New classes of examples satisfying the three matrix analog of Gerstenhaber's theorem

JOURNAL OF ALGEBRA

卷:516

Article

Rajchgot, Jenna¹  Satriano, Matthew² 

[1] Univ Saskatchewan, Dept Math & Stat, Saskatoon, SK S7N 5E6, Canada

[2] Univ Waterloo, Dept Pure Math, Waterloo, ON N2L 3G1, Canada

关键词: Commuting matrices;    Gerstenhaber's theorem;    Commuting scheme;   

DOI  :  10.1016/j.jalgebra.2018.09.020

来源: Elsevier

【 摘 要 】

In 1961, Gerstenhaber proved the following theorem: if k is a field and X and Y are commuting d x d matrices with entries in k, then the unital k-algebra generated by these matrices has dimension at most d. The analog of this statement for four or more commuting matrices is false. The three matrix version remains open. We use commutative-algebraic techniques to prove that the three matrix analog of Gerstenhaber's theorem is true for some new classes of examples. In particular, we translate this three commuting matrix statement into an equivalent statement about certain maps between modules, and prove that this commutative-algebraic reformulation is true in special cases. We end with ideas for an inductive approach intended to handle the three matrix analog of Gerstenhaber's theorem more generally. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free