2 research outputs found
Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions
We present a systematic approach to studying the geometric aspects of Vinberg
theta-representations. The main idea is to use the Borel-Weil construction for
representations of reductive groups as sections of homogeneous bundles on
homogeneous spaces, and then to study degeneracy loci of these vector bundles.
Our main technical tool is to use free resolutions as an "enhanced" version of
degeneracy loci formulas. We illustrate our approach on several examples and
show how they are connected to moduli spaces of Abelian varieties. To make the
article accessible to both algebraists and geometers, we also include
background material on free resolutions and representation theory.Comment: 41 pages, uses tabmac.sty, Dedicated to David Eisenbud on the
occasion of his 65th birthday; v2: fixed some typos and added reference
There is no "Theory of Everything" inside E8
We analyze certain subgroups of real and complex forms of the Lie group E8,
and deduce that any "Theory of Everything" obtained by embedding the gauge
groups of gravity and the Standard Model into a real or complex form of E8
lacks certain representation-theoretic properties required by physical reality.
The arguments themselves amount to representation theory of Lie algebras in the
spirit of Dynkin's classic papers and are written for mathematicians.Comment: Final version, to appear in Communications in Mathematical Physics.
The main difference, from the previous version, is the creation of a new
section, containing a response to Lisi's objections to our wor