Sanghoon Baek, Erhard Neher, Kirill Zainoulline: Basic polynomial invariants, fundamental representations and the Chern class map

Submission: 2011, Jun 30

Consider a crystallographic root system together with its Weyl group $W$ acting on the weight lattice $M$. Let $Z[M]^W$ and $S^*(M)^W$ be the $W$-invariant subrings of the integral group ring $Z[M]$ and the symmetric algebra $S^*(M)$ respectively. A celebrated theorem of Chevalley says that $Z[M]^W$ is a polynomial ring over $Z$ in classes of fundamental representations $w_1,...,w_n$ and $S^*(M)^{W}$ over rational numbers is a polynomial ring in basic polynomial invariants $q_1,...,q_n$, where $n$ is the rank. In the present paper we establish and investigate the relationship between $w_i$'s and $q_i$'s over the integers.

2010 Mathematics Subject Classification: 13A50; 14L24; 20F55

Keywords and Phrases: Dynkin index, polynomial invariant, fundamental representation, invariant theory, Chern class map, finite reflection group

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