Generalized Littlewood–Richardson coefficients for branching rules of GL<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>and extremal weight crystals

Abstract

Following the methods used by Derksen–Weyman in [] and Chindris in [], we use quiver theory to represent the generalized Littlewood–Richardson coefficients for the branching rule for the diagonal embedding of GL(n) as the dimension of a weight space of semi-invariants. Using this, we prove their saturation and investigate when they are nonzero. We also show that for certain partitions the associated stretched polynomials satisfy the same conjectures as single Littlewood–Richardson coefficients. We then provide a polytopal description of this multiplicity and show that its positivity may be computed in strongly polynomial time. Finally, we remark that similar results hold for certain other generalized Littlewood–Richardson coefficients.

Locations

• Algebraic Combinatorics - View - PDF
• arXiv (Cornell University) - View - PDF
• MOspace Institutional Repository (University of Missouri) - View - PDF
• French digital mathematics library (Numdam) - View - PDF