Exterior powers in Iwasawa theory

Exterior Powers

Let R be a commutative ring. Unless indicated otherwise, all modules are R-modules and all tensor products are taken over R, so we abbreviate ⊗R to ⊗. A bilinear function out of M1 × M2 turns into a linear function out of the tensor product M1 ⊗ M2. In a similar way, a multilinear function out of M1 × · · · ×Mk turns into a linear function out of the k-fold tensor product M1 ⊗ · · · ⊗Mk. We wil...

Exterior Powers of the Reflection Representation in Springer Theory

Let H∗(Be) be the total Springer representation of W for the nilpotent element e in a simple Lie algebra g. Let ∧V denote the exterior powers of the reflection representation V of W . The focus of this paper is on the algebra of W -invariants in H∗(Be)⊗ ∧ ∗V and we show that it is an exterior algebra on the subspace (H∗(Be) ⊗ V ) in some new cases. This was known previously for e = 0 by a resul...

Which Exterior Powers are Balanced?

A signed graph is a graph whose edges are given ±1 weights. In such a graph, the sign of a cycle is the product of the signs of its edges. A signed graph is called balanced if its adjacency matrix is similar to the adjacency matrix of an unsigned graph via conjugation by a diagonal ±1 matrix. For a signed graph Σ on n vertices, its exterior kth power, where k = 1, . . . , n−1, is a graph ∧k Σ w...

Introduction to Iwasawa Theory

Minors of Symmetric and Exterior Powers

We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski's theory of complete ideals and of representation theory. Let R be a commutative ring. The determinantal ideals attached to matrices with entries in R play ubiquitous roles in the study of the syzygies of R{modules. In this note, we describe so...