M. Florence | Searcharxiv

arXiv2026

Weyl's Polarization in Classical Invariant Theory: A Primer, with Worked Examples

Hermann Weyl's The Classical Groups is a landmark work connecting classical invariant theory with modern representation theory. It shows how polynomial invariants of the general linear, orthogonal, and symplectic groups can be systematically understood through linear representations and tensor methods. The current note is primarily based on a personal reading of the book of Weyl and of the more accessible books Classical Invariant Theory by Kraft and Procesi and by Olver. It is neither exhaustive, nor original, nor state of the art. We focus on a few selected aspects, aiming for an elementary and concrete approach. We work over the field of reals R with the classical groups GL(n), SL(n), O(n), and SO(n). Most of our efforts have been devoted to carefully worked examples, introducing just enough of the general theory to handle them effectively.

math.RT↗

arXiv2017

Lifting Theorems and Smooth Profinite Groups

This work is motivated by the search for an "explicit" proof of the Bloch-Kato conjecture in Galois cohomology, proved by Voevodsky. Our concern here is to lay the foundation for a theory that, we believe, will lead to such a proof- and to further applications. Let p be a prime number. Let k be a perfect field of characteristic p. Let m be a positive integer. Our first goal is to provide a canonical process for "lifting" a module M, over the ring of Witt vectors $W_m(k)$ (of length m), to a $W_{m+1}(k)$-module, in a way that deeply respects Pontryagin duality. These are our big, medium and small Omega powers, each of which naturally occurs as a direct factor of the previous one. In the case where M is a k-vector space, they come equipped with Verschiebung and Frobenius operations. If moreover the field k is finite, Omega powers are endowed with a striking extra operation: the Transfer, to shifted Omega powers of finite-codimensional linear subspaces. To show how this formalism fits into Galois theory, we first offer an axiomatized approach to Hilbert's Theorem 90 (or more precisely, to its consequence for cohomology with finite coefficients: Kummer theory). In the context of profinite group cohomology, we thus define the notions a cyclotomic G-module, and of a smooth profinite group. We bear in mind that the fundamental example is that of an absolute Galois group, together with the Tate module of roots of unity. We then define the notion of exact sequences of G-modules of Kummer type. To finish, we give applications of this formalism. The first ones are the Stable Lifting Theorems, enabling the lifting to higher torsion in the cohomology of smooth profinite groups, with p-primary coefficients. We finish by an application to p-adic deformations. We state and prove a general descent statement, for the quotient map $\mathbb{Z}/p^2\mathbb{Z}\rightarrow\mathbb{Z}/p\mathbb{Z}$.

math.AG↗