}}({X_1}, \ldots ,{X_{2n}})$ vanishes for all specializations of $ {X_1}, \ldots ,{X_{2n}}$ to elements of $ {M_n}(F)$. Now, with respect to the transpose, let $ M_n^ - (F)$ be the set of antisymmetric elements and let $ M_n^ + (F)$ be the set of symmetric elements. Kostant has shown using Lie group theory that for n even $ {S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ vanishes for all specializations of $ {X_1}, \ldots ,{X_{2n - 2}}$ to elements of $ M_n^ - (F)$. By strictly elementary methods we have obtained the following strengthening of Kostant's theorem:

$ {S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ vanishes for all specializations of $ {X_1}, \ldots ,{X_{2n - 2}}$ to elements of $ M_n^ - (F)$, for all n.

$ {S_{2n - 1}}({X_1}, \ldots ,{X_{2n - 1}})$ vanishes for all specializations of $ {X_1}, \ldots ,{X_{2n - 2}}$ to elements of $ M_n^ - (F)$ and of $ {X_{2n - 1}}$ to an element of $ M_n^ + (F)$, for all n.

$ {S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ vanishes for all specializations of $ {X_1}, \ldots ,{X_{2n - 3}}$ to elements of $ M_n^ - (F)$ and of $ {X_{2n - 2}}$ to an element of $ M_n^ + (F)$, for n odd.

These are the best possible results if F has characteristic 0; a complete analysis of the problem is also given if F has characteristic 2.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 15A30

Retrieve articles in all journals with MSC: 15A30


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0349715-3
PII: S 0002-9947(1974)0349715-3
Keywords: Antisymmetric, involution, matrix algebra, polynomial identity, standard identity, symmetric, transpose
Article copyright: © Copyright 1974 American Mathematical Society



Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia