Fake degrees for reflection actions on roots

2011/12/29 by Victor Reiner, Reiner, Victor, Zhiwei Yun +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Coding theory and cryptography #Finite Group Theory Research #math.CO #msc:13A50 #msc:17B22 #msc:20F55 #msc:51F15

paper · pdf · doi:10.48550/arxiv.1201.0032

Added discussion of overlap with prior work by J. Stembridge

arxiv created 2012/01/27 · arxiv updated 2012/01/30

Abstract

A finite irreducible real reflection group of rank l and Coxeter number h has root system of cardinality h*l. It is shown that the fake degree for the permutation action on its roots is divisible by [h]_q = 1+q+q²+...+q^h-1, and that in simply-laced types, it equals [h]_q times the summation of q^{e_i - 1} where e_i runs through the exponents, so that e_i - 1 are the codegrees.

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