Degrees
Shephard and Todd proved that a finite group acting on a complex vector space is a complex reflection group if and only if its ring of invariants is a polynomial ring (Chevalley–Shephard–Todd theorem). For being the rank of the reflection group, the degrees of the generators of the ring of invariants are called degrees of W and are listed in the column above headed "degrees". They also showed that many other invariants of the group are determined by the degrees as follows:
- The center of an irreducible reflection group is cyclic of order equal to the greatest common divisor of the degrees.
- The order of a complex reflection group is the product of its degrees.
- The number of reflections is the sum of the degrees minus the rank.
- An irreducible complex reflection group comes from a real reflection group if and only if it has an invariant of degree 2.
- The degrees di satisfy the formula
Read more about this topic: Complex Reflection Group
Famous quotes containing the word degrees:
“Complete courage and absolute cowardice are extremes that very few men fall into. The vast middle space contains all the intermediate kinds and degrees of courage; and these differ as much from one another as men’s faces or their humors do.”
—François, Duc De La Rochefoucauld (1613–1680)
“When a thought of Plato becomes a thought to me,—when a truth that fired the soul of Pindar fires mine, time is no more. When I feel that we two meet in a perception, that our two souls are tinged with the same hue, and do as it were run into one, why should I measure degrees of latitude, why should I count Egyptian years?”
—Ralph Waldo Emerson (1803–1882)
“I was by degrees awakened as from a dream, and feared that my whole life could properly be counted nothing else but a fantastic vision.”
—Sarah Fielding (1710–1768)