2010/07/08 by J. Matthew Douglass, Gerhard Roehrle, Douglass, J. Matthew +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Algebra and Geometry #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.1007.1350
Suppose that W is a finite, unitary, reflection group acting on the complex\nvector space V and X is a subspace of V. Define N to be the setwise stabilizer\nof X in W, Z to be the pointwise stabilizer, and C=N/Z. Then restriction\ndefines a homomorphism from the algebra of W-invariant polynomial functions on\nV to the algebra of C-invariant functions on X. In this note we consider the\nspecial case when W is a Coxeter group, V is the complexified reflection\nrepresentation of W, and X is in the lattice of the arrangement of W, and give\na simple, combinatorial characterization of when the restriction mapping is\nsurjective in terms of the exponents of W and C. As an application of our\nresult, in the case when W is the Weyl group of a semisimple, complex, Lie\nalgebra, we complete a calculation begun by Richardson in 1987 and obtain a\nsimple combinatorial characterization of regular decomposition classes whose\nclosure is a normal variety.\n