Solvable Groups and Modular Representation Theory

Modular Representation Theory of Blocks with Trivial Intersection Defect Groups

We show that Uno’s refinement of the projective conjecture of Dade holds for every block whose defect groups intersect trivially modulo the maximal normal p-subgroup. This corresponds to the block having p-local rank one as defined by Jianbei An and Eaton. An immediate consequence is that Dade’s projective conjecture, Robinson’s conjecture, Alperin’s weight conjecture, the Isaacs– Navarro conje...

Groups and Representation Theory

Ghosts in Modular Representation Theory

A ghost over a finite p-group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis—the statement that ghosts between finite-dimensional G-representations factor through a projective—we define the ghost number of kG to be the smallest integer l such that the composite of any l ghosts between finite-dimensiona...

Quantum Groups and Representation Theory

I. QUANTUM GROUPS AND QUANTUM INTEGRABLE SYSTEMS The mathematical theory of solitons started with the invention of the Inverse Scattering Method (ISM) [1]. ISM is based on the introduction of the Lax pair of linear equations vx = U(x, λ, t)v, vt = V (x, λ, t)v (1) instead of an original integrable nonlinear evolution equation ut = K(u, ux, uxx, . . .). (2) Here v = v(x, t) is an m-dimensional v...

Modular Representation Theory and the CDE Triangle