Geometric Complexity Theory IV

Jonah Blasiak, Drexel University, Philadelphia, PA, USA. Ketan D. Mulmuley, The University of Chicago, IL, USA. Milind Sohoni, Indian Institute of Technology, Mumbai, India.

Introduction
Basic concepts and notation
Hecke algebras and canonical bases
The quantum group $GL_q(V)$
Bases for $GL_q(V)$ modules
Quantum Schur-Weyl duality and canonical bases
Notation for $GL_q(V) /times GL_q(W)$
The nonstandard coordinate algebra $/mathscr{O}(M_q(/check{X}))$
Nonstandard determinant and minors
The nonstandard quantum groups $GL_q(/check{X})$ and $/texttt{U}_q(/check{X})$
The nonstandard Hecke algebra $/check{/mathscr{H}}_r$
Nonstandard Schur-Weyl duality
Nonstandard representation theory in the two-row case
A canonical basis for $/check{Y}_/alpha$
A global crystal basis for two-row Kronecker coefficients
Straightened NST and semistandard tableaux}

A Kronecker graphical calculus and applications
Explicit formulae for Kronecker coefficients
Future work
Appendix A. Reduction system for ${/mathscr{O}}(M_q(/check{X}))$
Appendix B. The Hopf algebra ${/mathscr{O}}_{q}^/tau$
Bibliography