Invariant theory in exterior algebras and Amitsur–Levitzki type theorems

Minoru Itoh

doi:10.1016/j.aim.2015.09.033

Abstract

This article discusses invariant theories in some exterior algebras, which are closely related to Amitsur–Levitzki type theorems.First we consider the exterior algebra on the vector space of square matrices of size n, and look at the invariants under conjugations. We see that the algebra of these invariants is isomorphic to the exterior algebra on an n-dimensional vector space. Moreover we give a Cayley–Hamilton type theorem for these invariants (the anticommutative version of the Cayley–Hamilton theorem). This Cayley–Hamilton type theorem can also be regarded as a refinement of the Amitsur–Levitzki theorem.We discuss two more Amitsur–Levitzki type theorems related to invariant theories in exterior algebras. One is a famous Amitsur–Levitzki type theorem due to Kostant and Rowen, and this is related to O(V)-invariants in Λ(Λ2(V)). The other is a new Amitsur–Levitzki type theorem, and this is related to GL(V)-invariants in Λ(Λ2(V)⊕S2(V⁎)).

Highlights

In this article, we discuss invariant theory in exterior algebras on some matrix spaces, and give several Cayley–Hamilton type relations for invariants in these exterior algebras as consequences of the second fundamental theorem of invariant theory for vector invariants
We discuss invariant theory in exterior algebras on some matrix spaces, and give several Cayley–Hamilton type relations for invariants in these exterior algebras as consequences of the second fundamental theorem of invariant theory for vector invariants. These Cayley–Hamilton type relations are all closely related to Amitsur–Levitzki type theorems
We first consider GL(V )-invariants in the exterior algebra Λ(V ⊗V ∗), where V is an n-dimensional complex vector space, and V ∗ is its linear dual

Summary

Introduction

We discuss invariant theory in exterior algebras on some matrix spaces, and give several Cayley–Hamilton type relations for invariants in these exterior algebras as consequences of the second fundamental theorem of invariant theory for vector invariants. Bn−1 are complex symmetric matrices of size n This new Amitsur–Levitzki type theorem is related to invariant theory in the exterior algebra Λ(Λ2(V ) ⊕ S2(V ∗)) on the direct product of the second antisymmetric tensor Λ2(V ) of V and the second symmetric tensor S2(V ∗) of V ∗, where V is an n-dimensional complex vector space. For this exterior algebra, we give two results. (6) As written in Introduction, Theorem 2.1 was given in [BPS] independently of this article (see [DPP] and [Pr])

Relation to the Amitsur–Levitzki theorem

Amitsur–Levitzki type theorem due to Kostant and Rowen

Amitsur–Levitzki type theorem for alternating and symmetric matrices